Intuitive Explanation of Runge Kutta Integration

This is an intuitive explanation of how Runge Kutta integration works. RK integration is a numerical method for solving arbitrary first-order differential equations and is widely used by scientists, engineers, economists, video game programmers, and anyone else needing to numerically solve differential equations. It’s not always the best way, but it’s pretty good most of the time, and often can be done in real time.

This article assumes you already know calculus, understand derivatives, slopes, and have encountered basic differential equations before.

Suppose we want to predict or simulate some physical system measured by some quantity y varying with time, y(t), for which we have been given an equation for its rate of change

For any time t and any value of y, this equation describes a unique slope. The function f(t,y) may be a simple algebraic function of t and y, or it may be a messier expression with exponentials and square roots and many terms. It may involve interpolating values given in a table, perhaps using experimental data.

Graph paper with many needles at various angles suggesting a field of slopes defined at all (t,y) locations

Slopes are given at every point in (t,y) space.

With this kind of first-order differential equation, we are typically given an initial value y1 at some start time t1. We want to know the value of y(t) at some later time t2…