Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation

Abstract

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes that converge uniformly with respect to the perturbation parameter ɛ, ɛ ∈ (0, 1]. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ɛ-uniformly in the maximum norm at the rate O (N−2 ln2N), where N + 1 is the number of nodes in the grid used; for fixed values of the parameter ɛ, the scheme converges at the rate O(N−2). Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges ɛ-uniformly in the maximum norm at the rate O(N−4 ln4N).