Alternating direction implicit method for singularly perturbed 2D parabolic convection–diffusion–reaction problem with two small parameters

ABSTRACT

In this article, we construct and analyse an Alternating Direction Implicit (ADI) scheme for singularly perturbed 2D parabolic convection–diffusion–reaction problems with two small parameters. We consider the operator-splitting ADI finite difference scheme for time stepping on a uniform mesh and a simple upwind-difference scheme for spatial discretization on a specially designed piecewise-uniform Shishkin mesh. The resulting scheme is proved to be uniformly convergent of order $O(N^{-1}\ln N+M^{-1})$, where N, M are the spatial and temporal parameters respectively. Numerical experiments confirm the theoretical results and the effectiveness of the proposed method.

KEYWORDS:

• Singularly perturbed 2D parabolic convection–reaction–diffusion problem
• alternating direction implicit scheme
• finite difference scheme
• Shishkin meshes
• stability
• uniform error estimate

AMS CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15

Acknowledgements

The authors wish to acknowledge the anonymous referees for carefully reading the manuscript and providing their valuable comments and suggestions, which really helped to improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.