4.7 Article

Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction-Diffusion System

期刊

MATHEMATICS
卷 11, 期 12, 页码 -

出版社

MDPI
DOI: 10.3390/math11122685

关键词

singular perturbation; time-dependent reaction-diffusion; boundary layers phenomena; system of equations; Shishkin mesh; parameter-uniform convergence

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This article develops and analyzes a numerical scheme to solve a singularly perturbed parabolic system of n reaction-diffusion equations. The scheme considers m equations with a perturbation parameter and the rest without it. It uses finite difference approximations on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable. Convergence properties and error analyses are derived, and numerical experiments are presented to support the theoretical results.
This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction-diffusion equations where m of the equations (with m < n) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter e ? 0, proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results.

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