Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation

For a model Dirichlet problem to a singularly perturbed ordinary differential convection-diffusion equation, we discuss a “standard” approach to the construction of difference schemes that use standard grid approximations on uniform grids, the step-size of which is chosen sufficiently small for small values of a perturbation parameter $$\varepsilon $$ , $$\varepsilon \in (0,1]$$ . It is shown that such a scheme, under its convergence in the maximum norm theoretically proved, is not $$\varepsilon $$ -uniformly stable to perturbations in the data of the discrete problem. When perturbations take place and the parameter $$\varepsilon $$ decreases, the actual accuracy of the computed solutions may deteriorate up to a full accuracy loss for sufficiently small values of $$\varepsilon $$ , namely, under the condition $$t = \mathcal{O}(\ln {\varepsilon }^{-1})$$ , where t is the number of computer word digits.