Parameter uniform numerical method for a system of two coupled singularly perturbed parabolic convection-diffusion equations

Abstract

In this paper, we propose a numerical scheme for a system of two linear singularly perturbed parabolic convection-diffusion equations. The presented numerical scheme consists of a classical backward-Euler scheme on a uniform mesh for the time discretization and an upwind finite difference scheme on an arbitrary nonuniform mesh for the spatial discretization. Then, for the time semidiscretization scheme, an a priori and an a posteriori error estimations in the maximum norm are obtained. It should be pointed out that the a posteriori error bound is suitable to design an adaptive algorithm, which is used to generate an adaptive spatial grid. It is proved that the method converges uniformly in the discrete maximum norm with first-order time and spatial accuracy, respectively, for the fully discrete scheme. At last, some numerical results are given to validate the theoretical results.

Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction–Diffusion System

“…A numerical method composed of the backward Euler method together with the HODIE (High Order via Differential Identity Expansion) scheme, classical central difference scheme, and a Shishkin mesh has appeared recently in [12] for a system of SPPDEs of Convection-diffusion (CD) type in which all the higher order spatial derivatives are multiplied by a parameter ε. In [13], a numerical method which is a combination of the backward Euler scheme, the upwind finite difference scheme, and a nonuniform mesh based on a monitor function is devised for the aforementioned system with two equations in which the higher order spatial derivatives are multiplied by different perturbation parameters. For this same system, a numerical method consisting of the backward Euler method, a hybrid scheme, and a Shishkin mesh is developed in [14] with higher order convergence in space.…”