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

Manikandan Mariappan, Chandru Muthusamy and Higinio Ramos ()
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Manikandan Mariappan: Department of Mathematics, School of Engineering, Presidency University, Bengaluru 560064, Karnataka, India
Chandru Muthusamy: Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India
Higinio Ramos: Department of Applied Mathematics, Scientific Computing Group, University of Salamanca, Plaza de la Merced, 37008 Salamanca, Spain

Mathematics, 2023, vol. 11, issue 12, 1-17

Abstract: 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 ε → 0 , proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results.

Keywords: singular perturbation; time-dependent reaction–diffusion; boundary layers phenomena; system of equations; Shishkin mesh; parameter-uniform convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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