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

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.

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

Bibliographic databases:

UDC: 519.624

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} 

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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
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
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
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
5. M. N. Nazarov, “Novyi podkhod k modelirovaniyu sistem reaktsii-diffuzii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 4, 84–94
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
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
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
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