Difference scheme of highest accuracy order for a singularly perturbed reaction–diffusion equation based on the solution decomposition method

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 whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter e, e ∈ (0, 1] (i.e., e-uniformly) with order of accuracy significantly greater than the ultimate achievable accuracy order for the Richardson method on piecewise uniform grids. The main point of this approach is that uniform grids are used to solve grid subproblems for the regular and singular components of the discrete solution. Using the asymptotic construction technique, a basic difference scheme of the solution decomposition method is constructed that converges e-uniformly in the maximum norm at the rate O(N −2 ln2 N), where N + 1 is the number of nodes in the uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges e-uniformly in the maximum norm at the rate O(N −6 ln6 N).