Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations

ABSTRACT

This article proposes a second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer. To handle this layer phenomenon, the problem is solved on a priori special mesh by using the implicit-Euler scheme for the discretization of the time derivative and the upwind scheme for the spatial derivatives which results almost first-order convergence, that is, $O(N^{-1}\ln N+\mathrm{\Delta }t)$. It is shown that, the implementation of Richardson extrapolation technique enhanced the order of convergence to $O(N^{-2}{\ln }^{2}N+\mathrm{\Delta }{t}^2)$. To support the theoretical results, numerical experiments are carried out by applying the proposed technique on two test examples.

KEYWORDS:

• Singularly perturbed delay parabolic convection-diffusion problems
• upwind scheme
• piecewise-uniform Shishkin mesh
• uniform convergence
• Richardson extrapolation technique

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable comments and suggestions, which helped to improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Srinivasan Natesan  http://orcid.org/0000-0001-7527-1989