A Fitted Numerical Method for Interior-Layer Parabolic Convection–Diffusion Problems

Abstract

The aim of this paper is to construct and analyze a fitted mesh finite difference method (FMFDM) for a class of time-dependent singularly perturbed convection–diffusion–reaction problems with a turning point. While such problems have been widely studied in the case of boundary layers, little has been achieved for interior layer problems. In addition, most of works have considered the coefficient functions as the functions of space only. In this work, we study problems where the coefficient functions are dependent on both space and time variables, and their solutions exhibit an interior layer. We derive the solution bounds and their derivatives. Then we use the classical implicit Euler method to discretize the time variable with a constant step-size. This method consists of two upwind finite difference schemes defined on an appropriate non-uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. We prove that the method is almost first-order uniformly convergent with respect to the perturbation parameter. To improve the accuracy of the proposed method, we apply Richardson extrapolation technique. Numerical simulations are provided to confirm the theoretical findings.