1.9 Runge-Kutta Methods

1.9.2 Four-stage Runge-Kutta Method

Measurable Outcome 1.16, Measurable Outcome 1.17, Measurable Outcome 1.19

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, \(p=4\).