Selçuk Journal of Applied Mathematics

 Selçuk Journal of
  Applied Mathematics

  Winter-Spring, 2001
  Volume  2
  Number 1

Research Center of 
  Applied Mathematics

 SJAM Winter-Spring 2001, Volume 2 - Number 1

Runge-Kutta Methods, Trees, and Maple
On a Simple Proof of Butcher's Theorem and the Automatic Generation of Order Conditions

Folkmar Bornemann  

Center of Mathematical Sciences, Munich University of Technology, 80290 Munich, Germany

Received: February 9, 2001

This paper presents a simple and elementary proof of Butcher's theorem on the order conditions of Runge-Kutta methods. It is based on a recursive definition of rooted trees and avoids combinatorial tools such as labelings and Faà di Bruno's formula. This strictly recursive approach can easily and elegantly be implemented using modern computer algebra systems like Maple for automatically generating the order conditions. The full, but short source code is presented and applied to some instructive examples.

Key words
numerical solution of ODEs, Runge-Kutta methods, recursive representation of rooted trees, Butcher's theorem, automatic generation of order conditions, computer algebra systems

Mathematics Subject Classification (1991): 65-01, 65L06, 65Y99

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