A robust numerical method for a coupled system of singularly perturbed parabolic delay problems

Abstract

PurposeThe purpose of this paper is to design and analyze a robust numerical method for a coupled system of singularly perturbed parabolic delay partial differential equations (PDEs).Design/methodology/approachSome a priori bounds on the regular and layer parts of the solution and their derivatives are derived. Based on these a priori bounds, appropriate layer adapted meshes of Shishkin and generalized Shishkin types are defined in the spatial direction. After that, the problem is discretized using an implicit Euler scheme on a uniform mesh in the time direction and the central difference scheme on layer adapted meshes of Shishkin and generalized Shishkin types in the spatial direction.FindingsThe method is proved to be robust convergent of almost second-order in space and first-order in time. Numerical results are presented to support the theoretical error bounds.Originality/valueA coupled system of singularly perturbed parabolic delay PDEs is considered and some a priori bounds are derived. A numerical method is developed for the problem, where appropriate layer adapted Shishkin and generalized Shishkin meshes are considered. Error analysis of the method is given for both Shishkin and generalized Shishkin meshes.