fipy.terms package¶

Submodules¶

fipy.terms.abstractBinaryTerm module¶

fipy.terms.abstractConvectionTerm module¶

fipy.terms.abstractDiffusionTerm module¶

fipy.terms.abstractUpwindConvectionTerm module¶

fipy.terms.advectionTerm module¶

class fipy.terms.advectionTerm.AdvectionTerm(coeff=None)¶

Bases: FirstOrderAdvectionTerm

The AdvectionTerm object constructs the b vector contribution for the advection term given by

$u \abs{\nabla \phi}$

from the advection equation given by:

$\frac{\partial \phi}{\partial t} + u \abs{\nabla \phi} = 0$

The construction of the gradient magnitude term requires upwinding as in the standard FirstOrderAdvectionTerm. The higher order terms are incorporated as follows. The formula used here is given by:

$u_P \abs{\nabla \phi}_P = \max \left( u_P , 0 \right) \left[ \sum_A \min \left( D_{AP}, 0 \right)^2 \right]^{1/2} + \min \left( u_P , 0 \right) \left[ \sum_A \max \left( D_{AP}, 0 \right)^2 \right]^{1/2}$

where,

$D_{AP} = \frac{ \phi_A - \phi_P } { d_{AP}} - \frac{ d_{AP} } {2} m \left(L_A, L_P \right)$

and

$m\left(x, y\right) &= x \qquad \text{if $\abs{x} \le \abs{y} \forall xy \ge 0$} \\ m\left(x, y\right) &= y \qquad \text{if $\abs{x} > \abs{y} \forall xy \ge 0$} \\ m\left(x, y\right) &= 0 \qquad \text{if $xy < 0$}$

also,

$L_A &= \frac{\phi_{AA} + \phi_P - 2 \phi_A}{d_{AP}^2} \\ L_P &= \frac{\phi_{A} + \phi_{PP} - 2 \phi_P}{d_{AP}^2}$

Here are some simple test cases for this problem:

>>> from fipy.meshes import Grid1D
>>> from fipy.solvers import *
>>> SparseMatrix = LinearPCGSolver()._matrixClass
>>> mesh = Grid1D(dx = 1., nx = 3)

Trivial test:

>>> from fipy.variables.cellVariable import CellVariable
>>> coeff = CellVariable(mesh = mesh, value = numerix.zeros(3, 'd'))
>>> v, L, b = AdvectionTerm(0.)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, numerix.zeros(3, 'd'), atol = 1e-10)) 
True

Less trivial test:

>>> coeff = CellVariable(mesh = mesh, value = numerix.arange(3))
>>> v, L, b = AdvectionTerm(1.)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((0., -1., -1.)), atol = 1e-10)) 
True

Even less trivial

>>> coeff = CellVariable(mesh = mesh, value = numerix.arange(3))
>>> v, L, b = AdvectionTerm(-1.)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((1., 1., 0.)), atol = 1e-10)) 
True

Another trivial test case (more trivial than a trivial test case standing on a harpsichord singing “trivial test cases are here again”)

>>> vel = numerix.array((-1, 2, -3))
>>> coeff = CellVariable(mesh = mesh, value = numerix.array((4, 6, 1)))
>>> v, L, b = AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, -vel * numerix.array((2, numerix.sqrt(5**2 + 2**2), 5)), atol = 1e-10)) 
True

Somewhat less trivial test case:

>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2)
>>> vel = numerix.array((3, -5, -6, -3))
>>> coeff = CellVariable(mesh = mesh, value = numerix.array((3, 1, 6, 7)))
>>> v, L, b = AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)
>>> answer = -vel * numerix.array((2, numerix.sqrt(2**2 + 6**2), 1, 0))
>>> print(numerix.allclose(b, answer, atol = 1e-10)) 
True

For the above test cases the AdvectionTerm gives the same result as the AdvectionTerm. The following test imposes a quadratic field. The higher order term can resolve this field correctly.

$\phi = x^2$

The returned vector b should have the value:

$-\abs{\nabla \phi} = -\left|\frac{\partial \phi}{\partial x}\right| = - 2 \abs{x}$

Build the test case in the following way,

>>> mesh = Grid1D(dx = 1., nx = 5)
>>> vel = 1.
>>> coeff = CellVariable(mesh = mesh, value = mesh.cellCenters[0]**2)
>>> v, L, b = __AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)

The first order term is not accurate. The first and last element are ignored because they don’t have any neighbors for higher order evaluation

>>> print(numerix.allclose(CellVariable(mesh=mesh,
... value=b).globalValue[1:-1], -2 * mesh.cellCenters.globalValue[0][1:-1]))
False

The higher order term is spot on.

