Geodesic
Effective solving of three-dimensional gas dynamics problems with the Runge-Kutta discontinuous Galerkin method
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 3, pp. 465-475 
B. A. Korneev; V. D. Levchenko. Effective solving of three-dimensional gas dynamics problems with the Runge-Kutta discontinuous Galerkin method. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 3, pp. 465-475. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_3_a13/
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     author = {B. A. Korneev and V. D. Levchenko},
     title = {Effective solving of three-dimensional gas dynamics problems with the {Runge-Kutta} discontinuous {Galerkin} method},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we present the Runge–Kutta discontinuous Galerkin method (RKDG method) for the numerical solution of the Euler equations of gas dynamics. The method is being tested on a series of Riemann problems in the one-dimensional case. For the implementation of the method in the three-dimensional case, a DiamondTorre algorithm is proposed. It belongs to the class of the locally recursive non-locally asynchronous algorithms (LRnLA). With the help of this algorithm a significant increase of speed of calculations is achieved. As an example of the three-dimensional computing, a problem of the interaction of a bubble with a shock wave is considered.

[1] Tirskii G. A. (red.), Giperzvukovaya aerodinamika i teploobmen spuskaemykh kosmicheskikh apparatov i planetnykh zondov, Fizmatlit, M., 2011

[2] Popov Yu. P., Samarskii A. A., Raznostnye metody resheniya zadach gazovoi dinamiki, Ucheb. posobie dlya vuzov po spets. prikl. matematike, Nauka, M., 1992 | MR

[3] V. S. Avduevskii (red.), Matematicheskoe modelirovanie konvektivnogo teplomassoobmena na osnove uravnenii Nave–Stoksa, Nauka, M., 1987

[4] A. V. Gasnikov (red.), Vvedenie v matematicheskoe modelirovanie transportnykh potokov, Uchebnoe posobie, MTsNMO, M., 2012

[5] B. Cockburn, C. Shu (executor), Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, Rep., NASA, 2000

[6] Gryngarten L. D., Menon S., “Shock-bubble interaction simulations using a new two-phase discontinuous Galerkin method”, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, FL, 2011, AIAA-2011-294

[7] Hejazialhosseini B., Rossinelli D., Koumoutsakos P., 3D shock-bubble interactions at Mach 3, 2012, arXiv: 1210.3822

[8] Tishkin V. F., Nikishin V. V., Popov I. V., Favorskii A. P., “Raznostnye skhemy trekhmernoi gazovoi dinamiki dlya zadachi o razvitii neustoichivosti Rikhtmaiera–Meshkova”, Matem. modelirovanie, 7:5 (1995), 15–25 | MR

[9] Niederhaus J. H. J., Greenough J. A., Oakley J. G. et al., “A computational parameter study for the three-dimensional shock-bubble interaction”, J. Fluid Mech., 594 (2008), 85–124 | DOI | Zbl

[10] Zakirov A., Levchenko V., “The effective 3D modeling of electromagnetic waves evolution in photonic crystals and metamaterials”, PIERS Proceedings, session 3AP, 2009, 580–584

[11] Levchenko V. D., “Asinkhronnye parallelnye algoritmy kak sposob dostizheniya effektivnosti vychislenii”, Informats. tekhn. i vychislit. sistemy, 1 (2005), 68–87 | Zbl

[12] Goryachev I. A., “Otsenka vychislitelnoi proizvoditelnosti lokalno-rekursivnykh nelokalno-asinkhronnykh algoritmov na geterogennykh sistemakh”, Nauchnyi servis v seti Internet: poisk novykh reshenii, Tezisy dokladov Mezhdunarodnoi superkompyuternoi konferentsii, 2012, 676–678

[13] Perepelkina A. Yu., Levchenko V. D., Goryachev I. A., “Issledovanie roli veibelevskoi neustoichivosti vo vzaimodeistvii lazernogo izlucheniya so sverkhkriticheskoi plazmoi pri pomoschi chislennogo modelirovaniya”, Prepr. IPM im. M. V. Keldysha RAN, 2014, 026, 19 pp.

[14] Gottlieb S., Shu C.-W., “Total variation diminishing Runge–Kutta schemes”, Math. Comp. Amer. Math. Sci., 67:221 (1998), 73–85 | DOI | MR | Zbl

[15] Jun Zhu, Jianxian Qiu, Tie-gang Liu, Boo Cheong Khoo, “RKDG methods with WENO type limiters and conservative interfacial procedure for one-dimensional compressible multi-medium flow simulations”, Appl. Numer. Math., 61:4 (2011), 554–580 | DOI | MR | Zbl

[16] Toro E., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, Springer, 2009 | MR | Zbl

[17] Korneev B. A., Levchenko V. D., “Modelirovanie uravnenii gazovoi dinamiki razryvnym metodom Galerkina s pomoschyu algoritmov LRnLA”, Prepr. IPM. im. M.B. Keldysha RAN, 2013, 028, 17 pp. | Zbl

[18] Perepelkina A. Yu., Levchenko V. D., Goryachev I. A., “Trekhmernyi kineticheskii kod CFHall dlya modelirovaniya zamagnichennoi plazmy”, Matem. modelirovanie, 25 (2013), 98–110 | MR

[19] Rybakin B., Goryachev V., “Reshenie i vizualizatsiya mnogomernykh zadach gazodinamiki na mnogoyadernykh protsessorakh i GPU”, Nauchnyi servis v seti Internet: poisk novykh reshenii, Tezisy dokladov Mezhdunarodnoi superkompyuternoi konferentsii, 2013, 381–387