A Robust computational method for singularly perturbed coupled system of reaction–diffusion boundary-value problems

Abstract

In this paper, we devise a robust computational method for singularly perturbed system of reaction–diffusion boundary-value problems (BVPs). We use cubic spline on nonuniform mesh to obtain the difference scheme, and apply it to the system of BVPs on a piecewise uniform Shishkin mesh on the whole domain. We observed that the cubic spline scheme is not stable in the regular (outer) region, where the mesh is coarse. To overcome this difficulty, we use the classical central difference scheme only for that portion, and the cubic spline scheme elsewhere. This newly proposed scheme is uniformly stable throughout the domain and provides second-order uniform convergence results. We derive the uniform error estimate for this scheme, and apply it to a test problem to verify the efficiency and accuracy of the method.