Computer difference scheme for a singularly perturbed convection-diffusion equation

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 8, pp. 1256-1269

Résumé

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter $\varepsilon$ (that takes arbitrary values from the half-open interval $(0, 1]$) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an $O(\delta_{st})$ rate, where $\delta_{st}=(\varepsilon+N^{-1})^{-1}N^{-1}$ and $N+1$ is the number of grid nodes; the scheme is not $\varepsilon$-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing $\varepsilon$ down to its complete loss for small $\varepsilon$ (namely, for $\varepsilon=O(\delta^{-2}\operatorname{max}_{i,j}|\delta a_i^j|+\delta^{-1}\operatorname{max}_{i,j}|\delta b_i^j|$), where $\delta=\delta_{st}$ and $\delta a_i^j$, $\delta b_i^j$ are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for $\varepsilon\in (0, 1]$ at the same rate as the standard scheme in the absence of perturbations.

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