Use of Standard Difference Scheme on Uniform Grids for Solving Singularly Perturbed Problems Under Computer Perturbations

Abstract

A Dirichlet problem is considered for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter $\varepsilon$ ($\varepsilon \in (0,1]$) multiplying the highest-order derivative in the equation. This problem is approximated by the standard monotone finite difference scheme on a uniform grid. Such a scheme does not converge $\varepsilon$-uniformly in the maximum norm when the number of grid nodes grows. Moreover, under its convergence, the scheme is not $\varepsilon$-uniformly well conditioned and stable to data perturbations of the discrete problem and/or computer perturbations. For small values of $\varepsilon$, perturbations of the grid solution can significantly exceed (and even in order of magnitude) the error in the unperturbed solution. For a computer difference scheme (the standard scheme in the presence of computer perturbations), technique is developed for theoretical and experimental study of convergence of perturbed grid solutions. For computer perturbations, conditions are obtained (depending on the parameter $\varepsilon$ and the number of grid intervals N), for which the solution of the computer scheme converges in the maximum norm with the same order as the solution of the standard scheme in the absence of perturbations.