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 Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 568–576 (Mi mais523)

Numerical study of an initial-boundary value Neumann problem for a singularly perturbed parabolic equation

L. P. Shishkina

N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Yekaterinburg 620990, Russia

Abstract: For a singularly perturbed one-dimensional parabolic equation with a perturbation parameter $\varepsilon$ multiplying the highest-order derivative in the equation, $\varepsilon \in (0,1]$, an initial-boundary value Neumann problem is considered on a segment. In this problem, when the parameter $\varepsilon$ tends to zero, boundary layers appear in neighborhoods of the lateral boundary. In the paper, convergence of the problem solution and its regular and singular components are studied. It is shown that standard finite difference schemes on uniform grids used to numerically solve this problem do not converge $\varepsilon$-uniformly. The error in the grid solution grows unboundedly when the parameter $\varepsilon \rightarrow 0$. The use of a special difference scheme on Shishkin grid which is a piecewise-uniform mesh with respect to $x$ condensing in neighborhoods of boundary layers and a uniform mesh in $t$, constructed by using monotone grid approximations of the differential problems, allows us to find a numerical solution of this problem convergent in the maximum norm $\varepsilon$-uniformly. The results of the numerical experiments confirm the theoretical results.

Keywords: initial-boundary value Neumann problem, singularly perturbed parabolic equation, piecewise-uniform grid, maximum norm, $\varepsilon$-uniform convergence.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00727_à This research was partially supported by the Russian Foundation for Basic Research under grant ¹ 16-01-00727.

DOI: https://doi.org/10.18255/1818-1015-2016-5-568-576

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UDC: 519.63

Citation: L. P. Shishkina, “Numerical study of an initial-boundary value Neumann problem for a singularly perturbed parabolic equation”, Model. Anal. Inform. Sist., 23:5 (2016), 568–576

Citation in format AMSBIB
\Bibitem{Shi16} \by L.~P.~Shishkina \paper Numerical study of an initial-boundary value Neumann problem for a singularly perturbed parabolic equation \jour Model. Anal. Inform. Sist. \yr 2016 \vol 23 \issue 5 \pages 568--576 \mathnet{http://mi.mathnet.ru/mais523} \crossref{https://doi.org/10.18255/1818-1015-2016-5-568-576} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3569853} \elib{http://elibrary.ru/item.asp?id=27202306}