Nonlinear Instability in Dissipative Finite Difference Schemes
- Creators
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Stuart, Andrew
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.
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.Attached Files
Published - 1031048.pdf
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Additional details
- Eprint ID
- 78139
- Resolver ID
- CaltechAUTHORS:20170613-075253765
- Created
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2017-06-13Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field
- Other Numbering System Name
- Andrew Stuart
- Other Numbering System Identifier
- J9