A Brief History of Runge-Kutta Methods

Butcher's "A History of Runge-Kutta Methods" contains a beautiful account of the early days of this now ubiquitous method for solving ordinary differential equations.

Carl Runge, a German mathematician and spectroscopist, toyed with the idea of improving upon Euler's simple time stepping approximation for solving ODEs. In a 1895 paper, he wrote three down three different "Runge-Kutta" schemes to solve a particular differential equation that arose in the study of atomic spectra.

He used so much mathematics in his research that physicists thought he was a mathematician, and he did so much physics that mathematicians thought he was a physicist.

Two other German mathematicians, Karl Heun and Martin Kutta made additional contributions. In 1900, Heun wrote down RK methods up to order 4. In 1901, Kutta extended it to order 5, and systematized most of the prior work. His contribution included the classical 4th order RK method, which is the ODE analog of Simpson's method for numerical integration.

Numerical Recipes has this to say about this century old algorithm:

For many scientific users, fourth-order Runge-Kutta is not just the first word on ODE integrators, but the last word as well. In fact, you can get pretty far on this old workhorse, especially if you combine it with an adaptive stepsize algorithm.

By 1925, Nystrom had extended and cleaned up most of the RK methods developed previously.

Over the years, higher order methods were developed, stability properties were studied, and implicit RK methods capable of solving stiff problems were developed.