
This article is cited in 3 scientific papers (total in 3 papers)
Grid approximation of singularly perturbed parabolic reactiondiffusion 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 initialboundary value problem for singularly perturbed parabolic reactiondiffusion equations with the perturbation parameter $\varepsilon^2$ multiplying the higher order derivative is considered. The parameter $\varepsilon$ takes arbitrary values in the halfopen 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 reactiondiffusion equation, grid approximation, formal and constructive difference schemes, $\varepsilon$uniform convergence.
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Computational Mathematics and Mathematical Physics, 2006, 46:11, 1953–1971
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UDC:
519.633 Received: 12.05.2006
Citation:
G. I. Shishkin, “Grid approximation of singularly perturbed parabolic reactiondiffusion 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
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\paper Grid approximation of singularly perturbed parabolic reactiondiffusion equations on large domains with respect to the space and time variables
\jour Zh. Vychisl. Mat. Mat. Fiz.
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\issue 11
\pages 20452064
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\jour Comput. Math. Math. Phys.
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\pages 19531971
\crossref{https://doi.org/10.1134/S096554250611011X}
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This publication is cited in the following articles:

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

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

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

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