On $$\varepsilon $$-Uniform Higher Order Accuracy of New Efficient Numerical Method and Its Extrapolation for Singularly Perturbed Parabolic Problems with Boundary Layer

Abstract

This article aims to achieve higher-order numerical approximation to the solutions of a class of singularly perturbed parabolic problems which can consist of the time-dependent convection coefficient and generally possess regular boundary layer. In order to fulfill the aim, at first we develop and analyze an efficient numerical method by discretizing the model problem using a new finite difference scheme on an appropriate layer-adapted mesh in the spatial direction, and the time derivative using the backward-Euler method on an equidistant mesh. We adopt the two-stage discretization process to establish the parameter-uniform estimate in the discrete supremum norm; and provide stability analysis in both the temporal and spatial discretization cases. Afterwards, we apply the Richardson extrapolation technique solely in the temporal direction for enhancing the temporal accuracy. We finally show that the resulting numerical solution is globally second-order convergent with respect to both the spatial and the temporal variables. At the end, numerous numerical results are presented to corroborate the theoretical findings; and also to demonstrate the computational efficiency and the accuracy of the present numerical method in comparison with the existing numerical method. Besides this, we extend the computational experiment by solving the singularly perturbed semi-linear parabolic problem.