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Selçuk Journal
of
Applied Mathematics
Winter-Spring, 2001
Volume 2
Number 1
Research Center
of
Applied Mathematics
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SJAM
Winter-Spring 2001, Volume 2 - Number 1
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Runge-Kutta Methods, Trees, and Maple
On
a Simple Proof of Butcher's Theorem and the Automatic
Generation of Order Conditions
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Folkmar Bornemann |
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Center
of Mathematical Sciences, Munich University of Technology, 80290
Munich, Germany
email: bornemann@ma.tum.de
Received: February 9, 2001
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Summary
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.
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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
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Mathematics Subject Classification (1991):
65-01,
65L06, 65Y99
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Article
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