Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.

The fourth-order Runge-Kutta method

The Runge-Kutta methods are one group of predictor-corrector methods. The name "Runge-Kutta" can be applied to an infinite variety of specific integration techniques -- including Euler's method -- but we'll focus on just one in particular: a fourth-order scheme which is widely used.

The basic idea of all Runge-Kutta methods is to move from step yi to yi+1 by multiplying some estimated slope by a timestep. The difference between particular implementations involve how one estimates the slope. In the fourth-order Runge-Kutta method we will study, the basic idea is to combine 4 preliminary estimates to get one really good slope.

In the diagram below, we start at a location yi at a time ti, and we want to figure out the value of y at the time ti+1. We make 4 estimates of the slope within this time interval. I'll label these estimates with letters k1 through k4, as some textbooks do.

Once we have these four slopes, we can combine them in a weighted fashion to create an average slope, and use that average slope to compute the next position.

Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.