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 Zh. Vychisl. Mat. Mat. Fiz., 2011, Volume 51, Number 6, Pages 1091–1120 (Mi zvmmf9466)

Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

G. I. Shishkin, L. P. Shishkina

Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990 Russia

Abstract: In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction–diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter $\varepsilon\in(0,1]$; the normalized derivatives are $\varepsilon$-uniformly bounded. The key idea of this approach to the construction of $\varepsilon$-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge $\varepsilon$-uniformly at the rate of $O(N^{-2}\ln^2N)$, where $N+1$ is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge $\varepsilon$-uniformly in the maximum norm at the same rate of $O(N^{-4}\ln^4N)$.

Key words: singularly perturbed boundary value problem, ordinary differential reaction–diffusion equation, decomposition of grid solution, asymptotic construction technique, finite difference scheme of the solution decomposition method, uniform grids, $\varepsilon$-uniform convergence, maximum norm, the Richardson technique, improved scheme of the solution decomposition method, improved approximation of derivatives.

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English version:
Computational Mathematics and Mathematical Physics, 2011, 51:6, 1020–1049

Bibliographic databases:

UDC: 519.633

Citation: 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”, Zh. Vychisl. Mat. Mat. Fiz., 51:6 (2011), 1091–1120; Comput. Math. Math. Phys., 51:6 (2011), 1020–1049

Citation in format AMSBIB
\Bibitem{ShiShi11} \by G.~I.~Shishkin, L.~P.~Shishkina \paper Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2011 \vol 51 \issue 6 \pages 1091--1120 \mathnet{http://mi.mathnet.ru/zvmmf9466} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2859177} \transl \jour Comput. Math. Math. Phys. \yr 2011 \vol 51 \issue 6 \pages 1020--1049 \crossref{https://doi.org/10.1134/S0965542511060169} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000291601900011} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79958777451} 

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This publication is cited in the following articles:
1. 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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