A Godunov-type method based on an exact solution to the Riemann problem for the shallow-water equations

V. V. Belikov   ; A. Yu. Semenov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 37 (1997) no. 8, pp. 1006-1019

Bibliographie

[1] Stoker Dzh., Volny na vode. Matematicheskaya teoriya i prilozheniya, Izd-vo inostr. lit., M., 1959

[2] Kochin N. E., Kibel I. A., Roze N. V., Teoreticheskaya gidromekhanika, v. 1, Fizmatgiz, M., 1963 | Zbl

[3] Voltsinger N. E., Pyaskovskii R. V., Teoriya melkoi vody. Okeanologicheskie zadachi i chislennye metody, Gidrometeoizdat, L., 1977

[4] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR

[5] Marchuk A. G., Chubarov L. B., Shokin Yu. I., Chislennoe modelirovanie tsunami, Nauka, Novosibirsk, 1983

[6] Rusanov V. V., “Raschet vzaimodeistviya nestatsionarnykh udarnykh voln s prepyatstviyami”, Zh. vychisl. matem. i matem. fiz., 1:2 (1961), 267–279 | MR

[7] Lyatkher V. M., Militeev A. N., “Issledovanie plana techenii v nizhnem befe gidrotekhnicheskikh sooruzhenii chislennymi metodami”, Gidrotekhn. str-vo, 1978, no. 6, 27–32

[8] MacCormac R. W., The effect of viscosity in hypervelocity impact cratering, AIAA Paper No 69-354, 1969, 7 pp.

[9] Pandolfi M., “Chislennye eksperimenty pri dvizhenii vody so svobodnoi poverkhnostyu i stupenchatymi volnami”, Chisl. reshenie zadach gidromekhan., Mir, M., 1977, 64–75

[10] Shokin Yu. I., Yanenko H. H., Metod differentsialnogo priblizheniya. Primenenie k gazovoi dinamike, Nauka, Novosibirsk, 1985 | MR | Zbl

[11] Fedotova Z. I., “O primenenii invariantnoi raznostnoi skhemy k raschetu kolebanii zhidkosti v basseine”, Chisl. metody mekhan. sploshnoi sredy, 9, no. 3, VTsSO AN SSSR, Novosibirsk, 1978, 137–146

[12] Courant R., Isaacson E., Rees M., “On the solution of nonlinear hyperbolic differential equations by finite differences”, Communs Pure and Appl. Math., 5:3 (1952), 243–255 | DOI | MR | Zbl

[13] Papa L., “Application of the Courant-Isaacson-Rees method to solve the shallow-water hydrodynamic equations”, Appl. Math. and Comput., 15:1 (1984), 85–92 | DOI | MR | Zbl

[14] Godunov S. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii gidrodinamiki”, Matem. sb., 47 (89):3 (1959), 271–306 | MR | Zbl

[15] Alalykin G. V., Godunov S. K., Kireeva I. L., Pliner L. A., Reshenie odnomernykh zadach gazovoi dinamiki v podvizhnykh setkakh, Nauka, M., 1970 | MR | Zbl

[16] Godunov S. K., Zabrodin A. V., Ivanov M. Ya. i dr., Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976 | MR | Zbl

[17] Louaked M., Hanich L., “A high resolution algorithm for the shallow water equations in general coordinates”, A Collection of Techn. Papers, The Sixth Internat. Symp. on CFD (Lake Tahoe, Nevada, USA, Sept. 4–8, 1995), v. 2, 725–731

[18] Roe P. L., “Approximate Riemann solvers, parameter vectors and difference schemes”, J. Comput. Phys., 43:2 (1981), 357–372 | DOI | MR | Zbl

[19] Semenov A. Yu., “Primenenie metoda Godunova k uravneniyam teorii melkoi vody s uchetom relefa dna”, Tr. VIII konf. molodykh uchenykh Mosk. fiz.-tekhn. in-ta (27 marta–7 aprelya 1983), v. 1, 150–157

