1.9 Runge-Kutta Methods | 1.9 Runge-Kutta Methods | Unit 1: Numerical Integration of Ordinary Differential Equations | Computational Methods in Aerospace Engineering | Aeronautics and Astronautics

1.9.2 Four-stage Runge-Kutta Method

The most popular form of a four-stage Runge-Kutta method is:

$\displaystyle a$	$\displaystyle =$	$\displaystyle {\Delta t}f(v^ n, t^ n)$	(1.145)
$\displaystyle b$	$\displaystyle =$	$\displaystyle {\Delta t}f(v^ n+a/2, t^ n + {\Delta t}/2)$	(1.146)
$\displaystyle c$	$\displaystyle =$	$\displaystyle {\Delta t}f(v^ n+b/2, t^ n + {\Delta t}/2)$	(1.147)
$\displaystyle d$	$\displaystyle =$	$\displaystyle {\Delta t}f(v^ n+c, t^ n + {\Delta t})$	(1.148)
$\displaystyle v^{n+1}$	$\displaystyle =$	$\displaystyle v^ n + \frac{1}{6}(a + 2b + 2c + d)$	(1.149)

Note that this method requires four evaluations of $f$ per iteration. This method is fourth-order accurate, $p=4$ .