Robust computational method for singularly perturbed system of parabolic convection-diffusion problems with interior layers

In this article, we present the convergence analysis of an upwind finite difference scheme for singularly perturbed system of parabolic convection-diffusion initial-boundary-value problems with discontinuous convection coefficient and source term. The proposed numerical scheme is constructed by using the implicit-Euler scheme for the time derivative on the uniform mesh, and the upwind finite difference scheme for the spatial derivatives on a layer-resolving piecewise-uniform Shishkin mesh. It is shown that the numerical solution obtained by the proposed scheme converges uniformly with respect to the perturbation parameter. The proposed numerical 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.