examples.diffusion.coupled

Solve the biharmonic equation as a coupled pair of diffusion equations.

FiPy has only first order time derivatives so equations such as the biharmonic wave equation written as

\frac{\partial^4 v}{\partial x^4} + \frac{\partial^2 v}{\partial t^2} &= 0

cannot be represented as a single equation. We need to decompose the biharmonic equation into two equations that are first order in time in the following way,

\frac{\partial^2 v_0}{\partial x^2} + \frac{\partial v_1}{\partial t} &= 0 \\
\frac{\partial^2 v_1}{\partial x^2} - \frac{\partial v_0}{\partial t} &= 0

Historically, FiPy required systems of coupled equations to be solved successively, “sweeping” the equations to convergence. As a practical example, we use the following system

\frac{\partial v_0}{\partial t} &= 0.01 \nabla^2 v_0 - \nabla^2 v_1 \\
\frac{\partial v_1}{\partial t} &= \nabla^2 v_0 + 0.01 \nabla^2 v_1

subject to the boundary conditions

\begin{align*}
v_0|_{x=0} &= 0 & v_0|_{x=1} &= 1 \\
v_1|_{x=0} &= 1 & v_1|_{x=1} &= 0
\end{align*}

This system closely resembles the pure biharmonic equation, but has an additional diffusion contribution to improve numerical stability. The example system is solved with the following block of code using explicit coupling for the cross-coupled terms.

>>> from fipy import Grid1D, CellVariable, TransientTerm, DiffusionTerm, Viewer
>>> m = Grid1D(nx=100, Lx=1.)
>>> v0 = CellVariable(mesh=m, hasOld=True, value=0.5)
>>> v1 = CellVariable(mesh=m, hasOld=True, value=0.5)
>>> v0.constrain(0, m.facesLeft)
>>> v0.constrain(1, m.facesRight)
>>> v1.constrain(1, m.facesLeft)
>>> v1.constrain(0, m.facesRight)
>>> eq0 = TransientTerm() == DiffusionTerm(coeff=0.01) - v1.faceGrad.divergence
>>> eq1 = TransientTerm() == v0.faceGrad.divergence + DiffusionTerm(coeff=0.01)
>>> vi = Viewer((v0, v1))
>>> from builtins import range
>>> for t in range(100):
...     v0.updateOld()
...     v1.updateOld()
...     res0 = res1 = 1e100
...     while max(res0, res1) > 0.1:
...         res0 = eq0.sweep(var=v0, dt=1e-5)
...         res1 = eq1.sweep(var=v1, dt=1e-5)
...     if t % 10 == 0:
...         vi.plot()

The uncoupled method still works, but it can be advantageous to solve the two equations simultaneously. In this case, by coupling the equations, we can eliminate the explicit sources and dramatically increase the time steps:

>>> v0.value = 0.5
>>> v1.value = 0.5
>>> eqn0 = TransientTerm(var=v0) == DiffusionTerm(0.01, var=v0) - DiffusionTerm(1, var=v1)
>>> eqn1 = TransientTerm(var=v1) == DiffusionTerm(1, var=v0) + DiffusionTerm(0.01, var=v1)
>>> eqn = eqn0 & eqn1
>>> from builtins import range
>>> for t in range(1):
...     v0.updateOld()
...     v1.updateOld()
...     eqn.solve(dt=1.e-3)
...     vi.plot()

It is also possible to pose the same equations in vector form:

>>> v = CellVariable(mesh=m, hasOld=True, value=[[0.5], [0.5]], elementshape=(2,))
>>> v.constrain([[0], [1]], m.facesLeft)
>>> v.constrain([[1], [0]], m.facesRight)
>>> eqn = TransientTerm([[1, 0],
...                      [0, 1]]) == DiffusionTerm([[[0.01, -1],
...                                                  [1, 0.01]]])
>>> vi = Viewer((v[0], v[1]))
>>> from builtins import range
>>> for t in range(1):
...     v.updateOld()
...     eqn.solve(var=v, dt=1.e-3)
...     vi.plot()

Whether you pose your problem in coupled or vector form should be dictated by the underlying physics. If v_0 and v_1 represent the concentrations of two conserved species, then it is natural to write two separate governing equations and to couple them. If they represent two components of a vector field, then the vector formulation is obviously more natural. FiPy will solve the same matrix system either way.

Last updated on Jan 14, 2021. Created using Sphinx 3.4.3.