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

• Published in: Engineering computations 2021-02, Vol.38 (2), p.964-988
• Main Authors: Kumar, Mukesh, Singh, Joginder, Kumar, Sunil
• Format: Article
• Language: English
• Subjects: Control theory; Delay; Error analysis; Hypotheses; Methods; Numerical analysis; Numerical methods; Ordinary differential equations; Partial differential equations; Robustness (mathematics)
• ISSN: 0264-4401; 1758-7077
• DOI: 10.1108/EC-04-2020-0191

Purpose The 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/approach Some 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. Findings The 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/value A 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.