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A uniformly convergent scheme for a system of reaction–diffusion equations

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Publication Date
Volume
206
Issue
1
Identifiers
DOI: 10.1016/j.cam.2006.06.005
Keywords
  • Singular Perturbation
  • Reaction–Diffusion Problems
  • Uniform Convergence
  • Coupled System
  • Shishkin Mesh

Abstract

Abstract In this work a system of two parabolic singularly perturbed equations of reaction–diffusion type is considered. The asymptotic behaviour of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution we consider the implicit Euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform Shishkin mesh. We prove that this scheme is uniformly convergent, with respect to the diffusion parameters, having first-order convergence in time and almost second-order convergence in space, in the discrete maximum norm. Numerical experiments illustrate the order of convergence proved theoretically.

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