RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Zh. Vychisl. Mat. Mat. Fiz.: Year: Volume: Issue: Page: Find

 Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 5, Pages 813–830 (Mi zvmmf139)

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.

Full text: PDF file (2319 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 519.633

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
\Bibitem{Shi08} \by G.~I.~Shishkin \paper Conditioning of finite difference schemes for a~singularly perturbed convection-diffusion parabolic equation \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2008 \vol 48 \issue 5 \pages 813--830 \mathnet{http://mi.mathnet.ru/zvmmf139} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2433642} \zmath{https://zbmath.org/?q=an:1164.35007} \transl \jour Comput. Math. Math. Phys. \yr 2008 \vol 48 \issue 5 \pages 769--785 \crossref{https://doi.org/10.1134/S0965542508050072} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000262334100007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-44149103402} 

• http://mi.mathnet.ru/eng/zvmmf139
• http://mi.mathnet.ru/eng/zvmmf/v48/i5/p813

 SHARE:

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. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Appl. Math. Comput., 217:8 (2010), 3641–3716
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
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
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
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
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
7. Shishkin G.I., “Standard Scheme For a Singularly Perturbed Parabolic Convection-Diffusion Equation With Computer Perturbations”, Dokl. Math., 91:3 (2015), 273–276
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
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
•  Number of views: This page: 241 Full text: 79 References: 34 First page: 2