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 Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 8, Pages 1416–1436 (Mi zvmmf4735)

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

G. I. Shishkin

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

Abstract: The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter $\varepsilon^2$, where $\varepsilon\in(0,1]$. When $\varepsilon$ is small, a boundary and an interior layer (with the characteristic width $\varepsilon$) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed $\varepsilon$, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges $\varepsilon$-uniformly at a rate of $O(N^{-2}\ln^2+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of the mesh points in $x$ and $t$, respectively. Based on the Richardson technique, a scheme that converges $\varepsilon$-uniformly at a rate of $ON^{-3}+N_0^{-2})$ is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in $\varepsilon$-uniformly in $x$ with an order greater than three.

Key words: singularly perturbed boundary value problem, parabolic reaction-diffusion equation, piecewise continuous initial condition, grid approximation, method of additive splitting of singularities, special grids, $\varepsilon$-uniform convergence, Richardson technique.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:8, 1348–1368

Bibliographic databases:

Document Type: Article
UDC: 519.633

Citation: G. I. Shishkin, “The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition”, Zh. Vychisl. Mat. Mat. Fiz., 49:8 (2009), 1416–1436; Comput. Math. Math. Phys., 49:8 (2009), 1348–1368

Citation in format AMSBIB
\Bibitem{Shi09} \by G.~I.~Shishkin \paper The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a~discontinuous initial condition \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2009 \vol 49 \issue 8 \pages 1416--1436 \mathnet{http://mi.mathnet.ru/zvmmf4735} \zmath{https://zbmath.org/?q=an:05649684} \transl \jour Comput. Math. Math. Phys. \yr 2009 \vol 49 \issue 8 \pages 1348--1368 \crossref{https://doi.org/10.1134/S0965542509080065} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000269218300006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70350584720} 

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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. G. I. Shishkin, L. P. Shishkina, “Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation”, Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S197–S214
2. G. I. Shishkin, L. P. Shishkina, “A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation”, Comput. Math. Math. Phys., 50:12 (2010), 2003–2022
3. G. I. Shishkin, L. P. Shishkina, “Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method”, Comput. Math. Math. Phys., 51:6 (2011), 1020–1049
4. Shishkin G.I., “Data Perturbation Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation”, Russ. J. Numer. Anal. Math. Model, 28:4 (2013), 381–417
5. G. I. Shishkin, L. P. Shishkina, “Difference scheme of highest accuracy order for a singularly perturbed reaction-diffusion equation based on the solution decomposition method”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 262–275
6. G. I. Shishkin, L. P. Shishkina, “A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation”, Comput. Math. Math. Phys., 55:3 (2015), 386–409
7. 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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