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Nonlinear Instability in Dissipative Finite Difference Schemes

Stuart, Andrew (1989) Nonlinear Instability in Dissipative Finite Difference Schemes. SIAM Review, 31 (2). pp. 191-220. ISSN 0036-1445. http://resolver.caltech.edu/CaltechAUTHORS:20170613-075253765

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Abstract

A unified analysis of reaction-diffusion equations and their finite difference representations is presented. The parallel treatment of the two problems shows clearly when and why the finite difference approximations break down. The approach used provides a general framework for the analysis and interpretation of numerical instability in approximations of dissipative nonlinear partial differential equations Continuous and discrete problems are studied from the perspective of bifurcation theory, and numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems. An asymptotic approach, due to Newell (SIAM J. Appl. Math., 33 (1977), 133–160), is used to investigate the instability phenomenon further. In particular, equations are derived that describe the interaction of the dynamics of the partial differential equation with the artefacts of the discretization.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/1031048DOIArticle
http://epubs.siam.org/doi/10.1137/1031048PublisherArticle
Additional Information:© 1989 Society for Industrial and Applied Mathematics. Received by the editors December 7, 1987; accepted for publication (in revised form) December 30, 1988. I am grateful to Professors L. N. Trefethen and J. M. Sanz-Serna for a number of comments and suggestions which improved earlier versions of this paper. The work presented here is based in part on a seminar given at the Numerical Analysis Group, Oxford University in 1986.
Subject Keywords:continuous and discrete problems, dissipation and nonlinearity, bifurcation and instability
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ9
Classification Code:AMS(MOS) subject classifications: 35A40, 35K57, 65M10
Record Number:CaltechAUTHORS:20170613-075253765
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170613-075253765
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78139
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 Jun 2017 16:06
Last Modified:22 Aug 2017 21:01

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