An efficient uniformly convergent method for multi-scaled two dimensional parabolic singularly perturbed systems of convection-diffusion type

In this work we solve initial-boundary value problems associated to coupled 2D parabolic singularly perturbed systems of convection-diffusion type. The analysis is focused on the cases where the diffusion parameters are small, distinct and also they may have different order of magnitude. In such cases, overlapping regular boundary layers appear at the outflow boundary of the spatial domain. The fully discrete scheme combines the classical upwind scheme defined on an appropriate Shishkin mesh to discretize the spatial variables, and the fractional implicit Euler method joins to a decomposition of the difference operator in directions and components to integrate in time. We prove that the resulting method is uniformly convergent of first order in time and of almost first order in space. Moreover, as only small tridiagonal linear systems must be solved to advance in time, the computational cost of our method is remarkably smaller than the corresponding ones to other implicit methods considered in the previous literature for the same type of problems. The numerical results, obtained for some test problems, corroborate in practice the good behavior and the advantages of the algorithm.