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A Priori Estimates for the Global Error Committed by Runge-Kutta Methods for a Nonlinear Oscillator

Published online by Cambridge University Press:  01 February 2010

Jitse Niesen
Jitse Niesen
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 OWAJ.Niesen@damtp.cam.ac.uk, http://www.damtp.cam.ac.uk/user/na/people/Jitse/index.html

Abstract

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The Alekseev–Gröbner lemma is combined with the theory of modified equations to obtain an a priori estimate for the global error of numerical integrators. This estimate is correct up to a remainder term of order h2p, where h denotes the step size and p the order of the method. It is applied to nonlinear oscillators whose behaviour is described by the Emden–Fowler equation y″+tνyn=0. The result shows explicitly that later terms sometimes blow up faster than the leading term of order hp, necessitating the whole computation. This is supported by numerical experiments.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 18 - 28
Copyright
Copyright © London Mathematical Society 2003

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