>>> v, L, b = AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(CellVariable(mesh=mesh,
... value=b).globalValue[1:-1], -2 * mesh.cellCenters.globalValue[0][1:-1]))
True

The AdvectionTerm will also resolve a circular field with more accuracy,

$\phi = \left( x^2 + y^2 \right)^{1/2}$

Build the test case in the following way,

>>> mesh = Grid2D(dx = 1., dy = 1., nx = 10, ny = 10)
>>> vel = 1.
>>> x, y = mesh.cellCenters
>>> r = numerix.sqrt(x**2 + y**2)
>>> coeff = CellVariable(mesh = mesh, value = r)
>>> v, L, b = __AdvectionTerm(1.)._buildMatrix(coeff, SparseMatrix)
>>> error = CellVariable(mesh=mesh, value=b + 1)
>>> ans = CellVariable(mesh=mesh, value=b + 1)
>>> ans[(x > 2) & (x < 8) & (y > 2) & (y < 8)] = 0.123105625618
>>> print((error <= ans).all())
True

The maximum error is large (about 12 %) for the first order advection.

>>> v, L, b = AdvectionTerm(1.)._buildMatrix(coeff, SparseMatrix)
>>> error = CellVariable(mesh=mesh, value=b + 1)
>>> ans = CellVariable(mesh=mesh, value=b + 1)
>>> ans[(x > 2) & (x < 8) & (y > 2) & (y < 8)] = 0.0201715476598
>>> print((error <= ans).all())
True

The maximum error is 2 % when using a higher order contribution.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.asymmetricConvectionTerm module¶

fipy.terms.binaryTerm module¶

fipy.terms.cellTerm module¶

class fipy.terms.cellTerm.CellTerm(coeff=1.0, var=None)¶

Bases: _NonDiffusionTerm

Attention

This class is abstract. Always create one of its subclasses.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.centralDiffConvectionTerm module¶

class fipy.terms.centralDiffConvectionTerm.CentralDifferenceConvectionTerm(coeff=1.0, var=None)¶

Bases: _AbstractConvectionTerm

This Term represents

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the central differencing scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.coupledBinaryTerm module¶

fipy.terms.diffusionTerm module¶

class fipy.terms.diffusionTerm.DiffusionTerm(coeff=(1.0,), var=None)¶

Bases: DiffusionTermNoCorrection

This term represents a higher order diffusion term. The order of the term is determined by the number of coeffs, such that:

DiffusionTerm(D1)

represents a typical 2nd-order diffusion term of the form

$\nabla\cdot\left(D_1 \nabla \phi\right)$

and:

DiffusionTerm((D1,D2))

represents a 4th-order Cahn-Hilliard term of the form

$\nabla \cdot \left\{ D_1 \nabla \left[ \nabla\cdot\left( D_2 \nabla \phi\right) \right] \right\}$

and so on.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.diffusionTerm.DiffusionTermCorrection(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.diffusionTerm.DiffusionTermNoCorrection(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.diffusionTermCorrection module¶

class fipy.terms.diffusionTermCorrection.DiffusionTermCorrection(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.diffusionTermNoCorrection module¶

class fipy.terms.diffusionTermNoCorrection.DiffusionTermNoCorrection(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.explicitDiffusionTerm module¶

class fipy.terms.explicitDiffusionTerm.ExplicitDiffusionTerm(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

The discretization for the ExplicitDiffusionTerm is given by

$\int_V \nabla \cdot (\Gamma\nabla\phi) dV \simeq \sum_f \Gamma_f \frac{\phi_A^\text{old}-\phi_P^\text{old}}{d_{AP}} A_f$

where $\phi_A^\text{old}$ and $\phi_P^\text{old}$ are the old values of the variable. The term is added to the RHS vector and makes no contribution to the solution matrix.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.explicitSourceTerm module¶

fipy.terms.explicitUpwindConvectionTerm module¶

class fipy.terms.explicitUpwindConvectionTerm.ExplicitUpwindConvectionTerm(coeff=1.0, var=None)¶

Bases: _AbstractUpwindConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P^\text{old} +(1-\alpha_f)\phi_A^\text{old}$ and $\alpha_f$ is calculated using the upwind scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.exponentialConvectionTerm module¶

class fipy.terms.exponentialConvectionTerm.ExponentialConvectionTerm(coeff=1.0, var=None)¶

Bases: _AsymmetricConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the exponential scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.faceTerm module¶

class fipy.terms.faceTerm.FaceTerm(coeff=1.0, var=None)¶

Bases: _NonDiffusionTerm

Attention

This class is abstract. Always create one of its subclasses.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.firstOrderAdvectionTerm module¶

class fipy.terms.firstOrderAdvectionTerm.FirstOrderAdvectionTerm(coeff=None)¶

Bases: _NonDiffusionTerm

The FirstOrderAdvectionTerm object constructs the b vector contribution for the advection term given by

