×

Dissipativity of numerical schemes. (English) Zbl 0734.65080

Authors’ summary: We show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations we construct solutions which blow up in finite time for two semi-discrete schemes. We 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 by the first and the fourth author to be dissipative, are analyzed. Both latter schemes are shown to have a stability condition which is independent of the initial data. A similar result is obtained for a fully discrete Galerkin scheme. While the results are stated for the Kuramoto-Sivashinsky equation, most naturally carry over to other dissipative partial differential equations.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI Link