## 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:

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

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, \(p=2\).