Numerical methods of Lagrange particle-points for one-dimensional gas dynamics wave equations

V. E. Troshchiev ; N. S. Bochkarev

Matematičeskoe modelirovanie, Tome 24 (2012) no. 6, pp. 91-108.

Résumé

In the paper we consider the construction of numerical methods of computational gas dynamics based on the approximation of the second order nonlinear wave equations (NWE). Thes approach of "NWE” allows one to construct finite difference and finite elements schemes with cells of balance (conservative cells) both in the “finite volume” and lagrange “particle-points” framework. Therefore, numerical methods based on the approximations of NWE are of the great interest for the solution of one- and multi-dimensional problems of computational gas dynamics. In this paper the construction and investigation of discrete models of ”NWE” is performed for one dimensional gas dynamics problems in the Lagrange form and the results of numerical experiments are discussed.

Détail

Keywords: gas dynamics equations, nonlinear wave equations, finite difference schemes, finite elements schemes, particle-points method.
Mots-clés : Lagrange variable

Bibliographie

[1] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1987, 592 pp. | MR

[2] Samarskii A. A., Popov Yu. I., Raznostnye metody resheniya zadach gazovoi dinamiki, Nauka, M., 2004, 423 pp. | MR

[3] Rikhtmaier R., Morton K., Raznostnye metody resheniya kraevykh zadach, Mir, M., 1972, 418 pp.

[4] Vyaznikov K. V., Tishkin V. F., Favorskii A. P., “Postroenie monotonnykh raznostnykh skhem povyshennogo poryadka approksimatsii dlya sistem uravnenii giperbolicheskogo tipa”, Matem. modelirovanie, 1:5 (1989), 95–120 | MR | Zbl

[5] Kurganov A., Tadmor E., “New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations”, J. Comp. Phys., 160 (2000), 241–280 | DOI | MR

[6] Goloviznin V. M., “Balansno-kharakteristicheskii metod chislennogo resheniya odnomernykh uravnenii gazovoi dinamiki v eilerovykh peremennykh”, Matem. modelir., 18:11 (2006), 14–30 | MR | Zbl

[7] Babenko K. I. (red.), Teoreticheskie osnovy i konstruirovanie chislennykh algoritmov zadach matematicheskoi fiziki, Nauka, M., 1979, 295 pp. | MR

[8] Belotserkovskii O. M., Davydov Yu. M., Metod krupnykh chastits v gazovoi dinamike, Nauka, M., 1982, 391 pp.

[9] Older B., Fernbakh S., Rotenberg M. (red.), Vychislitelnye metody v gidrodinamike, Mir, M., 1967, 384 pp.

[10] Godunov S. K., Zabrodin A. V., Ivanov M. Ya., Kraiko A. N., Prokopov G. P., Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976 | MR | Zbl

[11] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001 | MR

[12] Fedorenko R. P., Vvedenie v vychislitelnuyu fiziku, Izdatelskii Dom «Intellekt», M., 2008

[13] Kharlou F. Kh., “Chislennyi metod chastits v yacheikakh dlya zadach gidrodinamiki”, Vychislitelnye metody v gidrodinamike, Mir, M., 1967, 316–342

[14] Dyachenko V. F., “Ob odnom novom metode chislennogo resheniya nestatsionarnykh zadach gazovoi dinamiki s dvumya prostranstvennymi peremennymi”, ZhVMiMF, 5:4 (1965), 680–688 | MR

[15] Krouli U., “FLAG — svobodno-lagranzhevyi metod dlya chislennogo modelirovaniya gidrodinamicheskikh techenii v dvukh izmereniyakh”, Chislennye metody v mekhanike zhidkostei, Mir, M., 1973, 135–145

[16] Glagoleva Yu. P., Zhogov B. M., Kiryanov Yu. F., Malshakov V. D., Nesterenko L. V., Podlivaev I. F., Sofronov I. D., “Osnovy metodiki Meduza chislennogo rascheta dvumernykh nestatsionarnykh zadach gazodinamiki”, Chisl. metody mekhaniki sploshnoi sredy, 3:2 (1972), 18–55

[17] Diyankov O. V., “Metod nechetkikh setok dlya uravnenii v chastnykh proizvodnykh”, Matem. modelirovanie, 19:7 (2007), 101–111 | MR

[18] Troschiev V. E., Nelineinye volnovye uravneniya gazovoi dinamiki i voprosy postroeniya chislennykh modelei na ikh osnove, Preprint TRINITI 150-a, Troitsk, 2012, 19 pp.

[19] Troschiev V. E., Shagaliev R. M., “Problema sovmescheniya konechno-raznostnykh i konechno-elementnykh skhem v zadachakh gazovoi dinamiki s teploprovodnostyu”, Matem. modelirovanie, 12:2 (2000), 3–11

[20] Elesin V. A., Troschiev V. E., Yudintsev V. F., “Razvitie chislennykh metodov i programm rascheta odnomernykh spektralnykh zadach perenosa teplovogo izlucheniya vo VNIIEF”, Voprosy atomnoi nauki i tekhniki, ser. Matem. modelirov. fizicheskikh protsessov, 2002, no. 1, 11–28

[21] Troschiev V. E., Apollonova O. V., Postroenie i issledovanie konechno-raznostnykh i konechno-elementnykh skhem dlya odnomernykh uravnenii gazovoi dinamiki, Preprint TRINITI 0059, TsNIIATOMINFORM, 1999, 22 pp.

[22] Troschiev V. E., Bochkarev N. S., Nelineinye volnovye uravneniya gazovoi dinamiki i chislennye metody na ikh osnove dlya odnomernykh techenii, Preprint TRINITI 147-A, Troitsk, 2011, 23 pp.

[23] Troschiev V. E., “O divergentnosti skhemy «krest» chislennogo resheniya uravnenii gazovoi dinamiki”, Chislennye metody mekhaniki sploshnoi sredy, 1:5 (1970), 87–93 | MR

[24] Troschiev V. E., Baltaev I. F., “O preobrazovanii skhemy «krest» dlya uravnenii gazovoi dinamiki v skhemu «prediktor-korrektor»”, Nauchnaya sessiya NIYaU MIFI (2011)

[25] Sedov L. I., Metody podobiya i razmernosti v mekhanike, Nauka, M., 1977 | MR