Geodesic
An iterative-marching method for solving problems of fluid and gas mechanics problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 2, pp. 171-181.

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A possibility is studied of applying the idea of global iterations with aspect to the pressure for the complete Navie–Stokes equations for a fluid as well as for a gas, for stationary and unstationary problems, for two and three dimensional problems. We give a generalization for the results published earlier and present new results concerning stability and convergence of the iterative-marching method, and its testing on the problems of motion of a fluid (twist fluid flows with bubbles; internal and surface waves generation by an eddy pair) and gas motions (flows in a nozzle with a bubble; shock wave formation resulting from viscous effects). The main conclusion is as follows. The proposed method allows us to develop numerical algorithms for various problems of fluid and gas mechanics based on a common principle. These algorithms are simple because their basic element is a marching procedure. The above implies the possibility of developing rather universal programs. 
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L. I. Skurin. An iterative-marching method for solving problems of fluid and gas mechanics problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 2, pp. 171-181. http://geodesic.mathdoc.fr/item/SJVM_1998_1_2_a5/

[1] Vazov V., Forsait Dzh., Metody resheniya differentsialnykh uravnenii v chastnykh proizvodnykh, IL, M., 1963

[2] Davis R. T., “Numerical solution of the hypersonic viscous shock-layer equation”, AIAA J., 8:5 (1970), 843–851 | DOI | Zbl

[3] Vigneron Y. C., Rakich J. V., and Tannehill J. C., Calculation of supersonic viscous flow over delta wings with sharp subsonic leading edges, AIAA Paper, no 1137, 1978 | Zbl

[4] Israelli M. and Lin A., “Iterative numerical solutions and boundary conditions for the parabolized Navie–Stokes equations”, J. Comput. and Fluids, 13:4 (1985), 397–409 | DOI | MR

[5] Voinovich P. A., Fursenko A. A., “Metod globalnykh iteratsii dlya rascheta smeshannykh techenii vyazkogo gaza”, Differents. uravneniya, 20:7 (1984), 1151–1156 | MR

[6] Barnet M. and Davis R. T., “Calculation of supersonic flows with strong viscous-inviscid interaction”, AIAA J., 24:12 (1986), 1949–1955 | DOI

[7] Ramakrishnan S. V. and Rubin S. V., “Time-consistent pressure relaxation procedure for compressible reduced Navie–Stokes equations”, AIAA J., 25:7 (1987), 905–913 | DOI | MR

[8] Vasilevskii S. A., Tirskii G. A., Utyuzhnikov S. V., “Chislennyi metod resheniya uravnenii vyazkogo udarnogo sloya”, Zhurn. vychisl. matem. i mat. fiziki, 27:5 (1987), 741–750

[9] Golovachev Yu. P., Timofeev E. V., Chislennoe issledovanie sverkhzvukovogo obtekaniya tel vyazkim gazom s pomoschyu metoda globalnykh iteratsii, Preprint No 1254, RAN. FTI im. A. F. Ioffe, L., 1988

[10] Tolstykh A. I., Kompaktnye raznostnye skhemy i ikh primenenie v zadachakh aerogidrodinamiki, Nauka, M., 1990 | MR

[11] Polyanskii A. F., Skurin L. I., “K metodike rascheta s pomoschyu globalnykh iteratsii vyazkogo udarnogo sloya”, Vestn. SPbGU Ser. 1, 1993, no. 4, 71–75

[12] Arkhangelskaya L. A., Skurin L. I., “Raschet vnutrennego techeniya zhidkosti s otryvom metodom globalnykh iteratsii”, Vestn. SPbGU Ser. 1, 1993, no. 2, 78–83

[13] Kovalev V. L., Krupnov A. A., Tirskii G. A., “Metod globalnykh iteratsii resheniya zadach sverkhzvukovogo obtekaniya zatuplennykh tel idealnym gazom”, DAN, 339:3 (1994), 342–345 | Zbl

[14] Arkhangelskaya L. A., Skurin L. I., “Ispolzovanie metoda globalnykh iteratsii po davleniyu dlya resheniya uravnenii Nave–Stoksa”, Vestn. SPbGU Ser. 1, 1994, no. 3, 70–75

