Sequential and parallel domain decomposition methods for a singularly perturbed parabolic convection-diffusion equation

I. V. Tselischeva ; G. I. Shishkin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 1, pp. 202-220

Résumé

A boundary value problem for a singularly perturbed parabolic convection–diffusion equation is considered in a rectangular domain in $x$ and $t$; the perturbation parameter $\varepsilon$ multiplying the highest derivative takes arbitrary values in the half-open interval $(0,1]$. For the boundary value problem, we construct a scheme based on the method of lines in $x$ passing through $N_0+1$ points of the mesh with respect to $t$. To solve the problem on a set of intervals, we apply a domain decomposition method (on overlapping subdomains with the overlap width $\delta$), which is a modification of the Schwarz method. For the continual schemes of the decomposition method, we study how sequential and parallel computations, the order of priority in which the subproblems are sequentially solved on the subdomains, and the value of the parameter $\varepsilon$ (as well as the values of $N_0$, $\delta$) influence the convergence rate of the decomposition scheme (as $N_0\to\infty$), and also computational costs for solving the scheme and time required for its solution (unless a prescribed tolerance is achieved). For convection–diffusion equations, in contrast to reaction-diffusion ones, the sequential scheme turns out to be more efficient than the parallel scheme.

Bibliographie

[1] 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, World Scientific, Singapore, 1996 | MR

[2] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i paraboli- cheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[3] Roos H.-G., Stynes M., Tobiska L., Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, Springer, Berlin, 1996 | MR

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

[5] Quarteroni A., “Domain decomposition and parallel processing for the numerical solution of partial differential equations”, Surveys Math. Indust., 1:1 (1991), 75–118 | MR | Zbl

[6] Ewing R.E., Lazarov R.D., Pasciak J.E., Vassilevski P.S., “Domain decomposition type iterative techniques for parabolic problems on locally refined grids”, SIAM J. Numer. Anal., 30:6 (1993), 1537–1557 | DOI | MR | Zbl

[7] Le Tallec P., “Domain decomposition methods in computational mechanics”, Comput. Mech. Adv., 1:2 (1994), 121–220 | MR | Zbl

[8] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989 | MR

[9] MacMullen H., Miller J. J. H., O'Riordan E., Shishkin G. I., “A second order parameter-uniform overlapping Schwartz method for reaction-diffusion problems with boundary layers”, J. Comput. Appl. Math., 130:1–2 (2001), 231–244 | DOI | MR | Zbl

[10] MacMullen H., O'Riordan E., Shishkin G. I., “The convergence of classical Schwarz methods applied to convection-diffusion problems with regular boundary layers”, Appl. Numer. Math., 43:3 (2002), 297–313 | DOI | MR | Zbl

[11] Shishkin G. I., “Acceleration of the process of the numerical solution to singularly perturbed boundary value problems for parabolic equations on the basis of parallel computations”, Russ. J. Numer. Anal. Math. Modelling, 12:3 (1997), 271–291 | MR | Zbl

[12] Hemker P. W., Shishkin G. I., Shishkina L. P., “Distributing the numerical solution of parabolic singularly perturbed problems with defect correction over independent processes”, Sib. J. Numer. Math., 3:3 (2000), 229–258 | Zbl

[13] Ladyzhenskaya O. A., Colonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[14] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[15] Krylov V. I., Bobkov V. V., Monastyrnyi P. I., Vychislitelnye metody, T. 2, Nauka, M., 1977 | MR

[16] Shishkin G. I., “Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems”, Math. Modelling and Anal., 10:4 (2005), 393–412 | MR | Zbl

[17] Farrell P. A., Hemker P. W., Shishkin G. I., “Discrete approximations for singularly perturbed boundary value problems with parabolic layers”, Part 1, J. Comput. Math., 14:1 (1996), 71–97 | MR | Zbl

[18] Hemker P. W., Shishkin G. I., Shishkina L. P., “High-order time-accurate parallel schemes for parabolic singularly perturbed problems with convection”, Computing, 66:2 (2001), 139–161 | DOI | MR | Zbl

[19] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1966 | MR | Zbl