Dissipativity of numerical schemes

C. Foias, M. S. Jolly, I. G. Kevrekidis, E. S. Titi

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Abstract

The authors show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations they construct solutions which blow up in finite time for two semi-discrete schemes. They also show the existence of spurious steady states whose unstable manifolds, in some cases, contain solutions which explode. This connection between the blow-up phenomenon and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations. Two fully discrete finite difference schemes derived from a third semi-discrete scheme, reported to be dissipative, are analysed. Both latter schemes are shown to have a stability condition which is independent of the initial data.

Original languageEnglish (US)
Article number001
Pages (from-to)591-613
Number of pages23
JournalNonlinearity
Volume4
Issue number3
DOIs
StatePublished - Dec 1 1991

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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    Foias, C., Jolly, M. S., Kevrekidis, I. G., & Titi, E. S. (1991). Dissipativity of numerical schemes. Nonlinearity, 4(3), 591-613. [001]. https://doi.org/10.1088/0951-7715/4/3/001