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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 10, Pages 1706–1726 (Mi zvmmf232)

Necessary conditions for $\varepsilon$-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

G. I. Shishkin

Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia

Abstract: A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis $x$ in the positive direction. For small values of the parameter $\varepsilon$ (this is the coefficient of the higher order derivatives of the equation, $\varepsilon\in(0,1]$), a moving boundary layer appears in a neighborhood of the left lateral boundary $S_1^L$. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N+N_0)$, where $N$ and $N_0$ define the number of mesh points in $x$ and $t$. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition $N^{-1}+N_0^{-1}\ll\varepsilon$. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of $S_1^L$ with respect to $x$ and $t$, the convergence under the condition $N^{-1}+N_0^{-1}\le\varepsilon^{1/2}$ cannot be achieved. Examination of widths that are similar to Kolmogorov's widths makes it possible to establish necessary and sufficient conditions for the $\varepsilon$-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N+N_0)$.

Key words: boundary value problem for parabolic equations, perturbation parameter $\varepsilon$, parabolic reaction-diffusion equation, finite difference approximation, moving boundary layer, Kolmogorov's width, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:10, 1636–1655

Bibliographic databases:

UDC: 519.633

Citation: G. I. Shishkin, “Necessary conditions for $\varepsilon$-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers”, Zh. Vychisl. Mat. Mat. Fiz., 47:10 (2007), 1706–1726; Comput. Math. Math. Phys., 47:10 (2007), 1636–1655

Citation in format AMSBIB
\Bibitem{Shi07} \by G.~I.~Shishkin \paper Necessary conditions for $\varepsilon$-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2007 \vol 47 \issue 10 \pages 1706--1726 \mathnet{http://mi.mathnet.ru/zvmmf232} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2388622} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 10 \pages 1636--1655 \crossref{https://doi.org/10.1134/S0965542507100065} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-35648974883} 

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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. Shishkin G.I., “Grid approximation of singularly perturbed parabolic equations with moving boundary layers”, Math. Model. Anal., 13:3 (2008), 421–442
2. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Applied Mathematics and Computation, 217:8 (2010), 3641–3716
3. D. V. Lukyanenko, V. T. Volkov, N. N. Nefedov, L. Recke, K. Schneider, “Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of dynamic adapted meshes”, Model. i analiz inform. sistem, 23:3 (2016), 334–341
4. D. V. Luk'yanenko, V. T. Volkov, N. N. Nefedov, “Dynamically adapted mesh construction for the efficient numerical solution of a singular perturbed reaction-diffusion-advection equation”, Model. i analiz inform. sistem, 24:3 (2017), 322–338
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