OOF2: The Manual

Details

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 ${(\mathrm{\Delta }t)}^2$ 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 ${\varphi }_n$ of unknowns (i.e. Field values in OOF2) at time $t_n$, and the first order differential equation

the Backward Euler solution is given by solving

for ${\varphi }_{n+1}$, with $\mathrm{\Delta }t=t_{n+1}-t_n$. Because ${\varphi }_{n+1}$ appears on the right hand side, this is an implicit method and is relatively costly to compute, especially when $f$ is nonlinear.

BackwardEuler can be applied to equations with second order time derivatives via equation (6.142).

See Also