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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 -2  st ), 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.

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