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Runge-Kutta methods, trees, and Mathematica

Folkmar Bornemann
Organization: Munich University of Technology
Department: Center of Mathematical Sciences
Journal / Anthology

Selšuk Journal of Applied Mathematics (version using Maple)
Year: 2001
Volume: 2
Page range: 3-15

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.

*Mathematics > Calculus and Analysis > Differential Equations
*Mathematics > Discrete Mathematics > Combinatorics
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RungeKutta.nb (14.7 KB) - Mathematica Notebook