On the Behavior of Attractors Under Finite Difference Approximation

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C1 perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.

Original languageEnglish (US)
Pages (from-to)1155-1180
Number of pages26
JournalNumerical Functional Analysis and Optimization
Volume16
Issue number9-10
DOIs
StatePublished - Jan 1 1995
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

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