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 y_{i} to y_{i+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 y_{i} at a time t_{i}, and we want to figure out the value of y at the time t_{i+1}. We make 4 estimates of the slope within this time interval. I'll label these estimates with letters k_{1} through k_{4}, as some textbooks do.
We project this slope forward by HALF a timestep to determine an estimated position (green dot) at this future time
We project this slope forward from the starting point by HALF a timestep to determine an estimated position (brown dot) at this future time
We project this slope forward from the starting point by a FULL timestep to determine an estimated position (pink dot) at the next timestep
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.
Copyright © Michael Richmond. This work is licensed under a Creative Commons License.