OOF2: The Manual
| Backward Euler (BackwardEuler) | ||
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6.5.2. Subclasses |  |
Name
Backward Euler (BackwardEuler) — Fully implicit first order time stepping.
Synopsis
BackwardEuler()
Description
The Backward Euler method is a method of numerically integrating
ordinary differential equations. Like the ForwardEuler method, it is first order
in time, meaning that it makes an error of order
on each step. It is more stable than the
Forward Euler method, especially for diffusion equations, but
requires more effort to compute. It is less accurate than CrankNicolson, but in some cases less
prone to oscillations.
Given a vector 
of unknowns
(i.e. Field values in OOF2) at
time 
,
and the first order differential equation
![\[ \frac{d\phi}{dt} = f(\phi, t) \]](equations/6.5.2-eq-3.gif)
the Backward Euler solution is given by solving
![\[ \phi_{n+1} = \phi_n + \Delta t f(\phi_{n+1}, t_{n+1}) \]](equations/6.5.2-eq-4.gif)
for 
, with
. Because 
appears on the
right hand side, this is an implicit method and
is relatively costly to compute, especially when
is
nonlinear.
BackwardEuler can be applied to equations
with second order time derivatives via equation (6.141).
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| Back and Forth (BackAndForthAnimation) |  |
Badly Shaped Elements (BadlyShapedElements) |