$u \abs{\nabla \phi}$

from the advection equation given by:

$\frac{\partial \phi}{\partial t} + u \abs{\nabla \phi} = 0$

The construction of the gradient magnitude term requires upwinding. The formula used here is given by:

$u_P \abs{\nabla \phi}_P = \max \left( u_P , 0 \right) \left[ \sum_A \min \left( \frac{ \phi_A - \phi_P } { d_{AP}}, 0 \right)^2 \right]^{1/2} + \min \left( u_P , 0 \right) \left[ \sum_A \max \left( \frac{ \phi_A - \phi_P } { d_{AP}}, 0 \right)^2 \right]^{1/2}$

Here are some simple test cases for this problem:

>>> from fipy.meshes import Grid1D
>>> from fipy.solvers import *
>>> SparseMatrix = LinearLUSolver()._matrixClass
>>> mesh = Grid1D(dx = 1., nx = 3)
>>> from fipy.variables.cellVariable import CellVariable

Trivial test:

>>> var = CellVariable(value = numerix.zeros(3, 'd'), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(0.)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, numerix.zeros(3, 'd'), atol = 1e-10)) 
True

Less trivial test:

>>> var = CellVariable(value = numerix.arange(3), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(1.)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((0., -1., -1.)), atol = 1e-10)) 
True

Even less trivial

>>> var = CellVariable(value = numerix.arange(3), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(-1.)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((1., 1., 0.)), atol = 1e-10)) 
True

Another trivial test case (more trivial than a trivial test case standing on a harpsichord singing “trivial test cases are here again”)

>>> vel = numerix.array((-1, 2, -3))
>>> var = CellVariable(value = numerix.array((4, 6, 1)), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(vel)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, -vel * numerix.array((2, numerix.sqrt(5**2 + 2**2), 5)), atol = 1e-10)) 
True

Somewhat less trivial test case:

>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2)
>>> vel = numerix.array((3, -5, -6, -3))
>>> var = CellVariable(value = numerix.array((3, 1, 6, 7)), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(vel)._buildMatrix(var, SparseMatrix)
>>> answer = -vel * numerix.array((2, numerix.sqrt(2**2 + 6**2), 1, 0))
>>> print(numerix.allclose(b, answer, atol = 1e-10)) 
True

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.hybridConvectionTerm module¶

class fipy.terms.hybridConvectionTerm.HybridConvectionTerm(coeff=1.0, var=None)¶

Bases: _AsymmetricConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the hybrid scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.implicitDiffusionTerm module¶

fipy.terms.implicitDiffusionTerm.ImplicitDiffusionTerm¶: alias of DiffusionTerm

fipy.terms.implicitSourceTerm module¶

class fipy.terms.implicitSourceTerm.ImplicitSourceTerm(coeff=0.0, var=None)¶

Bases: SourceTerm

The ImplicitSourceTerm represents

$\int_V \phi S \,dV \simeq \phi_P S_P V_P$

where $S$ is the coeff value.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.nonDiffusionTerm module¶

fipy.terms.powerLawConvectionTerm module¶

class fipy.terms.powerLawConvectionTerm.PowerLawConvectionTerm(coeff=1.0, var=None)¶

Bases: _AsymmetricConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the power law scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.residualTerm module¶

class fipy.terms.residualTerm.ResidualTerm(equation, underRelaxation=1.0)¶

Bases: _ExplicitSourceTerm

The ResidualTerm is a special form of explicit SourceTerm that adds the residual of one equation to another equation. Useful for Newton’s method.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.sourceTerm module¶

class fipy.terms.sourceTerm.SourceTerm(coeff=0.0, var=None)¶

Bases: CellTerm

Attention

This class is abstract. Always create one of its subclasses.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.term module¶

class fipy.terms.term.Term(coeff=1.0, var=None)¶

Bases: object

Attention

This class is abstract. Always create one of its subclasses.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶: Return repr(self).

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.test module¶

fipy.terms.transientTerm module¶

class fipy.terms.transientTerm.TransientTerm(coeff=1.0, var=None)¶

Bases: CellTerm

The TransientTerm represents

$\int_V \frac{\partial (\rho \phi)}{\partial t} dV \simeq \frac{(\rho_{P} \phi_{P} - \rho_{P}^\text{old} \phi_P^\text{old}) V_P}{\Delta t}$

where $\rho$ is the coeff value.

