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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2015, Volume 55, Number 3, Pages 393–416
DOI: https://doi.org/10.7868/S0044466915030175
(Mi zvmmf10167)
 

This article is cited in 1 scientific paper (total in 1 paper)

A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

G. I. Shishkin, L. P. Shishkina

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia
Full-text PDF (340 kB) Citations (1)
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Abstract: An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge $\varepsilon$-uniformly in the maximum norm (where $\varepsilon$ is the perturbation parameter multiplying the highest order derivative, $\varepsilon\in(0, 1]$). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges $\varepsilon$-uniformly in the maximum norm at a rate of $\mathcal{O}(N^{-4}\ln^4N+N_0^{-2})$, where $N+1$ and $N_0+1$ are the numbers of nodes in uniform meshes in $a$ and $t$, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge $\varepsilon$-uniformly with an order close to the sixth in $x$ and equal to the third in $t$.
Key words: singularly perturbed initial-boundary value problem, parabolic reaction-diffusion equation, perturbation parameter $\varepsilon$, solution decomposition method, numerical-analytical scheme, improved Richardson difference scheme, $\varepsilon$-uniform convergence, maximum norm.
Received: 31.07.2014
English version:
Computational Mathematics and Mathematical Physics, 2015, Volume 55, Issue 3, Pages 386–409
DOI: https://doi.org/10.1134/S0965542515030161
Bibliographic databases:
Document Type: Article
UDC: 519.633
MSC: Primary 65M06; Secondary 35B25, 35K20, 35K57, 65M12
Language: Russian
Citation: G. I. Shishkin, L. P. Shishkina, “A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation”, Zh. Vychisl. Mat. Mat. Fiz., 55:3 (2015), 393–416; Comput. Math. Math. Phys., 55:3 (2015), 386–409
Citation in format AMSBIB
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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