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 Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 9, Pages 1617–1637 (Mi zvmmf414)

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

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 on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter $\varepsilon$ taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost $\varepsilon$-uniformly, i.e., with an error that weakly depends on the parameter $\varepsilon$: $|u(x,t)-z(x,t)|\le M[N_1^{-1}\ln^2N_1+N_0^{-1}\ln N_0+\varepsilon^{-1}N_1^{-K}\ln^{K-1}N_1]$, $(x,t)\in\bar G_h$, where $N_1+1$ è $N_0+1$ are the numbers of grid points in $x$ and $t$, respectively; $K$ is the number of refinement iterations (with respect to $x$) in the adapted grid; and $M=M(K)$. Outside the $\sigma$-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges $\varepsilon$-uniformly at a rate $O(N_1^{-1}\ln^2N_1+N_0^{-1}\ln N_0)$, ïðè÷åì $\sigma\le MN_1^{-K+1}\ln^{K-1}N_1$ for $K\ge2$.

Key words: singularly perturbed parabolic convection-diffusion equation, numerical embedded grid method, adapted grids, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:9, 1539–1559

Bibliographic databases:

UDC: 519.633

Citation: G. I. Shishkin, “The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids”, Zh. Vychisl. Mat. Mat. Fiz., 46:9 (2006), 1617–1637; Comput. Math. Math. Phys., 46:9 (2006), 1539–1559

Citation in format AMSBIB
\Bibitem{Shi06} \by G.~I.~Shishkin \paper The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2006 \vol 46 \issue 9 \pages 1617--1637 \mathnet{http://mi.mathnet.ru/zvmmf414} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2287662} \transl \jour Comput. Math. Math. Phys. \yr 2006 \vol 46 \issue 9 \pages 1539--1559 \crossref{https://doi.org/10.1134/S0965542506090077} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748995922} 

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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, “Setochnaya approksimatsiya singulyarno vozmuschennogo kvazilineinogo parabolicheskogo uravneniya konvektsii-diffuzii na apriorno adaptiruyuschikhsya setkakh”, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 149, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2007, 146–172
2. G. I. Shishkin, “Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation”, Comput. Math. Math. Phys., 48:5 (2008), 769–785
3. G. I. Shishkin, “Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: $\varepsilon$-uniformly convergent schemes”, Comput. Math. Math. Phys., 48:6 (2008), 956–974
4. Shishkin G.I., “A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation”, Numer. Math. Theory Methods Appl., 1:2 (2008), 214–234
5. 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
6. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Applied Mathematics and Computation, 217:8 (2010), 3641–3716
7. Shishkin G.I., “Difference scheme of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation”, Russian J Numer Anal Math Modelling, 25:3 (2010), 261–278
8. G. I. Shishkin, “A finite difference scheme of improved accuracy on a priori adapted grids for a singularly perturbed parabolic convection–diffusion equation”, Comput. Math. Math. Phys., 51:10 (2011), 1705–1728
9. Shishkin G.I., “Improved Scheme on Adapted Locally-Uniform Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Problem”, Bail 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods, Lecture Notes in Computational Science and Engineering, 81, eds. Clavero C., Gracia J., Lisbona F., Springer-Verlag Berlin, 2011, 207–215
10. G. I. Shishkin, “Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection–diffusion equation”, Comput. Math. Math. Phys., 52:6 (2012), 895–925
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