# 1.9 Runge-Kutta Methods

## 1.9.1 Two-stage Runge-Kutta Methods

Measurable Outcome 1.16, Measurable Outcome 1.17, Measurable Outcome 1.19

In the previous lectures, we have concentrated on multi-step methods. However, another powerful set of methods are known as multi-stage methods. Perhaps the best known of multi-stage methods are the Runge-Kutta methods. In this lecture, we give some of the most popular Runge-Kutta methods and briefly discuss their properties.

A popular two-stage Runge-Kutta method is known as the modified Euler method:

| \(\displaystyle a\) | \(\displaystyle =\) | \(\displaystyle {\Delta t}f(v^ n, t^ n)\) |  | (1.139)
| \(\displaystyle b\) | \(\displaystyle =\) | \(\displaystyle {\Delta t}f(v^ n + a/2, t^ n + {\Delta t}/2)\) |  | (1.140)
| \(\displaystyle v^{n+1}\) | \(\displaystyle =\) | \(\displaystyle v^ n + b\) |  | (1.141)

Another popular two-stage Runge-Kutta method is known as the Heun method:

| \(\displaystyle a\) | \(\displaystyle =\) | \(\displaystyle {\Delta t}f(v^ n, t^ n)\) |  | (1.142)
| \(\displaystyle b\) | \(\displaystyle =\) | \(\displaystyle {\Delta t}f(v^ n + a, t^ n + {\Delta t})\) |  | (1.143)
| \(\displaystyle v^{n+1}\) | \(\displaystyle =\) | \(\displaystyle v^ n + \frac{1}{2}(a+b)\) |  | (1.144)

As can been seen with either of these methods, \(f\) is evaluated twice in finding the new value of \(v^{n+1}\): once to determine \(a\) and once to determine \(b\). Both of these methods are second-order accurate, \(p=2\).