The following test case verifies that variable coefficients and old coefficient values work correctly. We will solve the following equation

$\frac{ \partial \phi^2 } { \partial t } = k.$

The analytic solution is given by

$\phi = \sqrt{ \phi_0^2 + k t },$

where $\phi_0$ is the initial value.

>>> phi0 = 1.
>>> k = 1.
>>> dt = 1.
>>> relaxationFactor = 1.5
>>> steps = 2
>>> sweeps = 8

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 1)
>>> from fipy.variables.cellVariable import CellVariable
>>> var = CellVariable(mesh = mesh, value = phi0, hasOld = 1)
>>> from fipy.terms.transientTerm import TransientTerm
>>> from fipy.terms.implicitSourceTerm import ImplicitSourceTerm

Relaxation, given by relaxationFactor, is required for a converged solution.

>>> eq = TransientTerm(var) == ImplicitSourceTerm(-relaxationFactor) \
...                            + var * relaxationFactor + k

A number of sweeps at each time step are required to let the relaxation take effect.

>>> from builtins import range
>>> for step in range(steps):
...     var.updateOld()
...     for sweep in range(sweeps):
...         eq.solve(var, dt = dt)

Compare the final result with the analytical solution.

>>> from fipy.tools import numerix
>>> print(var.allclose(numerix.sqrt(k * dt * steps + phi0**2)))
1

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.unaryTerm module¶

fipy.terms.upwindConvectionTerm module¶

class fipy.terms.upwindConvectionTerm.UpwindConvectionTerm(coeff=1.0, var=None)¶

Bases: _AbstractUpwindConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the upwind convection scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.vanLeerConvectionTerm module¶

class fipy.terms.vanLeerConvectionTerm.VanLeerConvectionTerm(coeff=1.0, var=None)¶

Bases: ExplicitUpwindConvectionTerm

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

Module contents¶

exception fipy.terms.AbstractBaseClassError(s="can't instantiate abstract base class")¶

Bases: NotImplementedError

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class fipy.terms.AdvectionTerm(coeff=None)¶

Bases: FirstOrderAdvectionTerm

The AdvectionTerm object constructs the b vector contribution for the advection term given by

$u \abs{\nabla \phi}$

from the advection equation given by:

$\frac{\partial \phi}{\partial t} + u \abs{\nabla \phi} = 0$

The construction of the gradient magnitude term requires upwinding as in the standard FirstOrderAdvectionTerm. The higher order terms are incorporated as follows. The formula used here is given by:

$u_P \abs{\nabla \phi}_P = \max \left( u_P , 0 \right) \left[ \sum_A \min \left( D_{AP}, 0 \right)^2 \right]^{1/2} + \min \left( u_P , 0 \right) \left[ \sum_A \max \left( D_{AP}, 0 \right)^2 \right]^{1/2}$

where,

$D_{AP} = \frac{ \phi_A - \phi_P } { d_{AP}} - \frac{ d_{AP} } {2} m \left(L_A, L_P \right)$

and

$m\left(x, y\right) &= x \qquad \text{if $\abs{x} \le \abs{y} \forall xy \ge 0$} \\ m\left(x, y\right) &= y \qquad \text{if $\abs{x} > \abs{y} \forall xy \ge 0$} \\ m\left(x, y\right) &= 0 \qquad \text{if $xy < 0$}$

also,

$L_A &= \frac{\phi_{AA} + \phi_P - 2 \phi_A}{d_{AP}^2} \\ L_P &= \frac{\phi_{A} + \phi_{PP} - 2 \phi_P}{d_{AP}^2}$

Here are some simple test cases for this problem:

>>> from fipy.meshes import Grid1D
>>> from fipy.solvers import *
>>> SparseMatrix = LinearPCGSolver()._matrixClass
>>> mesh = Grid1D(dx = 1., nx = 3)

Trivial test:

>>> from fipy.variables.cellVariable import CellVariable
>>> coeff = CellVariable(mesh = mesh, value = numerix.zeros(3, 'd'))
>>> v, L, b = AdvectionTerm(0.)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, numerix.zeros(3, 'd'), atol = 1e-10)) 
True

Less trivial test:

>>> coeff = CellVariable(mesh = mesh, value = numerix.arange(3))
>>> v, L, b = AdvectionTerm(1.)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((0., -1., -1.)), atol = 1e-10)) 
True

Even less trivial

>>> coeff = CellVariable(mesh = mesh, value = numerix.arange(3))
>>> v, L, b = AdvectionTerm(-1.)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((1., 1., 0.)), atol = 1e-10)) 
True

Another trivial test case (more trivial than a trivial test case standing on a harpsichord singing “trivial test cases are here again”)

