RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Zh. Vychisl. Mat. Mat. Fiz.: Year: Volume: Issue: Page: Find

 Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 3, Pages 458–478 (Mi zvmmf4843)

A Richardson scheme of an increased order of accuracy for a semilinear singularly perturbed elliptic convection-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: The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges $\varepsilon$-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges $\varepsilon$-uniformly with an improved order, namely, at the rate $O(N_1^{-2}\ln_1^2N+N_2^{-2})$, where $N_1+1$ and $N_2+1$ are the number of grid nodes along the $x_1$-axis and per unit interval of the $x_2$-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging $\varepsilon$-uniformly with an improved order. Both the basic and improved iterative schemes converge $\varepsilon$-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same $\varepsilon$-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging $\varepsilon$-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.

Key words: elliptic convection-diffusion equation, regular layer, boundary layer, Richardson technique, finite difference scheme, nonlinear scheme, linearized iterative scheme, truncated iterative scheme, $\varepsilon$-uniform convergence.

Full text: PDF file (313 kB)
References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2010, 50:3, 437–456

Bibliographic databases:

UDC: 519.632.4

Citation: G. I. Shishkin, L. P. Shishkina, “A Richardson scheme of an increased order of accuracy for a semilinear singularly perturbed elliptic convection-diffusion equation”, Zh. Vychisl. Mat. Mat. Fiz., 50:3 (2010), 458–478; Comput. Math. Math. Phys., 50:3 (2010), 437–456

Citation in format AMSBIB
\Bibitem{ShiShi10} \by G.~I.~Shishkin, L.~P.~Shishkina \paper A Richardson scheme of an increased order of accuracy for a semilinear singularly perturbed elliptic convection-diffusion equation \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2010 \vol 50 \issue 3 \pages 458--478 \mathnet{http://mi.mathnet.ru/zvmmf4843} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2681923} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2010CMMPh..50..437S} \transl \jour Comput. Math. Math. Phys. \yr 2010 \vol 50 \issue 3 \pages 437--456 \crossref{https://doi.org/10.1134/S0965542510030061} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000277337300006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77951830793} 

• http://mi.mathnet.ru/eng/zvmmf4843
• http://mi.mathnet.ru/eng/zvmmf/v50/i3/p458

 SHARE:

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., Shishkina L.P., “Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection-diffusion equation”, Russian J. Numer. Anal. Math. Modelling, 26:4 (2011), 427–445
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. 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
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
•  Number of views: This page: 998 Full text: 98 References: 43 First page: 2