Symbolic derivation of Runge-Kutta methods
type: Item
instance of: scholarly article
title: Symbolic derivation of Runge-Kutta methods
zbMATH Open document ID: 0819.65108
published in: Journal of Symbolic Computation
publication date: 1995-04-04T00:00:00Z
review text: This paper is concerned with a symbolic Mathematica package Butcher.m which automates the derivation of order conditions of Runge-Kutta (RK) methods for the numerical solution of ordinary differential equations. It must be noticed that there is a first version of Butcher.m written by Keiper around 1988. However the version under consideration here is new and incorporates substantial improvements not only in the derivation of the order conditions but also in the possibility to use different sets of simplifying assumptions to get more easily different families of RK formulas. The paper explains the techniques used by the author to generate efficiently the set of all rooted trees and the elementary weights and densities associated to the trees which are essential to derive the order conditions in the RK methods. It follows from the remarks of the author that computer algebra has been considerably reduced over the first version of Butcher.m. Next, the problem of solving the set of order conditions for a given number of stages and order is considered. At this point the use of simplifying assumptions and the more general technique of Gröbner bases is examined. Some examples are presented to show the potentiality of this package. In conclusion we have a new tool which can be very useful in the investigation of RK schemes, thus for a given formula (even of higher order) we may compute straightforwardly the coefficients of the elementary differentials in the local error. However, due to the complexity, the derivation of e.g. high order RK formulas (or pairs of) is far from being solved with this package.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 740428
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: symbolic Mathematica package Butcher.m
zbMATH Keywords: order conditions
zbMATH Keywords: computer algebra
zbMATH Keywords: Gröbner bases
zbMATH Keywords: complexity
author: Mark Sofroniou
reviewed by: Manuel Calvo
describes a project that uses: Mathematica
describes a project that uses: butcher
MaRDI profile type: Publication
full work available at URL: jsco.1994.1048
OpenAlex ID: W2122336811
DOI: 10.1006/JSCO.1994.1048
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description: scientific article; zbMATH DE number 740428