A parallel implementation of the algebraic multigrid method for solving problems in dynamics of viscous incompressible fluid

K. N. Volkov   ; A. S. Kozelkov   ; S. V. Lashkin   ; N. V. Tarasova   ; A. V. Yalozo

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 12, pp. 2079-2097

Résumé

An algorithm for improving the scalability of the multigrid method used for solving the system of difference equations obtained by the finite volume discretization of the Navier–Stokes equations on unstructured grids with an arbitrary cell topology is proposed. It is based on the cascade assembly of the global level; the cascade procedure gradually decreases the number of processors involved in the computations. Specific features of the proposed approach are described, and the results of solving benchmark problems in the dynamics of viscous incompressible fluid are discussed; the scalability and efficiency of the proposed method are estimated. The advantages of using the global level in the parallel implementation of the multigrid method which sometimes makes it possible to speed up the computations by several fold.

Bibliographie

[1] Kozelkov A. S., Kurulin V. V., “Chislennaya skhema dlya modelirovaniya turbulentnykh techenii neszhimaemoi zhidkosti s ispolzovaniem vikhrerazreshayuschikh podkhodov”, Zh. vychisl. matem. i matem. fiz., 55:7 (2015), 135–146

[2] Saad Y., Iterative methods for sparse linear systems, SIAM, Philadelphia, 2003, 568 pp. | MR

[3] Volkov K. N., Deryugin Yu. N., Emelyanov V. N., Karpenko A. G., Kozelkov A. S., Teterina I. V., Metody uskoreniya gazodinamicheskikh raschetov na nestrukturirovannykh setkakh, Fizmatlit, M., 2013, 536 pp.

[4] Kozelkov A. S., Shagaliev R. M., Kurulin V. V., Yalozo A. V., Lashkin S. V., “Issledovanie potentsiala superkompyuterov dlya masshtabiruemogo chislennogo modelirovaniya zadach gidrodinamiki v industrialnykh prilozheniyakh”, Zh. vychisl. matem. i matem. fiz., 56:8 (2016), 1524–1535 | DOI

[5] Volkov K. N., Deryugin Yu. N., Emelyanov V. N., Kozelkov A. S., Teterina I. V., “Algebraicheskii mnogosetochnyi metod v zadachakh vychislitelnoi fiziki”, Vychisl. metody i programmirovanie, 15:1 (2014), 183–200

[6] Fedorenko R. P., “Relaksatsionnyi metod resheniya raznostnykh ellipticheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 1:5 (1961), 922–927

[7] Bakhvalov N. V., “O skhodimosti odnogo relaksatsionnogo metoda dlya ellipticheskogo operatora s estestvennymi ogranicheniyami”, Zh. vychisl. matem. i matem. fiz., 6:5 (1966), 101–135

[8] Brandt A., “Guide to multigrid development”, Lect. Notes in Math., 960, 1982, 220–312 | DOI | MR

[9] Stuben K., Trottenberg U., “Multigrid methods: fundamental algorithms, model problem analysis and applications”, Lect. Notes in Math., 960, Springer, Berlin, 1982, 1–176 | DOI | MR

[10] Ruge J., Stüben K., “Algebraic multigrid (AMG)”, Multigrid Methods, Frontiers in Appl. Math., 3, ed. S.F. McCormick, SIAM, Philadelphia, 1987, 73–130 | MR

[11] Wagner C., Introduction to algebraic multigrid, Course Notes of an Algebraic Multigrid Course, University of Heidelberg, 1999, 128 pp.

[12] Cleary A. J., Falgout R. D., Henson V. E., Jones J. E., Manteu T. A., McCormick S. F., Miranda G. N., Ruge J. W., “Robustness and scalability of algebraic multigrid”, SIAM J. Sci. Statistical Comput, 21:6 (2000), 18861908 | MR

[13] Stüben K., “A review of algebraic multigrid”, J. Comput. and Appl. Math., 128:1–2 (2001), 281–309 | DOI | MR

[14] Stüben K., “An introduction to algebraic multigrid”, Multigrid, eds. U. Trottenberg, C. Oosterlee, A. Schüller, Academic Press, London, 2001, 413–532 | MR

[15] Kozelkov A. S., Deryugin Yu. N., Lashkin S. V., Silaev D. P., Simonov P. G., Tyatyushkina E. S., “Realizatsiya metoda rascheta vyazkoi neszhimaemoi zhidkosti s ispolzovaniem mnogosetochnogo metoda na osnove algoritma SIMPLE v pakete programm LOGOS”, Vopr. atomnoi nauki i tekhn. Matem. modelirovanie fiz. protsessov, 4 (2013), 44–56

[16] Vanek P., Mandel J., Brezina M., “Algebraic multigrid based on smoothed aggregation for second and fourth order problems”, Comput., 56:2 (1996), 179–196 | DOI | MR

[17] Farris C., Misra M., “Distributed algebraic multigrid for finite element computations”, Math. and Comput. Modelling, 27:8 (1998), 41–67 | DOI | MR

