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Trudy Inst. Mat. i Mekh. UrO RAN, 2015, Volume 21, Number 1, Pages 280–293 (Mi timm1164)  

Difference scheme of highest accuracy order for 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 of the Russian Academy of Sciences, Ekaterinburg

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 whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter $\varepsilon$, $\varepsilon \in (0,1]$ (i.e., $\varepsilon$-uniformly) with order of accuracy significantly greater than the achievable accuracy order for the Richardson method on piecewise-uniform grids. Important in this approach is the use of uniform grids for solving grid subproblems for regular and singular components of the grid solution. Using the asymptotic construction technique, a basic 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 uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges $\varepsilon$-uniformly in the maximum norm at the rate ${\mathcal O} (N^{-6} \ln^6 N)$.

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; maximum norm; Richardson extrapolation technique; difference scheme of highest accuracy order.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2016, 292, suppl. 1, 262–275

Bibliographic databases:

UDC: 519.624
Received: 15.12.2014

Citation: 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”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 1, 2015, 280–293; Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 262–275

Citation in format AMSBIB
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\by G.~I.~Shishkin, L.~P.~Shishkina
\paper Difference scheme of highest accuracy order for a singularly perturbed reaction-diffusion equation based on the solution decomposition method
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2015
\vol 21
\issue 1
\pages 280--293
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3407901}
\elib{http://elibrary.ru/item.asp?id=23137997}
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2016
\vol 292
\issue , suppl. 1
\pages 262--275
\crossref{https://doi.org/10.1134/S0081543816020231}
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