An improved uniformly convergent scheme in space for 1D parabolic reaction-diffusion systems

In this paper the numerical approximation of 1D parabolic singularly perturbed systems with two equations of reaction-diffusion type is considered. These problems typically exhibit two overlapping boundary layers at both end points of the spatial domain. A decomposition of the exact solution into its regular and singular part is established, given appropriate bounds for the partial derivatives of the exact solution up to sixth order. These bounds are crucial to prove the uniform convergence of a numerical method that combines the classical backward Euler method and a hybrid finite difference scheme defined on a special nonuniform mesh condensing in the layer regions. The numerical method is uniformly convergent in the discrete maximum norm, and it has first and third order of convergence in time and space, respectively. Numerical results for some test problems are showed, illustrating in practice the order of convergence theoretically proved.