Runge-Kutta methods, trees, and Mathematica
arXiv:math/0211049
Abstract
A simple and elementary proof of Butcher's theorem on the order conditions of Runge-Kutta methods is presented. It is based on a recursive definition of rooted trees and avoids combinatorial tools such as labelings and Faa di Bruno's formula. This strictly recursive approach can easily and elegantly be implemented using modern computer algrebra systems like Mathematica. The full but short source code is included and applied to some instructive examples.
arXiv comment9 pages, source files include a Mathematica notebook