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Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 11, Pages 2045–2064 (Mi zvmmf385)  

This article is cited in 3 scientific papers (total in 3 papers)

Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

G. I. Shishkin

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

Abstract: In an unbounded (with respect to $x$ and $t$) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter $\varepsilon^2$ multiplying the higher order derivative is considered. The parameter $\varepsilon$ takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge $\varepsilon$-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of $\varepsilon$ in the prescribed bounded subdomains that can expand as the number of grid points increases. As $\varepsilon\to0$, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge $\varepsilon$-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of $\varepsilon$ and the subdomains may expand as the number of nodes in the underlying grids increases.

Key words: singularly perturbed parabolic reaction-diffusion equation, grid approximation, formal and constructive difference schemes, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:11, 1953–1971

Bibliographic databases:

UDC: 519.633
Received: 12.05.2006

Citation: G. I. Shishkin, “Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables”, Zh. Vychisl. Mat. Mat. Fiz., 46:11 (2006), 2045–2064; Comput. Math. Math. Phys., 46:11 (2006), 1953–1971

Citation in format AMSBIB
\Bibitem{Shi06}
\by G.~I.~Shishkin
\paper Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2006
\vol 46
\issue 11
\pages 2045--2064
\mathnet{http://mi.mathnet.ru/zvmmf385}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2304076}
\transl
\jour Comput. Math. Math. Phys.
\yr 2006
\vol 46
\issue 11
\pages 1953--1971
\crossref{https://doi.org/10.1134/S096554250611011X}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33845315907}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. G. I. Shishkin, “Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity”, Comput. Math. Math. Phys., 49:10 (2009), 1748–1764  mathnet  crossref  isi
    2. Shishkin G.I., “Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity”, Russian Journal of Numerical Analysis and Mathematical Modelling, 24:6 (2009), 591–617  crossref  mathscinet  zmath  isi  scopus
    3. 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  crossref  mathscinet  zmath  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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