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 Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 12, Pages 2113–2133 (Mi zvmmf4977)

A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

G. I. Shishkin, L. P. Shishkina

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

Abstract: For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon$, where $\varepsilon\in(0,1]$, the grid approximation of the Dirichlet problem on a rectangular domain in the $(x,t)$-plane is examined. For small $\varepsilon$, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of $\varepsilon$-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges $\varepsilon$-uniformly in the maximum norm at the rate of $O(N^{-2}\ln^2N+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme converges $\varepsilon$-uniformly at the rate of $O(N^{-4}\ln^4N+N_0^{-2})$. For fixed values of the parameter, the convergence rate is $O(N^{-4}+N_0^{-2})$.

Key words: parabolic reaction-diffusion equation, boundary layer, decomposition of grid solution, uniform grids, asymptotic construction technique, Richardson extrapolation technique, higher order finite difference scheme, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:12, 2003–2022

Bibliographic databases:

UDC: 519.633
Revised: 15.06.2010

Citation: G. I. Shishkin, L. P. Shishkina, “A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation”, Zh. Vychisl. Mat. Mat. Fiz., 50:12 (2010), 2113–2133; Comput. Math. Math. Phys., 50:12 (2010), 2003–2022

Citation in format AMSBIB
\Bibitem{ShiShi10} \by G.~I.~Shishkin, L.~P.~Shishkina \paper A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2010 \vol 50 \issue 12 \pages 2113--2133 \mathnet{http://mi.mathnet.ru/zvmmf4977} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2010CMMPh..50.2003S} \transl \jour Comput. Math. Math. Phys. \yr 2010 \vol 50 \issue 12 \pages 2003--2022 \crossref{https://doi.org/10.1134/S0965542510120043} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78650625886} 

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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 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
2. G. I. Shishkin, “Obuslovlennost raznostnoi skhemy metoda dekompozitsii resheniya dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii”, Tr. IMM UrO RAN, 18, no. 2, 2012, 291–304
3. G. I. Shishkin, “Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation”, Comput. Math. Math. Phys., 53:4 (2013), 431–454
4. Clavero C., Gracia J.L., “A Higher Order Uniformly Convergent Method with Richardson Extrapolation in Time for Singularly Perturbed Reaction-Diffusion Parabolic Problems”, J. Comput. Appl. Math., 252 (2013), 75–85
5. 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
6. Shishkin G., Shishkina L., Luis Gracia J., Clavero C., “On a Numerical Technique To Study Difference Schemes For Singularly Perturbed Parabolic Reaction-Diffusion Equations”, Int. J. Numer. Anal. Model., 11:2 (2014), 412–426
7. 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
8. 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
9. Salama A.A. Al-Amery D.G., “A Higher Order Uniformly Convergent Method For Singularly Perturbed Delay Parabolic Partial Differential Equations”, Int. J. Comput. Math., 94:12 (2017), 2520–2546
10. Singh M.K., Natesan S., “Richardson Extrapolation Technique For Singularly Perturbed System of Parabolic Partial Differential Equations With Exponential Boundary Layers”, Appl. Math. Comput., 333 (2018), 254–275
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