A stable standard difference scheme for a singularly perturbed convection-diffusion equation in the presence of computer perturbations

A Dirichlet problem approximated by the standard monotone difference scheme on a uniform grid is considered for a singularly perturbed ordinary differential convection-diffusion equation with perturbation parameter $\varepsilon$ ($\varepsilon\in(0,1]$) multiplying the highest-order derivative. Such a scheme does not converge $\varepsilon$-uniformly and, moreover, in the case of its convergence, it is not $\varepsilon$-uniformly well-conditioned and stable to computer perturbations. In this paper, a technique is developed to study solutions of the standard difference scheme in the presence of computer perturbations. Conditions are derived under which the standard finite difference scheme becomes stable to perturbations, necessary and sufficient conditions are obtained for the convergence of computer solutions as the number of grid nodes tends to infinity, and estimates are given for the number of grid nodes (depending on the parameter $\varepsilon$ and computer perturbations $\vartriangle$ defined by the number of computer word digits) for which the error of the numerical solution is smallest.