[20] Belikov V. V., Semenov A. Yu., “Metod Godunova s modifikatsiei Kolgana dlya chislennogo resheniya dvumernykh uravnenii melkoi vody”, Tr. X konf. molodykh uchenykh Mosk. fiz.-tekhn. in-ta (23 marta–7 aprelya 1985), v. 1, 179–214

[21] Belikov V. V., Semenov A. Yu., Yavnyi chislennyi metod raspada razryvov dlya resheniya uravnenii melkoi vody, Preprint No 42, In-t obschei fiz. AN SSSR, M., 1988, 44 pp. | MR

[22] Belikov V. V., Semenov A. Yu., “Godunov's type method for a numerical solution of the two-dimensional shallow water equations”, Proc. 17th Session of Sci. and Methodol. Seminar on Ship Hydrodynamics (Bulgaria, Varna, Oct. 17–22, 1988), v. 2, 1988, 56/1–56/6

[23] Belikov V. V., Semenov A. Yu., “Godunov's type method for numerical solution of the shallow water equations”, Proc. of SU-Jpn Simp. on CFD (USSR, Khabarovsk, Sept. 9–16, 1988), v. 2, Comput. Centre USSR Acad. Sci., M., 1989, 177–183

[24] Voevodin A. F., Shugrin S. M., Chislennye metody rascheta odnomernykh sistem, Nauka, Novosibirsk, 1981 | MR

[25] Ovsyannikov L. V., Makarenko N. I., Nalimov V. I. i dr., Nelineinye problemy teorii poverkhnostnykh i vnutrennikh voln, Nauka, Novosibirsk, 1985

[26] Friedrichs K. O., “On the derivation of the shallow water theory”, Appendix to: J. J. Stoker, The formation of breakers and bores, Communs Pure and Appl. Math., 1:1 (1948), 81–85 | MR

[27] Stoker J. J., “The formation of breakers and bores”, Communs Pure and Appl. Math., 1:1 (1948), 1–87 | DOI | MR | Zbl

[28] Gladyshev M. T., “K zadache o raspade nachalnogo razryva v otkrytykh ruslakh”, Izv. vuzov. Ser. Energetika, 1968, no. 4, 81–88

[29] Baklanovskaya V. F., Paltsev B. V., Chechel I. I., “O kraevykh zadachakh dlya sistemy uravnenii Sen-Venana na ploskosti”, Zh. vychisl. matem. i matem. fiz., 19:3 (1979), 708–725 | MR | Zbl

[30] Nazhëstkina E. I., Rusanov V. V., “Approksimatsiya granichnykh uslovii v raznostnykh skhemakh”, Zh. vychisl. matem. i matem. fiz., 20:6 (1980), 1483–1499 | MR | Zbl

[31] Pogorelov N. V., Semënov A. Yu., “Modifikatsiya neotrazhayuschikh granichnykh uslovii pri gazodinamicheskom modelirovanii v astrofizike”, Zh. vychisl. matem. i matem. fiz., 36:3 (1996), 135–146 | MR | Zbl

[32] Kolgan V. P., “Primenenie printsipa minimalnykh znachenii proizvodnykh k postroeniyu konechno-raznostnykh skhem dlya rascheta razryvnykh reshenii gazovoi dinamiki”, Uch. zap. TsAGI, 3:6 (1972), 68–77

[33] Kopchenov V. I., Kraiko A. N., Postroenie monotonnoi konechno-raznostnoi skhemy vtorogo poryadka approksimatsii dlya uravnenii giperbolicheskogo tipa, Tekhn. otchet TsIAM, No 9108, M., 1980, 42 pp.

