Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki
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


Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2014, Volume 54, Number 8, Pages 1256–1269
DOI: https://doi.org/10.7868/S0044466914080146
(Mi zvmmf10073)
 

This article is cited in 3 scientific papers (total in 3 papers)

Computer difference scheme for a singularly perturbed convection-diffusion equation

G. I. Shishkin

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia
Full-text PDF (274 kB) Citations (3)
References:
Abstract: The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter $\varepsilon$ (that takes arbitrary values from the half-open interval $(0, 1]$) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an $O(\delta_{st})$ rate, where $\delta_{st}=(\varepsilon+N^{-1})^{-1}N^{-1}$ and $N+1$ is the number of grid nodes; the scheme is not $\varepsilon$-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing $\varepsilon$ down to its complete loss for small $\varepsilon$ (namely, for $\varepsilon=O(\delta^{-2}\operatorname{max}_{i,j}|\delta a_i^j|+\delta^{-1}\operatorname{max}_{i,j}|\delta b_i^j|$), where $\delta=\delta_{st}$ and $\delta a_i^j$, $\delta b_i^j$ are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for $\varepsilon\in (0, 1]$ at the same rate as the standard scheme in the absence of perturbations.
Key words: singularly perturbed boundary value problem, ordinary differential convection-diffusion equation, boundary layer, standard difference scheme on uniform meshes, perturbations in data of the grid problem, computer perturbations in computations, maximum norm, stability of a scheme to perturbations, conditioning of a scheme, computer difference scheme.
Received: 21.02.2014
English version:
Computational Mathematics and Mathematical Physics, 2014, Volume 54, Issue 8, Pages 1221–1233
DOI: https://doi.org/10.1134/S0965542514080120
Bibliographic databases:
Document Type: Article
UDC: 519.633
MSC: 35K57, 35K67
Language: Russian
Citation: G. I. Shishkin, “Computer difference scheme for a singularly perturbed convection-diffusion equation”, Zh. Vychisl. Mat. Mat. Fiz., 54:8 (2014), 1256–1269; Comput. Math. Math. Phys., 54:8 (2014), 1221–1233
Citation in format AMSBIB
\Bibitem{Shi14}
\by G.~I.~Shishkin
\paper Computer difference scheme for a singularly perturbed convection-diffusion equation
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2014
\vol 54
\issue 8
\pages 1256--1269
\mathnet{http://mi.mathnet.ru/zvmmf10073}
\crossref{https://doi.org/10.7868/S0044466914080146}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3250872}
\zmath{https://zbmath.org/?q=an:06391165}
\elib{https://elibrary.ru/item.asp?id=21803835}
\transl
\jour Comput. Math. Math. Phys.
\yr 2014
\vol 54
\issue 8
\pages 1221--1233
\crossref{https://doi.org/10.1134/S0965542514080120}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000341085500003}
\elib{https://elibrary.ru/item.asp?id=23990056}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84907314939}
Linking options:
  • https://www.mathnet.ru/eng/zvmmf10073
  • https://www.mathnet.ru/eng/zvmmf/v54/i8/p1256
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
    Statistics & downloads:
    Abstract page:391
    Full-text PDF :101
    References:77
    First page:11
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024