Computer difference scheme for a singularly perturbed elliptic convection-diffusion equation in the presence of perturbations

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 5, pp. 814-831

Résumé

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection-diffusion equation with a perturbation parameter $\varepsilon$, $\varepsilon\in(0, 1]$, multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge $\varepsilon$-uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if $N_1^{-1}N_2^{-1}\ll\varepsilon$, where $N_1$ and $N_2$ are the numbers of grid intervals in $x$ and $y$, respectively, the scheme is not -uniformly well-conditioned or $\varepsilon$-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on $\varepsilon$, $N_1$, $N_2$, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as $N_1$, $N_2\to\infty$, $\varepsilon\in(0, 1]$. The difference schemes constructed in the presence of the indicated perturbations that converges as $N_1$, $N_2\to\infty$ for fixed $\varepsilon$, $\varepsilon\in(0, 1]$, is called a computer difference scheme. Schemes converging $\varepsilon$-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.

Bibliographie

[1] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[2] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E., Shishkin G. I., Robust computational techniques for boundary layers, Chapman and Hall/CRC, Boca Raton, 2000 | MR | Zbl

[3] Roos H.-G., Stynes M., Tobiska L., Numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems, Springer Series in Computational Mathematics, 34, Second Edition, Springer-Verlag, Berlin, 2008 | MR

[4] Shishkin G. I., Shishkina L. P., Difference methods for singular perturbation problems, Chapman Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 140, CRC Press, Boca Raton, 2009 | MR | Zbl

[5] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions, Revised Edition, World Scientific, Singapore, 2012 | MR | Zbl

[6] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989

[7] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978

[8] Marchuk G. I., Shaidurov V. V., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979

[9] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989

[10] Shishkin G. I., “Obuslovlennost raznostnoi skhemy metoda dekompozitsii resheniya dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii”, Tr. IMM UrO RAN, 18, no. 2, 2012, 291–304

[11] Shishkin G. I., “Stability of difference schemes on uniform grids to pertubations of the data for a singularly perturbed convection-diffusion equation”, Russian J. Numer. Anal. and Math. Modelling, 28:4 (2013), 381–417 | DOI | MR

[12] Shishkin G. I., “Difference schemes of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation”, Russian J. Numer. Anal. and Math. Modelling, 25:3 (2010), 261–278 | DOI | MR | Zbl

[13] Shishkin G. I., “Obuslovlennost i ustoichivost raznostnykh skhem na ravnomernykh setkakh dlya singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii-diffuzii”, Zh. vychisl. matem. i matem. fiz., 53:4 (2013), 575–599 | DOI | Zbl

[14] Kalitkin N. N., Chislennye metody, Nauka, M., 1978

[15] Wesseling P., Principles of computational fluid dynamics, Springer-Verlag, Berlin, 2001 | MR

[16] Shishkin G., “Stability of difference schemes on uniform grids for a singularly perturbed convection-diffusion equation”, Numerical Mathematics and Advanced Applications 2011, Proc. ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications (Leicester, September 2011), Springer, Berlin, 2013, 293–302 | DOI | MR

[17] Shishkin G. I., “Ustoichivost standartnoi skhemy dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii”, Dokl. AN, 448:6 (2013), 648–650 | DOI | Zbl

[18] Shishkin G. I., Shishkina L. P., “Ustoichivaya standartnaya raznostnaya skhema dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii pri nalichii vozmuschenii”, Trudy IMM UrO RAN, 20, no. 1, 2014, 322–333

[19] Shishkin G. I., “Kompyuternaya raznostnaya skhema dlya singulyarno vozmuschennogo uravneniya konvektsii–diffuzii”, Zh. vychisl. matem. i matem. fiz., 54:8 (2014), 1256–1269 | DOI | Zbl

[20] Shishkin G. I., “Standartnaya skhema dlya singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii-diffuzii pri kompyuternykh vozmuscheniyakh”, Dokl. AN, 462:1 (2015), 26–29 | DOI | Zbl

[21] Shishkin G. I., “Raznostnaya skhema dlya singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii-diffuzii pri nalichii vozmuschenii”, Zh. vychisl. matem. i matem. fiz., 55:11 (2015), 74–90

[22] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennykh uravnenii s konvektivnymi chlenami pri vozmuschenii dannykh”, Zh. vychisl. matem. i matem. fiz., 41:5 (2001), 692–707 | Zbl

[23] Shishkin G. I., “Obuslennost raznostnykh skhem dlya singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii-diffuzii”, Zh. vychisl. matem. i matem. fiz., 48:5 (2008), 813–830 | Zbl

[24] Bakhvalov N. C., Chislennye metody, Nauka, M., 1973

[25] Bakhvalov N. C., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Laboratoriya bazovykh znanii, M., 2001

[26] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973