Another uniform convergence analysis technique of some numerical methods for parabolic singularly perturbed problems

Abstract

In this work we analyze the uniform convergence of some monotone finite difference schemes which are used to semidiscretize in space time dependent two dimensional singularly perturbed parabolic problems of convection–diffusion or reaction–diffusion type. The analysis technique combines a suitable semidiscrete maximum principle joint to some techniques used in the study of the convergence of well known numerical schemes developed to solve elliptic singularly perturbed problems. We focus our attention to standard central or upwind finite difference schemes defined on special nonuniform meshes of Shishkin type. Nevertheless, the technique can be easily extended to other classical discretization methods like, for example, fitted operator methods. We prove that the stiff initial value problems resulting of the spatial semidiscretization processes, have a unique solution which converges uniformly with respect to the singular perturbation parameter. To obtain an efficient numerical algorithm, such initial value problems are discretized by using appropriate time integrators; here, we have chosen the Implicit Euler method for doing this, obtaining an unconditional and uniformly convergent numerical method in a simple way. Finally, some numerical results are shown in order to illustrate the efficiency of the numerical methods, according to the theoretical results.