>>> vel = numerix.array((-1, 2, -3))
>>> coeff = CellVariable(mesh = mesh, value = numerix.array((4, 6, 1)))
>>> v, L, b = AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(b, -vel * numerix.array((2, numerix.sqrt(5**2 + 2**2), 5)), atol = 1e-10)) 
True

Somewhat less trivial test case:

>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2)
>>> vel = numerix.array((3, -5, -6, -3))
>>> coeff = CellVariable(mesh = mesh, value = numerix.array((3, 1, 6, 7)))
>>> v, L, b = AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)
>>> answer = -vel * numerix.array((2, numerix.sqrt(2**2 + 6**2), 1, 0))
>>> print(numerix.allclose(b, answer, atol = 1e-10)) 
True

For the above test cases the AdvectionTerm gives the same result as the AdvectionTerm. The following test imposes a quadratic field. The higher order term can resolve this field correctly.

$\phi = x^2$

The returned vector b should have the value:

$-\abs{\nabla \phi} = -\left|\frac{\partial \phi}{\partial x}\right| = - 2 \abs{x}$

Build the test case in the following way,

>>> mesh = Grid1D(dx = 1., nx = 5)
>>> vel = 1.
>>> coeff = CellVariable(mesh = mesh, value = mesh.cellCenters[0]**2)
>>> v, L, b = __AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)

The first order term is not accurate. The first and last element are ignored because they don’t have any neighbors for higher order evaluation

>>> print(numerix.allclose(CellVariable(mesh=mesh,
... value=b).globalValue[1:-1], -2 * mesh.cellCenters.globalValue[0][1:-1]))
False

The higher order term is spot on.

>>> v, L, b = AdvectionTerm(vel)._buildMatrix(coeff, SparseMatrix)
>>> print(numerix.allclose(CellVariable(mesh=mesh,
... value=b).globalValue[1:-1], -2 * mesh.cellCenters.globalValue[0][1:-1]))
True

The AdvectionTerm will also resolve a circular field with more accuracy,

$\phi = \left( x^2 + y^2 \right)^{1/2}$

Build the test case in the following way,

>>> mesh = Grid2D(dx = 1., dy = 1., nx = 10, ny = 10)
>>> vel = 1.
>>> x, y = mesh.cellCenters
>>> r = numerix.sqrt(x**2 + y**2)
>>> coeff = CellVariable(mesh = mesh, value = r)
>>> v, L, b = __AdvectionTerm(1.)._buildMatrix(coeff, SparseMatrix)
>>> error = CellVariable(mesh=mesh, value=b + 1)
>>> ans = CellVariable(mesh=mesh, value=b + 1)
>>> ans[(x > 2) & (x < 8) & (y > 2) & (y < 8)] = 0.123105625618
>>> print((error <= ans).all())
True

The maximum error is large (about 12 %) for the first order advection.

>>> v, L, b = AdvectionTerm(1.)._buildMatrix(coeff, SparseMatrix)
>>> error = CellVariable(mesh=mesh, value=b + 1)
>>> ans = CellVariable(mesh=mesh, value=b + 1)
>>> ans[(x > 2) & (x < 8) & (y > 2) & (y < 8)] = 0.0201715476598
>>> print((error <= ans).all())
True

The maximum error is 2 % when using a higher order contribution.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.CentralDifferenceConvectionTerm(coeff=1.0, var=None)¶

Bases: _AbstractConvectionTerm

This Term represents

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the central differencing scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.ConvectionTerm¶: alias of PowerLawConvectionTerm

class fipy.terms.DiffusionTerm(coeff=(1.0,), var=None)¶

Bases: DiffusionTermNoCorrection

This term represents a higher order diffusion term. The order of the term is determined by the number of coeffs, such that:

DiffusionTerm(D1)

represents a typical 2nd-order diffusion term of the form

$\nabla\cdot\left(D_1 \nabla \phi\right)$

and:

DiffusionTerm((D1,D2))

represents a 4th-order Cahn-Hilliard term of the form

$\nabla \cdot \left\{ D_1 \nabla \left[ \nabla\cdot\left( D_2 \nabla \phi\right) \right] \right\}$

and so on.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.DiffusionTermCorrection(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.DiffusionTermNoCorrection(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.ExplicitDiffusionTerm(coeff=(1.0,), var=None)¶

Bases: _AbstractDiffusionTerm

The discretization for the ExplicitDiffusionTerm is given by

$\int_V \nabla \cdot (\Gamma\nabla\phi) dV \simeq \sum_f \Gamma_f \frac{\phi_A^\text{old}-\phi_P^\text{old}}{d_{AP}} A_f$

where $\phi_A^\text{old}$ and $\phi_P^\text{old}$ are the old values of the variable. The term is added to the RHS vector and makes no contribution to the solution matrix.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

