Numerical solution of singularly perturbed linear parabolic system with discontinuous source term

In this work, we consider a parabolic system with an arbitrary number of linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms with a discontinuous source term. The diffusion term in each equation is multiplied by a small positive parameter, but these parameters may have different order of magnitude. The components of the solution have boundary and interior layers that overlap and interact. To obtain the approximate solution of the problem we construct a numerical method by combining the backward-Euler method on an uniform mesh in time direction, together with a central difference scheme on a variant of piecewise-uniform Shishkin mesh in space. We prove that the numerical method is uniformly convergent of first order in time and almost second order in spatial variable. Numerical experiments are presented to validate the theoretical results.