[18] De Sterck H, Yang U. M., Heys J. J., “Reducing complexity in parallel algebraic multigrid preconditioners”, SIAM J. Matrix Analys. and Appl., 27:6 (2006), 1019–1039 | DOI | MR

[19] Emans M., “Performance of parallel AMG-preconditioners in CFD-codes for weakly compressible flows”, Parallel Comput., 36:5–6 (2010), 326–338 | DOI | MR

[20] Emans M., “Benchmarking aggregation amg for linear systems in CFD simulations of compressible internal flows”, Electronic Transact. on Numerical Analys., 37 (2010), 351–366 | MR

[21] Weiss J. M., Maruszewski J. P., Smith W. A., “Implicit solution of preconditioned Navier-Stokes equations using algebraic multigrid”, AIAA Journal, 37:1 (1999), 29–36 | DOI

[22] Napov A., Notay Y., An algebraic multigrid method with guaranteed convergence rate, Technical Rept. University of Brussels, No GANMN 10-03, 2010

[23] Notay Y., “An aggregation-based algebraic multigrid method”, Electronic Transact. Numerical Analys., 37 (2010), 123–146 | MR

[24] Sala M., Tuminaro R. S., “A new Petrov-Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems”, SIAM J. Sci. Comput., 31:1 (2008), 143–166 | DOI | MR

[25] Lin P. T., Sala M., Shadid J. N., Tuminaro R. S., “Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport”, Internat. J. Numerical Meth. Eng., 67:2 (2006), 208–225 | DOI

[26] Gravemeier V., Gee M. W., Kronbichler M., Wall W. A., “An algebraic variational multiscale-multigrid method for large eddy simulation of turbulent flow”, Computer Meth. Appl. Mechan. and Eng., 199:13–16 (2010), 853–864 | DOI | MR

[27] Brezina M., Falgout R., MacLachlan S., Manteuffel T., McCormick S., Ruge J., “Adaptive algebraic multigrid”, SIAM J. Sci. Comput., 27:4 (2005), 1261–1286 | DOI | MR

[28] Brezina M., Falgout R., MacLachlan S., Manteuffel T., McCormick S., Ruge J., “Adaptive smoothed aggregation ($\alpha$SA)”, SIAM J. Sci. Comput., 25:6 (2004), 1896–1920 | DOI | MR

[29] van Emden H., Meier-Yang U., “BoomerAMG: a parallel algebraic multigrid solver and preconditioner”, Appl. Numerical Math., 41:1 (2001), 155–177 | MR

[30] Zhukov V. T., Krasnov M. M., Novikova N. D., Feodoritova O. B., “Parallelnyi mnogosetochnyi metod: sravnenie effektivnosti na sovremennykh vychislitelnykh arkhitekturakh”, Preprinty IPM im. M.V. Keldysha, 2014, 031, 22 pp.

[31] Lin P., Bettencourt M., Domino S., Fisher T., Hoemmen M., Hu J., Phipps E., Prokopenko A., Rajamanickam S., Siefert C., Cyr E., Kennon S., “Towards extreme-scale simulations with next-generation Trilinos: a low Mach fluid application case study”, Workshop on Large-Scale Parallel Proc., LSPP (23 May 2014, Phoeniz, AZ, USA), 10 pp.

[32] Braess D., “Towards algebraic multigrid for elliptic problems of second order”, Comput., 55:4 (1995), 379–393 | DOI | MR

[33] Guillard H., Janka A., Vanek P., “Analysis of an algebraic Petrov-Galerkin smoothed aggregation multigrid method”, Appl. Numerical Math., 58:12 (2008), 1861–1874 | DOI | MR

[34] Darwish M. S., Saad T., Hamdan Z., “A high scalability parallel algebraic multigrid solver”, Proc. European Conference on Comput. Fluid Dynamics, ECCOMAS CFD 2006 (5–8 September 2006, Egmond aan Zee, Netherlands), 2006, 16 pp.

[35] Yang U. M., “Parallel algebraic multigrid methods — high performance preconditioners”, Numerical Solution of Partial Differential Equations on Parallel Computers, Lecture Notes in Computational Science and Engineering, 51, eds. A.M. Bruaset, A. Tveito, Springer, 2006, 209–236 | DOI | MR

[36] Kozelkov A. S., Kurulin V. V., Tyatyushkina E. S., Puchkova O. L., “Modelirovanie turbulentnykh techenii vyazkoi neszhimaemoi zhidkosti na nestrukturirovannykh setkakh s ispolzovaniem modeli otsoedinennykh vikhrei”, Matem. modelirovanie, 26:8 (2014), 81–96

[37] Gergel V. P., Strongin R. G., Osnovy parallelnykh vychislenii dlya mnogoprotsessornykh vychislitelnykh sistem, NNGU, Nizhnii Novgorod, 2000, 121 pp.

[38] Ferziger J. H., Peric M., Computational methods for fluid dynamics, Springer, Berlin, 2002, 310 pp. | MR

[39] Vogel J. C., Eaton J. K., “Combined heat transfer and fluid dynamic measurements downstream of a backward-facing step”, J. Heat Transfer, 107 (1985), 922–929 | DOI