On a uniformly accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution

G. Beckett, J.A. Mackenzie

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Abstract

We examine the convergence properties of a finite difference approximation of a singularly perturbed reaction-diffusion boundary value problem using a nonuniform grid. The grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution. Analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter. The use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids. Numerical results are given that confirm the uniform convergence rates.
LanguageEnglish
Pages381-405
Number of pages24
JournalJournal of Computational and Applied Mathematics
Volume131
Issue number1-2
DOIs
Publication statusPublished - 2001

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Reaction-diffusion Problems
Equidistribution
Singularly Perturbed Problem
Finite Difference Approximation
Grid
Monitor
Boundary value problems
Non-uniform Grid
Reaction-diffusion
Singular Perturbation
Second derivative
Uniform convergence
Singularly Perturbed
Derivatives
Numerical Approximation
Convergence Properties
Convergence Rate
Linear Combination
Boundary Value Problem
Numerical Results

Keywords

  • uniform convergence
  • adaptivity
  • equidistribution
  • singular perturbation
  • reaction-diffusion
  • computational mathematics

Cite this

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