A Fitted Linear Multistep Approach for Singularly Perturbed Parabolic‐Type Reaction–Diffusion Problems Using Shishkin Meshes

Abstract

This paper presents a class of singularly perturbed parabolic‐type reaction diffusion problems. Due to the presence of a small parameter ε , (0 < ε ≪ 1) as a diffusion coefficient, the proposed problem exhibits twin boundary layers in the neighborhood of the end points of the spatial domain near x = 0 and x = 1. The solutions obtained in such layer regions have the properties of oscillations or abrupt changes. Because of these challenges, the classical numerical methods are inefficient to solve the problem. To address these challenges, we formulated and analyzed a parameter uniformly convergent scheme. To approximate the solution, we employ a fitted numerical method combining the implicit Euler scheme for time discretization on a uniform mesh and a linear multistep finite difference scheme for spatial discretization on Shishkin meshes. Richardson extrapolation is used to improve accuracy of numerical computation. The uniform convergence analysis confirmed that the proposed method is uniformly convergent fourth‐order accurate in the spatial direction and second‐order accurate in the temporal direction. Three model examples were presented for simulation in order to verify the applicability of the developed method, and the numerical results validated that the theoretical analyses coincide with the practical examples. Furthermore, the obtained numerical results demonstrate that the new strategy outperforms certain existing methods in the literature.