Abstract
We consider a Dirichlet problem 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. Moreover, under its convergence, it is not \(\varepsilon\)-uniformly well conditioned and stable to perturbations in the data of the discrete problem and/or computer perturbations. For a model boundary value problem in the case of computer perturbations, we discuss results of numerical experiments and their conformity to theoretical results.
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Notes
- 1.
The notation \(L_{(j.k)}\ (M_{(j.k)},\ G_{h(j.k)})\) means that these operators (constants, grids) are introduced in formula (j. k).
- 2.
By M (or m), we denote sufficiently large (small) positive constants independent of the parameter \(\varepsilon\) and of the discretization parameters.
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Acknowledgements
This research was supported by the Russian Foundation for Basic Research under grant No.13-01-00618.
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Shishkin, G., Shishkina, L., Petrenko, A. (2014). Standard Difference Scheme for a Singularly Perturbed Convection-Diffusion Equation in the Presence of Perturbations. In: Ansari, A. (eds) Advances in Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-06923-4_5
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DOI: https://doi.org/10.1007/978-3-319-06923-4_5
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