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 Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 11, Pages 1876–1892 (Mi zvmmf10298)

Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations

G. I. Shishkin

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

Abstract: An initial–boundary value problem is considered for a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter $\varepsilon$ $(\varepsilon\in(0, 1])$ multiplying the highest order derivative. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform mesh is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. The scheme does not converge $\varepsilon$-uniformly in the maximum norm as the number of its grid nodes is increased. When the solution of the difference scheme converges, which occurs if $N^{-1}\ll\varepsilon$ and $N_0^{-1}\ll1$, where $N$ and $N_0$ are the numbers of grid intervals in $x$ and $t$, respectively, the scheme is not $\varepsilon$-uniformly well conditioned or stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions on the “parameters” of the difference scheme and of the computer (namely, on $\varepsilon$, $N$, $N_0$, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions. Additionally, the conditions are obtained under which the perturbed numerical solution has the same order of convergence as the solution of the unperturbed standard difference scheme.

Key words: singularly perturbed initial–boundary value problem, parabolic convection–diffusion equation, boundary layer, standard difference scheme on uniform meshes, perturbations of data of the grid problem, computer perturbations in computations, maximum norm, stability of schemes to perturbations, conditioning of schemes, computer difference scheme.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00618_à

DOI: https://doi.org/10.7868/S0044466915110174

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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:11, 1842–1856

Bibliographic databases:

UDC: 519.633
MSC: Primary 65M06; Secondary 65M12, 65M50

Citation: G. I. Shishkin, “Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations”, Zh. Vychisl. Mat. Mat. Fiz., 55:11 (2015), 1876–1892; Comput. Math. Math. Phys., 55:11 (2015), 1842–1856

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. I. Shishkin, “Kompyuternaya raznostnaya skhema dlya singulyarno vozmuschennogo parabolicheskogo uravneniya reaktsii-diffuzii pri nalichii kompyuternykh vozmuschenii”, Model. i analiz inform. sistem, 23:5 (2016), 577–586
2. G. I. Shishkin, “Computer difference scheme for a singularly perturbed elliptic convection-diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 57:5 (2017), 815–832
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