Convergence Analysis of a Layer Resolving Numerical Techniquefor a Class of Coupled System of Singularly Perturbed Parabolic Convection-Diffusion Equations Having an Interface

In this article, we consider a time-dependent weakly coupled system of \( m(\ge 2) \) singularly perturbed convection-diffusion equations in the domain \(G:= \varOmega \times S\) that has an interface \( \varGamma _d:=\left\{ (d,t): t\in S\right\} , \) \( d\in \varOmega :=(0,~1) \) and \(S:=(0,~ T].\) The source terms in the system of equations have discontinuities along \(\varGamma _d\). Also, the second-order term of each equation is multiplied by a small positive parameter. These parameters can be arbitrarily small and different in magnitude due to which overlapping boundary and interior layers appear in the solution. An appropriate Shishkin mesh is used to discretize the domain. At the mesh points that are not on the interface line, the problem is discretized using an upwind central difference scheme. For the mesh points on the interface line, a particular upwind central difference scheme is used. An appropriate decomposition of exact and numerical solutions is made to analyze the parameters-uniform convergence of the considered numerical scheme. The numerical approximations yielded by this scheme are parameters-uniformly convergent of first-order in time and almost first-order in space concerning the perturbation parameters. Numerical results are presented to validate the theoretical results.