Data perturbation stability of difference schemes on uniform grids for a singularly perturbed convection–diffusion equation

Abstract

An approach to the study of the conditioning of difference schemes and their stability to data perturbations is developed for the Dirichlet problem for a singularly perturbed convection-diffusion ordinary differential equation with the perturbation parameter e, e 2 (0;1]. We consider a standard difference scheme, which is a monotone scheme on a uniform grid, and a special scheme of the grid solution decomposition method. Constructing the special scheme, we use a decomposition of the grid solution into a regular and a singular components that are the solutions to grid subproblems considered on uniform grids. The following facts are proved for the standard scheme: (a) the scheme converges only for N-1 = o(ε) at the rate (N-1 (ε +N-1)-1), , where N +1 is the number of grid nodes, (b) the scheme is not e-uniformly well conditioned and stable to perturbations, (c) the condition number of the scheme has the order (ε-1 δ-2, where δ is the accuracy of the grid solution, δ = δst ≈ N-1 (ε +N-1)-1. In the case of convergence of the standard scheme proved theoretically, the actual accuracy of the computed solution decreases with the decrease in the parameter e under the presence of perturbations and may completely vanish for a sufficiently small e (namely, under the condition t = (lnε-1 +lnδ-2st), where t is the number of digits in the machine word). At the same time, the special scheme of the solution decomposition method converges e-uniformly in the uniform norm at the rate (N-1 lnN); in the variables e and d, the special scheme is e-uniformly well conditioned and stable to grid problem data perturbations; the condition number of the scheme of the decomposition method is of the order (δ-2 lnδ-1), δ = δdec ≈ N-1 lnN.