OOF2: The Manual

`2nd order Runge-Kutta (RK2)`
	6.5.2. Subclasses

Name

2nd order Runge-Kutta (RK2) — Second order Runge-Kutta time stepping.

Synopsis

RK2()

Details

Base class: TimeStepper

Description

RK2 is a TimeStepper that implements the second order Runge-Kutta method for solving ordinary differential equations. The error on each step is of order $\Delta t^3$ . RK2 is also referred to as the midpoint method.

Given a vector $\phi_n$ of unknowns (i.e. Field values in OOF2) at time $t_n$ , and the first order differential equation

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

(6.155)

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

$\begin{align*} k_1 &= \Delta t f(\phi_n, t_n) \\ k_2 &= \Delta t f(\phi_n+\frac{k_1}{2}, t_n+\frac{\Delta t}{2}) \\ \phi_{n+1} &= \phi_n + k_2 \end{align*}$

(6.156)

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

RK2 can be applied to second order equations by using equation (6.141).

Because the method is explicit ( $\phi_{n+1}$ doesn't appear as an argument to $f$ ), equation (6.156) doesn't require a nonlinear solver even if $f$ is nonlinear.


`RGBData8 (RGBData8)`		`4th order Runge-Kutta (RK4)`