The effect of freezing and discretization to the asymptotic stability of relative equilibria
type: Item
instance of: scholarly article
title: The effect of freezing and discretization to the asymptotic stability of relative equilibria
zbMATH Open document ID: 1149.65077
author: V. Thümmler
published in: Journal of Dynamics and Differential Equations
publication date: 2008-06-03T00:00:00Z
review text: The purpose of the paper is to analyze numerical methods for the approximation of relative equilibria of parabolic systems in one space dimension: \[ u_t=Au_{xx}+f(u,u_x) \] In the first section the author gives a short introduction to the method of freezing relative equilibria and state conditions which ensure the asymptotic phase of these solutions. The second section concerns the numerical approximation. Thus, he introduces the finite difference approximation and states the main stability result for the solution of the discretized equations, given by the fact that nonlinear stability properties are inherited by the numerical approximation with finite differences on a finite equidistant grid with appropriate boundary conditions. The third section focuses on the stability of the nonlinear discretized system and proves the main stability result obtained in the previous section, using resolvent estimates. The fourth section proves resolvent estimates for the discretized system. The fifth section contains some numerical examples concerning the cubic quintic Ginzburg Landau equation, which shows a variety of coherent structures: stable pulse, fronts, sources, sinks. By a counterexample for the Nagumo equation the author shows that some of the assumptions on the boundary operators are sharp.
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 5281921
zbMATH Keywords: general evolution equations
zbMATH Keywords: equivariance
zbMATH Keywords: stability
zbMATH Keywords: Lie groups
zbMATH Keywords: partial differential algebraic equations
zbMATH Keywords: unbounded domains
zbMATH Keywords: finite differences
zbMATH Keywords: asymptotic stability
reviewed by: Romulus Militaru
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: s10884-007-9091-0
OpenAlex ID: W1967266235
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DOI: 10.1007/S10884-007-9091-0
description: scientific article; zbMATH DE number 5281921