[34] Kopchënov V. I., Kraiko A. N., “Monotonnaya raznostnaya skhema vtorogo poryadka dlya giperbolicheskikh sistem s dvumya nezavisimymi peremennymi”, Zh. vychisl. matem. i matem. fiz., 23:4 (1983), 848–859 | MR | Zbl

[35] Rodionov A. V., “Monotonnaya skhema vtorogo poryadka approksimatsii dlya skvoznogo rascheta neravnovesnykh techenii”, Zh. vychisl. matem. i matem. fiz., 27:4 (1987), 585–593 | MR | Zbl

[36] Rodionov A. V., “Povyshenie poryadka approksimatsii skhemy S. K. Godunova”, Zh. vychisl. matem. i matem. fiz., 27:12 (1987), 1853–1860 | MR | Zbl

[37] Menshov I. S., “Povyshenie poryadka approksimatsii skhemy Godunova na osnove resheniya obobschennoi zadachi Rimana”, Zh. vychisl. matem. i matem. fiz., 30:9 (1990), 1357–1371 | MR

[38] Harten A., Osher S., Engquist B., Chakravarthy S. R., “Some results on uniformly high-order accurate essentially nonoscillatory schemes”, Appl. Numer. Math., 2:3–5 (1986), 347–377 | DOI | MR | Zbl

[39] Harten A., Osher S., “Uniformly high-order accurate nonoscillatory shemes, I”, SIAM J. Numer. Analys., 24:2 (1987), 279–309 | DOI | MR | Zbl

[40] Harten A., Engquist B., Osher S., Chakravarthy S. B., “Uniformly high-order accurate nonoscillatory shemes, III”, J. Comput. Phys., 71:2 (1987), 231–303 | DOI | MR | Zbl

[41] Kamenetskii V. F., Semenov A. Yu., “Samosoglasovannoe vydelenie razryvov pri skvoznykh raschetakh gazodinamicheskikh techenii”, Zh. vychisl. matem. i matem. fiz., 34:10 (1994), 1489–1502 | MR

[42] Ovsyannikov L. V., Lektsii po osnovam gazovoi dinamiki, Nauka, M., 1981 | MR | Zbl

[43] Ivanov M. Ya., Koretskii V. V., Kurochkina N. Ya., “Issledovanie svoistv raznostnykh skhem skvoznogo scheta pervogo poryadka approksimatsii”, Chisl. metody mekhan. sploshnoi sredy, 11, no. 1, ITPM SO AN SSSR, Novosibirsk, 1980, 81–110 | MR

[44] Ivanov M. Ya., Koretskii V. V., Kurochkina N. Ya., “Issledovanie svoistv raznostnykh skhem skvoznogo scheta vtorogo poryadka approksimatsii”, Chisl. metody mekhan. sploshnoi sredy, 11, no. 2, ITPM SO AN SSSR, Novosibirsk, 1980, 41–63 | MR

[45] Ivanov M. Ya., Koretskii V. V., Kurochkina N. Ya., “Issledovanie svoistv raznostnykh skhem skvoznogo scheta povyshennogo poryadka approksimatsii”, Chisl. metody mekhan. sploshnoi sredy, 11, no. 4, ITPM SO AN SSSR, Novosibirsk, 1980, 83–103

[46] Belikov V. V., Volchenkov G. Ya., Peich Yu. L. i dr., Razrabotka metodiki opredeleniya deformatsii i optimalnykh razmerov ukreplenii na vykhodakh iz kosogornykh vodopropusknykh trub, Nauchno-tekhn. otchet TsNIIS. Tema No IP-XI-5-84/85, razd. 1, TsNIIS, M., 1985, 181 pp.

[47] Mishuev A. V., Sladkevich M. S., Silchenko A. S., “Analiz eksperimentalnykh rezultatov i chislennogo metoda rascheta dlinnykh voln na otkos”, Soveschanie po tsunami, Gorkii, 1984, 121–123

[48] Lyatkher V. M., Militeev A. N., Mishuev A. V., Sladkevich M. S., “Issledovanie nakata voln tsunami na berega chislennymi metodami”, Issled. tsunami. Vozniknovenie i rasprostranenie v okeane voln tsunami, 1, M., 1986, 110–119