Alternating direction numerical scheme for singularly perturbed 2D degenerate parabolic convection-diffusion problems

Abstract

In this article, we study the numerical solution of singularly perturbed 2D degenerate parabolic convection-diffusion problems on a rectangular domain. The solution of this problem exhibits parabolic boundary layers along x=0,y=0 and a corner layer in the neighborhood of (0, 0). First, we use an alternating direction implicit finite difference scheme to discretize the time derivative of the continuous problem on a uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problems, we apply the upwind finite difference scheme on a piecewise-uniform Shishkin mesh. We derive error estimate for the proposed numerical scheme, which shows that the scheme is ε-uniformly convergent of almost first-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.