Developing explicit Runge-Kutta formulas using open-source software

Abstract

Runge-Kutta formulas are some of the workhorses of numerical solving of differential equations. However, they are extremely difficult to generate; the algebra involved can be very complicated indeed. It is now standard, following the work of Butcher in the 1960's and 70's, to use the theory of trees to simplify the algebra. More recently, however, several authors have shown that it is quite feasible to use a computer algebra system to generate Runge-Kutta formulas. This article shows that, instead of using commercial systems as has been done previously, open-source systems can be used with equal effect. This has the advantage that anybody can experiment with the code.