On the global error of discretization methods for highly-oscillatory ordinary differential equations
type: Item
instance of: scholarly article
title: On the global error of discretization methods for highly-oscillatory ordinary differential equations
zbMATH Open document ID: 1027.65107
DOI: 10.1023/A:1022049814688
published in: BIT
publication date: 2003-05-14T00:00:00Z
review text: The author considers the numerical integration of linear highly-oscillating systems of the form \(y''+g(t)y=0\), where \(g(t) \to \infty\) for \(t \to \infty\), e.g. \(g(t)= t^2, g(t)=t\) or \(g(t)=\log(1+t)\) leading to Riccati's and Airy's equation and the log-oscillator, respectively. Using WKB analysis an explicit form of the global-error envelope is derived for Runge-Kutta and Magnus methods. The starting point for the analysis is a global-error expansion of an arbitrary time-stepping method dating back to P. Henrici. The error formula provides a thorough explanation for the superior performance of the Magnus methods compared to classical solvers for the type of ordinary differential equations considered. The author also proposes a modification of the Magnus method based on local linearisation. The theoretical results are illustrated by nicely chosen numerical examples.
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 1886391
zbMATH Keywords: highly-oscillatory differential equation
zbMATH Keywords: global error
zbMATH Keywords: WKB technique
zbMATH Keywords: Magnus expansion
zbMATH Keywords: Riccati equation
zbMATH Keywords: Runge-Kutta method
zbMATH Keywords: Airy's equation
zbMATH Keywords: log-oscillator
zbMATH Keywords: numerical examples
author: Arieh Iserles
reviewed by: Rolf Dieter Grigorieff
describes a project that uses: COLNEW
describes a project that uses: BNDSCO
MaRDI profile type: Publication
full work available at URL: a:1022049814688
OpenAlex ID: W1602423653
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description: scientific article; zbMATH DE number 1886391