An efficient numerical method for coupled systems of singularly perturbed parabolic delay problems

Abstract

In this work, a coupled system of two singularly perturbed delay parabolic reaction–diffusion problems is considered. In each equation the diffusion term is multiplied by a small parameter. The parameters can be of different magnitude due to which the problem exhibits overlapping layers. To approximate the solution of the problem, we consider two splitting schemes on uniform meshes for time discretization and the central difference scheme on Shishkin and generalised Shishkin meshes for space discretization. We prove that the proposed numerical method is uniformly convergent having convergence of order one in time and almost two in space. The splitting schemes are considered for the first time for coupled systems of singularly perturbed parabolic delay problems. They decouple the components of the approximate solution at each time level, and hence resulted in a reduced computational time. Further, we present numerical results for two test problems in support of our theoretical findings and to demonstrate that the splitting schemes are more efficient than the classical scheme. © 2021, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.