High order robust approximations for singularly perturbed semilinear systems

In this paper, a system of M (⩾2) singularly perturbed semilinear reaction–diffusion equations is considered. To obtain a high order approximation to the solution of this system, we propose a hybrid numerical method that employs a generalized Shishkin mesh with the Numerov discretization in the boundary layer regions and either a non-equidistant generalization of the Numerov discretization or classical central differences in the outer region. It is proved that the method is almost fourth order convergent in the maximum norm uniformly with respect to the perturbation parameter. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method.