A robust computational method for singularly perturbed system of 2D parabolic convection-diffusion problems

This article presents a numerical scheme to solve singularly perturbed system of 2D parabolic convection-diffusion problem exhibiting exponential boundary layers. The numerical scheme consists of a fractional implicit-Euler scheme on uniform mesh for time discretisation and the classical upwind scheme on a piecewise uniform Shishkin mesh for spatial discretisation. For the proposed scheme, the stability analysis is presented and parameter-uniform error estimates are derived. It is shown that the numerical scheme is uniformly convergent with respect to the singular perturbation parameter. The proposed method is applied to a test problem to verify theoretical results numerically.