Numerical simulation of differential systems displaying rapidly oscillating solutions
type: Item
instance of: scholarly article
title: Numerical simulation of differential systems displaying rapidly oscillating solutions
zbMATH Open document ID: 0923.65042
DOI: 10.1023/A:1004337014272
published in: Journal of Engineering Mathematics
publication date: 1999-10-15T00:00:00Z
review text: This paper is concerned with the numerical approximation of some systems of differential equations whose solutions contain components with very different time scales. The systems under consideration have the form \( \dot u + A u + \varepsilon^{-1} L u + B(u,u) = f \) where \( A\) is a symmetric positive definite matrix, \(L\) is either a symmetric positive definite or antisymmetric matrix that can be diagonalized, \(B\) is a quadratic operator and \( \varepsilon >0\) is a small parameter. To approximate the solutions of these systems the authors introduce a previous averaging process that leads to a new system whose solutions are slowly varying and therefore can be integrated numerically by a standard numerical integrator without severe restrictions on the size of the step. Then the solution of the original system can be expressed in terms of the solution of the averaged system. Some theoretical results are stated to justify the proposed approach. Finally some low-dimensional examples are presented to show the behaviour of the new approach. From these examples it follows that, for small \(\varepsilon\), the approximate solutions obtained are able to capture the main features of the solution with low computational effort.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E13
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 1282085
zbMATH Keywords: stiff oscillatory equations
zbMATH Keywords: rapidly oscillating solutions
zbMATH Keywords: systems
zbMATH Keywords: averaging process
author: Ioana Moise
author: Eric Simonnet
author: R. Temam
author: M. Ziane
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: a:1004337014272
OpenAlex ID: W4256161800
description: scientific article; zbMATH DE number 1282085