The Error Analysis of Finite Difference Approximation for a System of Singularly Perturbed Semilinear Reaction-Diffusion Equations with Discontinuous Source Term

Abstract

We consider a coupled system of two singularly perturbed semilinear reaction-diffusion equations with a discontinuous source term. The leading term in each equation is multiplied by a small positive parameter, but these parameters have different order of magnitude. The solution of these system of equations have overlapping and interacting boundary and interior layers. Based on the discrete Green’s function theory, the properties of the discretized operator are established. The error estimates are derived in the maximum norm for a central difference scheme on layer-adapted meshes, and the method is proved to be almost second order uniformly convergent independently of both the perturbation parameters. Numerical results validate the theoretical results.