# 1.9 Runge-Kutta Methods

## 1.9.1 Two-stage Runge-Kutta Methods

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, $$p=2$$.