Robust numerical schemes for singularly perturbed delay parabolic convection-diffusion problems with degenerate coefficient

Abstract

This article studies a Dirichlet boundary value problem for singularly perturbed time delay convection-diffusion equation with degenerate coefficient. A priori explicit bounds are established on the solution and its derivatives. For asymptotic analysis of the spatial derivatives the solution is decomposed into regular and singular parts. To approximate the solution a numerical method is considered which consists of backward Euler scheme for time discretization on uniform mesh and a combination of midpoint upwind and central difference scheme for the spatial discretization on modified Shishkin mesh. Further, the order of convergence is increased in time direction via implementing Richardson extrapolation. Stability analysis is carried out, numerical results are presented and comparison is done with upwind scheme on uniform mesh as well as upwind scheme on Shishkin mesh to demonstrate the effectiveness of the proposed methods. The convergence obtained in practical satisfies the theoretical predictions.

Keywords:

• Singular perturbation
• parabolic delay differential equations
• degenerate coefficient
• hybrid scheme
• Shishkin mesh
• Richardson extrapolation

2010 Mathematics Subject Classifications:

• 65N12
• 65N30
• 65N06
• 65N15

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is thankful to the UGC (University Grants Commission) Non-NET Fellowship for supporting financially vide Ref.No. Sch./139/Non-NET/Ext-152/2018-19/67.