On explicit multi-revolution Runge-Kutta schemes
type: Item
instance of: scholarly article
title: On explicit multi-revolution Runge-Kutta schemes
zbMATH Open document ID: 1115.65081
author: Manuel Calvo
author: J. I. Montijano
author: Luis Rández
published in: Advances in Computational Mathematics
publication date: 2007-04-26T00:00:00Z
review text: Let \(\varphi(y) = y+\varepsilon f(y)\) denote a near identity Poincaré map. The computation of \(\varphi^N\), for \(N\) an integer, is considered using a modified Runge-Kutta method with \(\varepsilon^N\) taking the role of stepsize and with the coefficients in the method depending on \(N\). It is assumed that an auxiliary integrator is available to evaluate \(\varphi\) values. An analysis of these ``Multi-revolution Runge-Kutta methods'' was presented by the authors and \textit{L. O. Jay} [Numer. Math. 97, No. 4, 635--666 (2004; Zbl 1067.65067)]. In the present paper, the order is analysed using Butcher series and this leads to a comprehensive study of fourth-order explicit methods with 4 steps. To obtain order 5, six steps are required, as for classical Runge-Kutta methods, and appropriate adaptation of standard simplifying assumptions are used in the derivation. Numerical experiments are presented showing the behaviour of a fifth order multi-revolution Runge-Kutta method derived in the paper, compared with the standard DOPRI5(4) method.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 5146131
zbMATH Keywords: initial value problems
zbMATH Keywords: almost periodic or highly oscillatory solutions
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: long term integration
zbMATH Keywords: multi-revolution methods
zbMATH Keywords: numerical experiments
zbMATH Keywords: Butcher series
reviewed by: J. C. Butcher
MaRDI profile type: Publication
full work available at URL: s10444-004-7209-z
OpenAlex ID: W2066869074
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DOI: 10.1007/S10444-004-7209-Z
description: scientific article; zbMATH DE number 5146131