OOF2: The Manual

`4th order Runge-Kutta (RK4)`
	6.5.2. Subclasses

Name

4th order Runge-Kutta (RK4) — Fourth order Runge-Kutta time stepping.

Synopsis

RK4()

Details

Base class: TimeStepper

Description

RK4 is a TimeStepper that implements the classic fourth order Runge-Kutta method for solving ordinary differential equations. The error on each step is of order $(\Delta t)^{5}$ .

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)

(6.158)

the fourth order Runge-Kutta estimate for $\phi_{n+1}$ is given by

	$\displaystyle k_{1}$	$\displaystyle=\Delta tf(\phi_{n},t_{n})$
	$\displaystyle k_{2}$	$\displaystyle=\Delta tf(\phi_{n}+\frac{k_{1}}{2},t_{n}+\frac{\Delta t}{2})$
	$\displaystyle k_{3}$	$\displaystyle=\Delta tf(\phi_{n}+\frac{k_{2}}{2},t_{n}+\frac{\Delta t}{2})$
	$\displaystyle k_{4}$	$\displaystyle=\Delta tf(\phi_{n}+k_{3},t_{n}+\Delta t)$
	$\displaystyle\phi_{n+1}$	$\displaystyle=\phi_{n}+\frac{k_{1}}{6}+\frac{k_{2}}{3}+\frac{k_{3}}{3}+\frac{k% _{4}}{6}$

(6.159)

where $\Delta t=t_{n+1}-t_{n}$ .

RK4 can be applied to second order equations by using equation (6.142).

Because the method is explicit ( $\phi_{n+1}$ doesn't appear as an argument to $f$ ), equation (6.159) doesn't require a nonlinear solver even if $f$ is nonlinear.


`2nd order Runge-Kutta (RK2)`		`Range (RangeOutput)`

OOF2: The Manual

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