OOF2: The Manual
| Crank-Nicolson (CrankNicolson) | ||
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6.5.2. Subclasses |  |
Name
Crank-Nicolson (CrankNicolson) — Semi-implicit first order time stepping, theta=0.5.
Synopsis
CrankNicolson()
Description
The Crank-Nicolson method is a method of
numerically integrating ordinary differential equations. It is
second order in time, meaning that it makes an error only of order
on each step, and is more accurate and more
stable than the ForwardEuler
method, but it is more expensive to compute.
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-5.gif)
the Crank-Nicolson estimate for 
is
![\[ \phi_{n+1} = \phi_n +
\frac12 \Delta t \left[f(\phi_n, t_n) + f(\phi_{n+1},t_{n+1})\right]
\]](equations/6.5.2-eq-6.gif)
where 
. The need to solve equation (6.129) for 
, which appears on
both sides, makes CrankNicolson a
semi-implicit method, requiring more CPU time than
an explicit method such as ForwardEuler, especially when
is
nonlinear.
CrankNicolson can be applied to equations
with second order time derivatives via equation (6.141).
Generalized Euler Methods
CrankNicolson is an example of a
Generalized Euler method, which is a
combination of the ForwardEuler
and BackwardEuler methods:
![\[ \phi_{n+1} = \phi_n +
\Delta t \left[ (1-\theta)f(\phi_n,t_n) + \theta f(\phi_{n+1},t_{n+1})
\right]
\]](equations/6.5.2-eq-7.gif)
where 
is a number between 0
and 1. 
gives the fully
explicit ForwardEuler
method. 
gives the fully
implicit BackwardEuler method.
Intermediate values give semi-implicit methods, such as CrankNicolson (
).
The error in the generalized Euler methods is of order
, except for CrankNicolson, which is
.
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| Continuum Profile (ContinuumProfileXTd) |  |
Cross Section (CrossSectionDomain) |