Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids

Abstract

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform grids. The nonuniform grids are obtained by equidistribution of a positive monitor function, which is a linear combination of a constant and the second-order spatial derivative of the singular component of the solution on every temporal level. Truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.

Keywords:

• singularly perturbed parabolic problem
• boundary layers
• uniform convergence
• equidistribution grid
• upwind finite difference scheme

2000 AMS Subject Classifications:

• 65M06
• 65M12
• 65M50

Acknowledgements

The authors express their sincere thanks to the referees whose valuable comments helped to improve the presentation.