OOF2: The Manual

`Crank-Nicolson (CrankNicolson)`
	6.5.2. Subclasses

Name

Crank-Nicolson (CrankNicolson) — Semi-implicit first order time stepping, theta=0.5.

Synopsis

CrankNicolson()

Details

Base class: TimeStepper

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 $(\Delta t)^{3}$ on each step, and is more accurate and more stable than the ForwardEuler method, but it is more expensive to compute.

Given a vector $\phi_{n}$ of unknowns (i.e. Field values in OOF2) at time $t_{n}$ , and the first order differential equation

\frac{d\phi}{dt}=f(\phi,t)

(6.129)

the Crank-Nicolson estimate for $\phi_{n+1}$ is

\phi_{n+1}=\phi_{n}+\frac{1}{2}\Delta t\left[f(\phi_{n},t_{n})+f(\phi_{n+1},t_% {n+1})\right]

(6.130)

where $\Delta t=t_{n+1}-t_{n}$ . The need to solve equation (6.130) for $\phi_{n+1}$ , which appears on both sides, makes CrankNicolson a semi-implicit method, requiring more CPU time than an explicit method such as ForwardEuler, especially when $f$ is nonlinear.

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

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]

(6.131)

where $\theta$ is a number between 0 and 1. $\theta=0$ gives the fully explicit ForwardEuler method. $\theta=1$ gives the fully implicit BackwardEuler method. Intermediate values give semi-implicit methods, such as CrankNicolson ( $\theta=0.5$ ).

The error in the generalized Euler methods is of order $(\Delta t)^{2}$ , except for CrankNicolson, which is $(\Delta t)^{3}$ .


`Continuum Profile (ContinuumProfileXTd)`		`Cross Section (CrossSectionDomain)`