Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay

Abstract

In this article, a parameter-uniform implicit scheme is constructed for a class of parabolic singularly perturbed reaction-diffusion initial-boundary value problems with large delay in the spatial direction. In general, the solution of these problems exhibits twin boundary layers and an interior layer (due to the presence of the delay in the reaction term). Crank-Nicolson difference formula (on a uniform mesh) is used in time to semi-discretize the given PDE, and then the standard finite difference scheme (on a piecewise-uniform mesh) is used for the system of ordinary differential equations obtained in the semi-discretization. The convergence analysis shows that the method is ε-uniformly convergent of order two in the temporal direction and almost first-order in the spatial direction. Two test examples are encountered to show the efficiency of the method, validate the computational results, and to confirm the predicted theory.