# 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, \(p=4\).