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



Trudy Inst. Mat. i Mekh. UrO RAN:
Year:
Volume:
Issue:
Page:
Find






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


Trudy Inst. Mat. i Mekh. UrO RAN, 2010, Volume 16, Number 1, Pages 255–271 (Mi timm542)  

This article is cited in 8 scientific papers (total in 8 papers)

Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation

G. I. Shishkin, L. P. Shishkina

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes convergent uniformly with respect to the perturbation parameter $\varepsilon$, $\varepsilon\in(0,1]$. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-2}\ln^{-2}N)$, where $N+1$ is the number of nodes in the grids used; for fixed values of the parameter $\varepsilon$, the scheme converges at the rate $\mathcal O(N^{-2})$. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-4 }\ln^{-4}N)$.

Keywords: singularly perturbed boundary value problem, ordinary differential reaction-diffusion equation, decomposition of a discrete solution, asymptotic construction technique, difference scheme of the solution decomposition method, uniform grids, $\varepsilon$-uniform convergence, Richardson technique, improved scheme of the solution decomposition method.

Full text: PDF file (247 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2011, 272, suppl. 1, S197–S214

Bibliographic databases:

UDC: 519.624
Received: 19.11.2009

Citation: G. I. Shishkin, L. P. Shishkina, “Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 1, 2010, 255–271; Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S197–S214

Citation in format AMSBIB
\Bibitem{ShiShi10}
\by G.~I.~Shishkin, L.~P.~Shishkina
\paper Improved difference scheme of the solution decomposition method for a~singularly perturbed reaction-diffusion equation
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2010
\vol 16
\issue 1
\pages 255--271
\mathnet{http://mi.mathnet.ru/timm542}
\elib{https://elibrary.ru/item.asp?id=13073004}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2011
\vol 272
\issue , suppl. 1
\pages S197--S214
\crossref{https://doi.org/10.1134/S0081543811020155}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000289527400015}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79954571661}


Linking options:
  • http://mi.mathnet.ru/eng/timm542
  • http://mi.mathnet.ru/eng/timm/v16/i1/p255

    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, L. P. Shishkina, “Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method”, Comput. Math. Math. Phys., 51:6 (2011), 1020–1049  mathnet  crossref  mathscinet  isi
    2. 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
    3. 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
    4. 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
    5. M. N. Nazarov, “Novyi podkhod k modelirovaniyu sistem reaktsii-diffuzii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 4, 84–94  mathnet
    6. G. I. Shishkin, L. P. Shishkina, “Difference scheme of highest accuracy order for a singularly perturbed reaction-diffusion equation based on the solution decomposition method”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 262–275  mathnet  crossref  mathscinet  isi  elib
    7. S. V. Tikhovskaya, “Issledovanie dvukhsetochnogo metoda povyshennoi tochnosti dlya ellipticheskogo uravneniya reaktsii–diffuzii s pogranichnymi sloyami”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2015, 60–74  mathnet  elib
    8. G. I. Shishkin, L. P. Shishkina, “A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation”, Comput. Math. Math. Phys., 55:3 (2015), 386–409  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
  • Trudy Instituta Matematiki i Mekhaniki UrO RAN
    Number of views:
    This page:442
    Full text:64
    References:38
    First page:8

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020