Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear reaction-diffusion system

2010/04/05 by Paramasivam, M., Valarmathi, S., John J. H. Miller +1
Engineering · Mathematics · #65L10 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Material Science and Thermodynamics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1004.0681

openalex publication_date 2010/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximations obtained with this method is essentially second order convergent uniformly with respect to all of the parameters.

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