An ϵ-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem

ABSTRACT

In this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.

KEYWORDS:

• Singularly perturbed degenerate parabolic convection–diffusion problem
• finite difference scheme
• piecewise-uniform Shishkin meshes
• error estimate
• uniform convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12

Acknowledgements

The authors wish to acknowledge 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