Anisotropy, propagation failure, and wave speedup in traveling waves of discretizaions of a Nagumo PDE

Document Type

Article

Publication Date

3-1-2003

Abstract

This article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic PDEs (Nagumo type) possessing piecewise linear bistable nonlinearities. Solution behavior is compared in terms of waveforms and in terms of the so-called (a, c) relationship where a is a parameter controlling the bistable nonlinearity by varying the potential energy difference of the two phases and c is the wave speed of the traveling wave. Uniform spatial discretizations and A(α) stable linear multistep methods in time are considered. Results obtained show that although the traveling wave solutions to parabolic PDEs are stationary for only one value of the parameter a, a0, spatial discretization of these PDEs produce traveling waves which are stationary for a nontrivial interval of a values which include a0, i.e., failure of the solution to propagate in the presence of a driving force. This is true no matter how wide the interface is with respect to the discretization. For temporal discretizations at large wave speeds the set of parameter a values for which there are traveling wave solutions is constrained. An analysis of a complete discretization points out the potential for nonuniqueness in the (a, c) relationship. © 2003 Published by Elsevier Science B.V.

Identifier

0037335295 (Scopus)

Publication Title

Journal of Computational Physics

External Full Text Location

https://doi.org/10.1016/S0021-9991(03)00004-4

ISSN

00219991

First Page

562

Last Page

582

Issue

2

Volume

185

Grant

DMS-0139824

Fund Ref

National Science Foundation

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