Propagation and its failure in coupled systems of discrete excitable cells
SIAM Journal on Applied Mathematics
Stability of traveling wavefronts for the discrete Nagumo equation
SIAM Journal on Mathematical Analysis
Jacobi iteration in implicit difference schemes for the wave equation
SIAM Journal on Numerical Analysis
Algorithms for phase field computation of the dendritic operating state at large supercoolings
Journal of Computational Physics
Mathematical physiology
Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice
SIAM Journal on Applied Mathematics
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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.