Analysis of a higher order uniformly convergent method for singularly perturbed parabolic delay problems

Abstract

We develop and analyze a higher-order uniformly convergent method for time dependent parabolic singularly perturbed convection-diffusion (C-D) problems with space dependent delay. Due to the presence of delay, there occurs an interior layer along with a boundary layer in the solution of the considered problem. A Bakhvalov-Shishkin mesh in space direction and a uniform mesh in time direction is constructed to discretize the domain. Numerical approximation is composed of the classical upwind difference scheme for space variable and the implicit-Euler scheme for time variable. The proposed scheme is proved to have ε-uniform convergence of O(K−1+Δt), where K and Δt denote the number of mesh- intervals in space direction and the step size in time direction, respectively. We further apply the Richardson extrapolation and establish that the resulting scheme has ε-uniform convergence of O(K−2+(Δt)2). Numerical results on two test examples are provided to demonstrate the effectiveness of the scheme.