Second order parameter-uniform convergence for a finite difference method for a partially singularly perturbed linear parabolic system

Abstract

A linear system of $n$ second order differential equations of parabolic reaction-diffusion type with initial and boundary conditions is considered. The first $k$ equations are singularly perturbed. Each of the leading terms of the first $m$ equations, $m\leq k$, is multiplied by a small positive parameter and these parameters are assumed to be distinct. The leading terms of the next $k-m$ equations are multiplied by the same perturbation parameter $\varepsilon_m$. Since the components of the solution exhibit overlapping layers, Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters.