Runge-Kutta methods adapted to the numerical integration of oscillatory problems
type: Item
instance of: scholarly article
title: Runge-Kutta methods adapted to the numerical integration of oscillatory problems
zbMATH Open document ID: 1057.65043
author: J. M. Franco
published in: Applied Numerical Mathematics
publication date: 2004-08-19T00:00:00Z
review text: The material is devoted to the construction of Runge-Kutta methods specially adapted to the numerical integration of initial value problems with oscillatory solutions. The coefficients of these methods are frequency-dependent such that certain particular oscillatory solutions are computed exactly. The first section is given by an introduction concerning this type of Runge-Kutta methods. Section two presents a detailed exposition of the ideas that lead to the formulation of the adapted Runge-Kutta methods. Explicit Runge-Kutta methods for oscillatory ordinary differential equations (ODEs) and order conditions are exposed. This part contains a modification of the classical Runge-Kutta algorithms such that the oscillatory linear problem is solved without truncation errors. Necessary and sufficient order conditions for the adapted Runge-Kutta methods are derived by using the B-series theory and the rooted trees. These conditions reduce to the classical order conditions for the classical Runge-Kutta methods when the parameter \(v= \omega h\to 0\), \(\omega\) being the main frequency of the oscillatory solutions and \(h\) the step-size. In section three one derives explicit adapted Runge-Kutta methods with order 3 and 4 as well as embedded pairs of adapted Runge-Kutta methods of orders 3 and 4. The fourth section presents some numerical experiments which support the efficiency of the adapted Runge-Kutta methods derived in the paper, when they are compared with other classical Runge-Kutta methods. Two types of comparisons with both fixed and variable step-size are exposed, each one for a linear model problem test, respectively a nonlinear model problem test. Thus the numerical results show the excellent behaviour of the new methods when they are compared with standard Runge-Kutta methods. Section five contains the conclusions and also some research perspective, like the extension of these methods to the case of semilinear ODE systems \((y'= Ay+ g(t, y))\), where the stiffn…
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 65L50
zbMATH DE Number: 2093399
zbMATH Keywords: adapted Runge-Kutta methods
zbMATH Keywords: oscillatory problems
zbMATH Keywords: B-series theory
zbMATH Keywords: step-size control
zbMATH Keywords: comparison of methods
zbMATH Keywords: oscillatory solutions
zbMATH Keywords: order conditions
zbMATH Keywords: rooted trees
zbMATH Keywords: numerical experiments
reviewed by: Romulus Militaru
MaRDI profile type: Publication
full work available at URL: j.apnum.2004.01.005
OpenAlex ID: W2006038957
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DOI: 10.1016/J.APNUM.2004.01.005
description: scientific article; zbMATH DE number 2093399