Analysis of an almost fourth-order parameter-uniformly convergent numerical method for singularly perturbed semilinear reaction-diffusion system with non-smooth source term

Abstract

A semilinear second-order system of m(≥2) reaction-diffusion equations is analyzed in a singularly-perturbed regime. In these equations, the right-hand side (source term) has a discontinuity at a point inside the domain, and the coefficient of the second-order derivative is a positive parameter. These parameters can be arbitrarily small and different in magnitude; this causes to form the boundary layers in the solution which may interact and overlap inside the domain. Due to discontinuity in the source term, the solution may also exhibit layers in the interior of the domain. The decomposition of the solution has been made to obtain sharper bounds on its derivatives. A higher-order finite difference scheme is constructed using an appropriate generalized Shishkin mesh and established that the computed solution is almost fourth-order parameter-uniform. Numerical experiments are performed to validate the theoretical findings.