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






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 2 scientific papers (total in 2 papers)

Difference scheme for a singularly perturbed parabolic convectiondiffusion 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 initialboundary value problem is considered for a singularly perturbed parabolic convectiondiffusion 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 initialboundary value problem, parabolic convectiondiffusion 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_


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

Full text: PDF file (171 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 519.633
MSC: Primary 65M06; Secondary 65M12, 65M50
Received: 07.04.2015

Citation: G. I. Shishkin, “Difference scheme for a singularly perturbed parabolic convectiondiffusion 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
\Bibitem{Shi15}
\by G.~I.~Shishkin
\paper Difference scheme for a singularly perturbed parabolic convectiondiffusion equation in the presence of perturbations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2015
\vol 55
\issue 11
\pages 1876--1892
\mathnet{http://mi.mathnet.ru/zvmmf10298}
\crossref{https://doi.org/10.7868/S0044466915110174}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3423049}
\elib{http://elibrary.ru/item.asp?id=24730745}
\transl
\jour Comput. Math. Math. Phys.
\yr 2015
\vol 55
\issue 11
\pages 1842--1856
\crossref{https://doi.org/10.1134/S0965542515110159}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000365036400007}
\elib{http://elibrary.ru/item.asp?id=24971308}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84947263624}


Linking options:
  • http://mi.mathnet.ru/eng/zvmmf10298
  • http://mi.mathnet.ru/eng/zvmmf/v55/i11/p1876

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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  mathnet  crossref  mathscinet  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
  •      Computational Mathematics and Mathematical Physics
    Number of views:
    This page:141
    Full text:14
    References:44
    First page:15

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019