Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Shishkin mesh

In this article, we study the numerical solution of a singularly perturbed 2D delay parabolic convection-diffusion problem. First, we discretize the domain with a uniform mesh in the temporal direction and a special mesh in the spatial directions. The numerical scheme used to discretize the continuous problem, consists of the implicit-Euler scheme for the time derivative and the classical upwind scheme for the spatial derivatives. Stability analysis is carried out, and parameter-uniform error estimates are derived. The proposed scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.