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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2008, Volume 48, Number 5, Pages 813–830 (Mi zvmmf139)  

This article is cited in 9 scientific papers (total in 9 papers)

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

G. I. Shishkin

Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia
References:
Abstract: In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an $\varepsilon$-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an $\varepsilon$-uniform bound $O(\delta^{-2}\ln\delta_1^{-1}+\delta_0^{-1})$ is obtained, where $\delta_1$ and $\delta_0$ are the error components due to the approximation of the derivatives with respect to $x$ and $t$, respectively. Thus, this scheme is $\varepsilon$-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate $O(\varepsilon^{-1}\delta_1^{-2}+\delta_0^{-1})$; this scheme is not $\varepsilon$-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as $\varepsilon\to0$, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of $O(\varepsilon^{-1}\delta_1^{-2}+\delta_0^{-1})$. Neither the matrix of the $\varepsilon$-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is $\varepsilon$-uniformly well-conditioned.
Key words: boundary value problem, perturbation parameter $\varepsilon$, parabolic convection-diffusion equation, finite difference approximation, $\varepsilon$-uniform convergence, $\varepsilon$-uniform good conditioning of a scheme, conditioning of a matrix.
Received: 01.10.2007
English version:
Computational Mathematics and Mathematical Physics, 2008, Volume 48, Issue 5, Pages 769–785
DOI: https://doi.org/10.1134/S0965542508050072
Bibliographic databases:
Document Type: Article
UDC: 519.633
Language: Russian
Citation: G. I. Shishkin, “Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation”, Zh. Vychisl. Mat. Mat. Fiz., 48:5 (2008), 813–830; Comput. Math. Math. Phys., 48:5 (2008), 769–785
Citation in format AMSBIB
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\paper Conditioning of finite difference schemes for a~singularly perturbed convection-diffusion parabolic equation
\jour Zh. Vychisl. Mat. Mat. Fiz.
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\vol 48
\issue 5
\pages 813--830
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\jour Comput. Math. Math. Phys.
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\pages 769--785
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  • This publication is cited in the following 9 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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    References:70
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