Second-order parameter-uniform finite difference scheme for singularly perturbed parabolic problem with a boundary turning point

Abstract

In this article, we consider a one-dimensional time-dependent singularly perturbed problem, whose solution possesses a parabolic boundary layer in the neighbourhood of the left lateral surface. A priori estimates are derived for the classical solution and its derivatives, which are useful for the stability and convergence analysis of the proposed difference scheme. The discussed scheme consists of an implicit Euler method on uniform mesh in time and a simple upwind scheme on piece-wise uniform mesh in space. Then, to improve the order of convergence, we apply the Richardson extrapolation scheme in both space and time direction. Theoretically, we prove that the proposed scheme is almost second-order parameter uniform convergent and verify it by doing some numerical experiments.