__neg__()¶

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.ExplicitUpwindConvectionTerm(coeff=1.0, var=None)¶

Bases: _AbstractUpwindConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P^\text{old} +(1-\alpha_f)\phi_A^\text{old}$ and $\alpha_f$ is calculated using the upwind scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

exception fipy.terms.ExplicitVariableError(s='Terms with explicit Variables cannot mix with Terms with implicit Variables.')¶

Bases: Exception

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class fipy.terms.ExponentialConvectionTerm(coeff=1.0, var=None)¶

Bases: _AsymmetricConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the exponential scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.FirstOrderAdvectionTerm(coeff=None)¶

Bases: _NonDiffusionTerm

The FirstOrderAdvectionTerm object constructs the b vector contribution for the advection term given by

$u \abs{\nabla \phi}$

from the advection equation given by:

$\frac{\partial \phi}{\partial t} + u \abs{\nabla \phi} = 0$

The construction of the gradient magnitude term requires upwinding. The formula used here is given by:

$u_P \abs{\nabla \phi}_P = \max \left( u_P , 0 \right) \left[ \sum_A \min \left( \frac{ \phi_A - \phi_P } { d_{AP}}, 0 \right)^2 \right]^{1/2} + \min \left( u_P , 0 \right) \left[ \sum_A \max \left( \frac{ \phi_A - \phi_P } { d_{AP}}, 0 \right)^2 \right]^{1/2}$

Here are some simple test cases for this problem:

>>> from fipy.meshes import Grid1D
>>> from fipy.solvers import *
>>> SparseMatrix = LinearLUSolver()._matrixClass
>>> mesh = Grid1D(dx = 1., nx = 3)
>>> from fipy.variables.cellVariable import CellVariable

Trivial test:

>>> var = CellVariable(value = numerix.zeros(3, 'd'), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(0.)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, numerix.zeros(3, 'd'), atol = 1e-10)) 
True

Less trivial test:

>>> var = CellVariable(value = numerix.arange(3), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(1.)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((0., -1., -1.)), atol = 1e-10)) 
True

Even less trivial

>>> var = CellVariable(value = numerix.arange(3), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(-1.)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, numerix.array((1., 1., 0.)), atol = 1e-10)) 
True

Another trivial test case (more trivial than a trivial test case standing on a harpsichord singing “trivial test cases are here again”)

>>> vel = numerix.array((-1, 2, -3))
>>> var = CellVariable(value = numerix.array((4, 6, 1)), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(vel)._buildMatrix(var, SparseMatrix)
>>> print(numerix.allclose(b, -vel * numerix.array((2, numerix.sqrt(5**2 + 2**2), 5)), atol = 1e-10)) 
True

Somewhat less trivial test case:

>>> from fipy.meshes import Grid2D
>>> mesh = Grid2D(dx = 1., dy = 1., nx = 2, ny = 2)
>>> vel = numerix.array((3, -5, -6, -3))
>>> var = CellVariable(value = numerix.array((3, 1, 6, 7)), mesh = mesh)
>>> v, L, b = FirstOrderAdvectionTerm(vel)._buildMatrix(var, SparseMatrix)
>>> answer = -vel * numerix.array((2, numerix.sqrt(2**2 + 6**2), 1, 0))
>>> print(numerix.allclose(b, answer, atol = 1e-10)) 
True

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.HybridConvectionTerm(coeff=1.0, var=None)¶

Bases: _AsymmetricConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the hybrid scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

fipy.terms.ImplicitDiffusionTerm¶: alias of DiffusionTerm

class fipy.terms.ImplicitSourceTerm(coeff=0.0, var=None)¶

Bases: SourceTerm

The ImplicitSourceTerm represents

$\int_V \phi S \,dV \simeq \phi_P S_P V_P$

where $S$ is the coeff value.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

exception fipy.terms.IncorrectSolutionVariable(s='The solution variable is incorrect.')¶

Bases: Exception

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class fipy.terms.PowerLawConvectionTerm(coeff=1.0, var=None)¶

Bases: _AsymmetricConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the power law scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.ResidualTerm(equation, underRelaxation=1.0)¶

Bases: _ExplicitSourceTerm

The ResidualTerm is a special form of explicit SourceTerm that adds the residual of one equation to another equation. Useful for Newton’s method.

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

exception fipy.terms.SolutionVariableNumberError(s='Different number of solution variables and equations.')¶

Bases: Exception

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

exception fipy.terms.SolutionVariableRequiredError(s='The solution variable needs to be specified.')¶

Bases: Exception

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

exception fipy.terms.TermMultiplyError(s='Must multiply terms by int or float.')¶

Bases: Exception

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class fipy.terms.TransientTerm(coeff=1.0, var=None)¶

Bases: CellTerm

The TransientTerm represents

$\int_V \frac{\partial (\rho \phi)}{\partial t} dV \simeq \frac{(\rho_{P} \phi_{P} - \rho_{P}^\text{old} \phi_P^\text{old}) V_P}{\Delta t}$

where $\rho$ is the coeff value.