[15] Arkhangelskaya L. A., Skurin L. I., “Primenenie metoda globalnykh iteratsii po davleniyu dlya resheniya sistemy uravnenii Nave–Stoksa”, Vychislitelnye tekhnologii. RAN Sib. otd-nie. — Novosibirsk, 4:12 (1995), 29–37

[16] Skurin L. I., “Issledovanie skhodimosti metoda globalnykh iteratsii po davleniyu dlya dvukh- i trekhmernykh zadach gidrodinamiki”, Vestn. SPbGU Ser. 1, 1995, no. 4, 80–84 | Zbl

[17] Arkhangelskaya L. A., Skurin L. I., “Osobennosti techeniya zhidkosti v slede v usloviyakh vozdeistviya elektromagnitnykh sil na pristennyi sloi zhidkosti”, Vestn. SPbGU. Ser. 1, 1996, no. 2, 57–61

[18] Polyanskii A. F., Skurin L. I., “Matematicheskoe modelirovanie otryvnykh techenii v zakruchennykh potokakh zhidkosti”, Vestn. SPbGU Ser. 1, 1996, no. 2, 77–82

[19] Skurin L. I., “O vozmozhnosti resheniya nestatsionarnykh zadach mekhaniki zhidkosti i gaza s ispolzovaniem globalnykh iteratsii po davleniyu”, Vestn. SPbGU Ser. 1, 1997, no. 3, 78–84 | Zbl

[20] Polyanskii A. F., Skurin L. I., “Raschetnoe issledovanie kharakteristik vikhrevogo generatora zvuka”, Akust. zhurn., 43:6 (1997), 835–839.

[21] Polyanskii A. F., Skurin L. I., “Issledovanie zakruchennykh i otryvnykh techenii zhidkosti i gaza”, Tez. dokl. 17 vseross. seminara “Techeniya gaza i plazmy v soplakh, struyakh i sledakh” (SPb, 1997), 57

[22] Skurin L. I., “Metod matematicheskogo modelirovaniya techenii zhidkosti, gaza i plazmy”, Tez. dokl. 17 vseross. seminara “Techeniya gaza i plazmy v soplakh, struyakh i sledakh” (SPb, 1997), 60

[23] Polyanskii A. F., Skurin L. I., “Iteratsionno-marshevyi metod integrirovaniya sistemy uravnenii Nave–Stoksa dlya gaza”, Vestn. SPbGU Ser. 1, 1998, no. 1, 87–92

[24] Godunov S. K., Uravneniya matematicheskoi fiziki, Nauka, M., 1971 | MR

[25] Kovenya V. M., Yanenko N. N., Metod rasschepleniya v zadachakh gazovoi dinamiki, Nauka, Novosibirsk, 1981 | MR | Zbl

[26] Godunov S. K., Ryabenkii B. C., Raznostnye skhemy, Nauka, M., 1977

[27] Kovenya V. M., Tarnavskii G. A., Chernyi S. G., Primenenie metoda rasschepleniya v zadachakh aerodinamiki, Nauka, Novosibirsk, 1990

[28] Fletcher K., Vychislitelnye metody v dinamike zhidkostei, v. 1, Mir, M., 1991

[29] Fletcher K., Vychislitelnye metody v dinamike zhidkostei, v. 2, Mir, M., 1991

[30] Nartov V. P., Chernykh G. G., O chislennom modelirovanii techeniya, voznikayuschego pri kollapse zony smesheniya v stratifitsirovannoi srede, Preprint No 15-82, RAN. Sib. otd-nie. ITPM, Novosibirsk, 1982

[31] Guschin V. A., “Metod rasschepleniya dlya zadach dinamiki neodnorodnoi vyazkoi neszhimaemoi zhidkosti”, Zhurn. vychisl. matem. i mat. fiziki, 21:4 (1981), 1003–1017 | MR

[32] Miln-Tomson L. M., Teoreticheskaya gidrodinamika, Mir, M., 1964

[33] Laitkhill Dzh., Volny v zhidkostyakh, Mir, M., 1981 | MR

[34] Aslanov T. D., Byrkin A. P., Schennikov V. V., “Chislennyi raschet vnutrennikh techenii vyazkogo gaza s ispolzovaniem uravnenii Nave–Stoksa”, Uch. zap. TsAGI, 3:3 (1981), 44–54

[35] Tanaki T., Tomida I., and Yamane R., “A study of pseudo-schok (1-st report)”, Bull. ISME, 13:55 (1970), 47–58