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Zh. Vychisl. Mat. Mat. Fiz., 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

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.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:5, 769–785

Bibliographic databases:

UDC: 519.633
Received: 01.10.2007

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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\jour Comput. Math. Math. Phys.
\yr 2008
\vol 48
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\pages 769--785
\crossref{https://doi.org/10.1134/S0965542508050072}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Appl. Math. Comput., 217:8 (2010), 3641–3716  crossref  mathscinet  zmath  isi  elib  scopus
    2. Golbabai A., Arabshahi M.M., “A numerical method for diffusion-convection equation using high-order difference schemes”, Comput. Phys. Comm., 181:7 (2010), 1224–1230  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. G. I. Shishkin, “Obuslovlennost raznostnoi skhemy metoda dekompozitsii resheniya dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii”, Tr. IMM UrO RAN, 18, no. 2, 2012, 291–304  mathnet  elib
    4. G. I. Shishkin, “Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation”, Comput. Math. Math. Phys., 53:4 (2013), 431–454  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. Shishkin G.I., “Data Perturbation Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation”, Russ. J. Numer. Anal. Math. Model, 28:4 (2013), 381–417  crossref  mathscinet  isi  elib
    6. G. I. Shishkin, “Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 55:11 (2015), 1842–1856  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. Shishkin G.I., “Standard Scheme For a Singularly Perturbed Parabolic Convection-Diffusion Equation With Computer Perturbations”, Dokl. Math., 91:3 (2015), 273–276  crossref  mathscinet  zmath  isi  elib  scopus
    8. 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  mathnet  crossref  mathscinet  elib
    9. 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  mathnet  crossref  crossref  mathscinet  isi  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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