The following test case verifies that variable coefficients and old coefficient values work correctly. We will solve the following equation

$\frac{ \partial \phi^2 } { \partial t } = k.$

The analytic solution is given by

$\phi = \sqrt{ \phi_0^2 + k t },$

where $\phi_0$ is the initial value.

>>> phi0 = 1.
>>> k = 1.
>>> dt = 1.
>>> relaxationFactor = 1.5
>>> steps = 2
>>> sweeps = 8

>>> from fipy.meshes import Grid1D
>>> mesh = Grid1D(nx = 1)
>>> from fipy.variables.cellVariable import CellVariable
>>> var = CellVariable(mesh = mesh, value = phi0, hasOld = 1)
>>> from fipy.terms.transientTerm import TransientTerm
>>> from fipy.terms.implicitSourceTerm import ImplicitSourceTerm

Relaxation, given by relaxationFactor, is required for a converged solution.

>>> eq = TransientTerm(var) == ImplicitSourceTerm(-relaxationFactor) \
...                            + var * relaxationFactor + k

A number of sweeps at each time step are required to let the relaxation take effect.

>>> from builtins import range
>>> for step in range(steps):
...     var.updateOld()
...     for sweep in range(sweeps):
...         eq.solve(var, dt = dt)

Compare the final result with the analytical solution.

>>> from fipy.tools import numerix
>>> print(var.allclose(numerix.sqrt(k * dt * steps + phi0**2)))
1

Create a Term.

Parameters:: coeff (float or CellVariable or FaceVariable) – Coefficient for the term. FaceVariable objects are only acceptable for diffusion or convection terms.

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.UpwindConvectionTerm(coeff=1.0, var=None)¶

Bases: _AbstractUpwindConvectionTerm

The discretization for this Term is given by

$\int_V \nabla \cdot (\vec{u} \phi)\,dV \simeq \sum_{f} (\vec{n} \cdot \vec{u})_f \phi_f A_f$

where $\phi_f=\alpha_f \phi_P +(1-\alpha_f)\phi_A$ and $\alpha_f$ is calculated using the upwind convection scheme. For further details see Numerical Schemes.

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

class fipy.terms.VanLeerConvectionTerm(coeff=1.0, var=None)¶

Bases: ExplicitUpwindConvectionTerm

Create a _AbstractConvectionTerm object.

>>> from fipy import *
>>> m = Grid1D(nx = 2)
>>> cv = CellVariable(mesh = m)
>>> fv = FaceVariable(mesh = m)
>>> vcv = CellVariable(mesh=m, rank=1)
>>> vfv = FaceVariable(mesh=m, rank=1)
>>> __ConvectionTerm(coeff = cv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = fv) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> __ConvectionTerm(coeff = vcv)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = vfv)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.]]), mesh=UniformGrid1D(dx=1.0, nx=2)))
>>> __ConvectionTerm(coeff = (1,))
__ConvectionTerm(coeff=(1,))
>>> ExplicitUpwindConvectionTerm(coeff = (0,)).solve(var=cv, solver=DummySolver()) 
Traceback (most recent call last):
    ...
TransientTermError: The equation requires a TransientTerm with explicit convection.
>>> (TransientTerm(0.) - ExplicitUpwindConvectionTerm(coeff = (0,))).solve(var=cv, solver=DummySolver(), dt=1.)

>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = 1)).solve(var=cv, solver=DummySolver(), dt=1.) 
Traceback (most recent call last):
    ...
VectorCoeffError: The coefficient must be a vector value.
>>> m2 = Grid2D(nx=2, ny=1)
>>> cv2 = CellVariable(mesh=m2)
>>> vcv2 = CellVariable(mesh=m2, rank=1)
>>> vfv2 = FaceVariable(mesh=m2, rank=1)
>>> __ConvectionTerm(coeff=vcv2)
__ConvectionTerm(coeff=_ArithmeticCellToFaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> __ConvectionTerm(coeff=vfv2)
__ConvectionTerm(coeff=FaceVariable(value=array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.]]), mesh=UniformGrid2D(dx=1.0, nx=2, dy=1.0, ny=1)))
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = ((0,), (0,)))).solve(var=cv2, solver=DummySolver(), dt=1.)
>>> (TransientTerm() - ExplicitUpwindConvectionTerm(coeff = (0, 0))).solve(var=cv2, solver=DummySolver(), dt=1.)

