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

This article is cited in 8 scientific papers (total in 8 papers)

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.

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

Bibliographic databases:

UDC: 519.632.4
Received: 22.09.2009

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
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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. 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  mathnet  crossref  isi  elib
    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  mathnet  crossref  adsnasa
    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  mathnet  crossref  mathscinet  isi
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  isi  elib
    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  mathnet  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  crossref  mathscinet  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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