Uniformly Convergent New Hybrid Numerical Method for Singularly Perturbed Parabolic Problems with Interior Layers

Abstract

In this article, we deal with a class of singularly perturbed parabolic convection–diffusion initial-boundary-value problems having discontinuous convection coefficient. Aiming to get better numerical approximation to the solutions of this class of problems, we devise a new hybrid finite difference scheme on a layer-resolving piecewise-uniform Shishkin mesh in the spatial direction, and the time derivative is discretized by the backward-Euler method in the temporal direction. We discuss the stability of the proposed method and establish the parameter-uniform error estimate. Numerical results are also displayed to support the theoretical findings and compared with the existing hybrid scheme to show the improvement in terms of spatial order of convergence. Further, we carry out numerical experiment for the semi-linear parabolic initial-boundary-value problems.