Parameters:: coeff (The Term’s coefficient value.) –

property RHSvector¶: Return the RHS vector calculated in solve() or sweep(). The cacheRHSvector() method should be called before solve() or sweep() to cache the vector.

__add__(other)¶

__and__(other)¶

__div__(other)¶

__eq__(other)¶: Return self==value.

__hash__()¶: Return hash(self).

__mul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__neg__()¶

Negate a Term.

>>> -__NonDiffusionTerm(coeff=1.)
__NonDiffusionTerm(coeff=-1.0)

__pos__()¶

__radd__(other)¶

__rand__(other)¶

__repr__()¶

The representation of a Term object is given by,

>>> print(__UnaryTerm(123.456))
__UnaryTerm(coeff=123.456)

__rmul__(other)¶

Multiply a term

>>> 2. * __NonDiffusionTerm(coeff=0.5)
__NonDiffusionTerm(coeff=1.0)

Test for ticket:291.

>>> from fipy import PowerLawConvectionTerm
>>> PowerLawConvectionTerm(coeff=[[1], [0]]) * 1.0
PowerLawConvectionTerm(coeff=array([[ 1.],
       [ 0.]]))

__rsub__(other)¶

__sub__(other)¶

__truediv__(other)¶

cacheMatrix()¶: Informs solve() and sweep() to cache their matrix so that matrix can return the matrix.

cacheRHSvector()¶: Informs solve() and sweep() to cache their right hand side vector so that getRHSvector() can return it.

copy()¶

getDefaultSolver(var=None, solver=None, *args, **kwargs)¶

justErrorVector(var=None, solver=None, boundaryConditions=(), dt=1.0, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the error as well as applying under-relaxation.

justErrorVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy.solvers import DummySolver
>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(DiffusionTerm().justErrorVector(v, solver=DummySolver())) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

error – The residual vector $\vec{e}=\mathsf{L}\vec{x}_\text{old} - \vec{b}$

Return type:

CellVariable

justResidualVector(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

justResidualVector returns the overlapping local value in parallel (not the non-overlapping value).

>>> from fipy import *
>>> m = Grid1D(nx=10)
>>> v = CellVariable(mesh=m)
>>> len(numerix.asarray(DiffusionTerm().justResidualVector(v))) == m.numberOfCells
True

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

property matrix¶: Return the matrix calculated in solve() or sweep(). The cacheMatrix() method should be called before solve() or sweep() to cache the matrix.

residualVectorAndNorm(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None)¶

Builds the Term’s linear system once.

This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.

Returns:

residual (~fipy.variables.cellVariable.CellVariable) – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$
norm (float) – The L2 norm of residual, $\|\vec{r}\|_2$

solve(var=None, solver=None, boundaryConditions=(), dt=None)¶

Builds and solves the Term’s linear system once. This method does not return the residual. It should be used when the residual is not required.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.

sweep(var=None, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None, cacheResidual=False, cacheError=False)¶

Builds and solves the Term’s linear system once. This method also recalculates and returns the residual as well as applying under-relaxation.

Parameters:

var (CellVariable) – Variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
solver (Solver) – Iterative solver to be used to solve the linear system of equations. The default sovler depends on the solver package selected.
boundaryConditions (tuple of BoundaryCondition) –
dt (float) – Timestep size.
underRelaxation (float) – Usually a value between 0 and 1 or None in the case of no under-relaxation
residualFn (function) – Takes var, matrix, and RHSvector arguments, used to customize the residual calculation.
cacheResidual (bool) – If True, calculate and store the residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$ in the residualVector member of Term
cacheError (bool) – If True, use the residual vector $\vec{r}$ to solve $\mathsf{L}\vec{e}=\vec{r}$ for the error vector $\vec{e}$ and store it in the errorVector member of Term

Returns:

residual – The residual vector $\vec{r}=\mathsf{L}\vec{x} - \vec{b}$

Return type:

CellVariable

exception fipy.terms.VectorCoeffError(s='The coefficient must be a vector value.')¶

Bases: TypeError

__cause__¶: exception cause

__context__¶: exception context

__delattr__(name, /)¶: Implement delattr(self, name).

__getattribute__(name, /)¶: Return getattr(self, name).

__reduce__()¶: Helper for pickle.

__repr__()¶: Return repr(self).

__setattr__(name, value, /)¶: Implement setattr(self, name, value).

__setstate__()¶

__str__()¶: Return str(self).

__suppress_context__¶

__traceback__¶

add_note()¶: Exception.add_note(note) – add a note to the exception

args¶

with_traceback()¶: Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

Last updated on Jun 27, 2023. Created using Sphinx 6.2.1.