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 Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 3, Pages 393–416 (Mi zvmmf10167)

A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

G. I. Shishkin, L. P. Shishkina

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

Abstract: An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge $\varepsilon$-uniformly in the maximum norm (where $\varepsilon$ is the perturbation parameter multiplying the highest order derivative, $\varepsilon\in(0, 1]$). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges $\varepsilon$-uniformly in the maximum norm at a rate of $\mathcal{O}(N^{-4}\ln^4N+N_0^{-2})$, where $N+1$ and $N_0+1$ are the numbers of nodes in uniform meshes in $a$ and $t$, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge $\varepsilon$-uniformly with an order close to the sixth in $x$ and equal to the third in $t$.

Key words: singularly perturbed initial-boundary value problem, parabolic reaction-diffusion equation, perturbation parameter $\varepsilon$, solution decomposition method, numerical-analytical scheme, improved Richardson difference scheme, $\varepsilon$-uniform convergence, maximum norm.

DOI: https://doi.org/10.7868/S0044466915030175

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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:3, 386–409

Bibliographic databases:

UDC: 519.633
MSC: Primary 65M06; Secondary 35B25, 35K20, 35K57, 65M12

Citation: G. I. Shishkin, L. P. Shishkina, “A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation”, Zh. Vychisl. Mat. Mat. Fiz., 55:3 (2015), 393–416; Comput. Math. Math. Phys., 55:3 (2015), 386–409

Citation in format AMSBIB
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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. Shishkina L., “Difference Schemes of High Accuracy Order on Uniform Grids For a Singularly Perturbed Parabolic Reaction-Diffusion Equation”, Boundary and Interior Layers, Computational and Asymptotic Methods - Bail 2014, Lecture Notes in Computational Science and Engineering, 108, ed. Knobloch P., Springer-Verlag Berlin, 2015, 281–291
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