Second-Order Uniformly Convergent Richardson Extrapolation Method for Singularly Perturbed Degenerate Parabolic PDEs

Abstract

This article studies the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of this problem exhibits a boundary layer in the neighborhood of left boundary of the domain. We discretize the domain by piecewise-uniform Shishkin mesh in the spatial direction and uniform mesh in the temporal direction. The numerical scheme consists of the implicit-Euler scheme for the time derivative and upwind finite difference scheme for the spatial derivatives. By applying the Richardson extrapolation technique, we improve the order of accuracy of the numerical solution from \(O (N^{-1} \ln ^2 N + \varDelta t)\) to \(O (N^{-2} \ln ^2 N + \varDelta t^2)\) (measured in the discrete supremum norm), where \(N+1\) mesh points are used in spatial direction and \(\varDelta t\) is the step size in the temporal direction. Error estimates are derived. Some numerical experiments are carried out to validate the theoretical results.