Monte Carlo methods in applied mathematics and computational aerodynamics

O. M. Belotserkovskii   ; Yu. I. Khlopkov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 8, pp. 1494-1518

Résumé

A survey of the Monte Carlo methods developed in the computational aerodynamics of rarefied gases is given, and application of these methods in unconventional fields is described. A short history of these methods is presented, and their advantages and drawbacks are discussed. A relationship of the direct statistical simulation of aerodynamical processes with the solution of kinetic equations is established; it is shown that the modern stage of the development of computational methods is impossible without the use of the complex approach to the development of algorithms with regard for all the specific features of the problem to be solved (its physical nature, mathematical model, the theory of computational mathematics, and stochastic processes). Possible directions of the development of the statistical simulation methods are discussed.

Bibliographie

[1] Hall A., “On an experiment determination of $\pi$”, Messeng. Math., 2 (1873), 113–114

[2] Metropolis N., Ulam S., “The Monte-Carlo method”, J. Amer. Stat. Assoc., 44:247 (1949), 335–341 | DOI | MR | Zbl

[3] Vladimirov B. C., Sobol I. M., “Raschet naimenshego kharakteristicheskogo chisla uravneniya Paierlsa metodom Monte-Karlo”, Vychisl. matem. i matem. fiz., 3, Izd-vo AN SSSR, M., 1958, 130–137 | MR

[4] Haviland J. K., Lavin M. D., “Application of the Monte-Carlo method to heat transfer in rarefied gases”, Phys. Fluids, 5:1 (1962), 279–286 | DOI | MR

[5] Bird G. A., “Shock-wave structure in rigid sphere gas”, Rarefied Gas Dynamics, v. 1, Acad. Press, New York, 1965, 25–31

[6] Ermakov S. M., Metod Monte-Karlo i smezhnye voprosy, Nauka, M., 1985 | MR

[7] Kats M., Veroyatnost i smezhnye voprosy v fizike, Mir, M., 1965 | Zbl

[8] Bhathnagar P. L., Gross E. P., Krook M., “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems”, Phys. Rev., 94:3 (1954), 511–525 | DOI

[9] Khlopkov Yu. I., Shakhov E. M., “Kineticheskie modeli i ikh rol v issledovanii techenii razrezhennogo gaza”, Chisl. metody v dinamike razrezhennogo gaza, 6, no. 3, VTs AN SSSR, M., 1972, 34–80

[10] Yanitskii V. E., “Statisticheskaya model techeniya idealnogo gaza i nekotorye ee osobennosti”, Chisl. metody mekhan. sploshnoi sredy, 6, no. 4, SO AN SSSR, Novosibirsk, 1975, 139–150

[11] Belotserkovskii O. M., Yanitskii V. E., “Statisticheskii metod chastits v yacheikakh dlya resheniya zadach dinamiki razrezhennogo gaza. Ch. I. Osnovy postroeniya metoda”, Zh. vychisl. matem. i matem. fiz., 15:5 (1975), 1195–1208 | MR

[12] Belotserkovskii O. M., “Vychislitelnyi eksperiment: pryamoe chislennoe modelirovanie slozhnykh techenii gazovoi dinamiki na osnove uravnenii Eilera, Nave–Stoksa i Boltsmana”, “Karmanovskaya lektsiya”, In-t dinamiki zhidkosti im. Karmana, Bryussel, 1976

[13] Belotserkovskii O. M., Yanitskii V. E., “Chislennye metody v dinamike razrezhennogo gaza”, Tr. IV Vses. konf. po dinamike razrezhennogo gaza, TsAGI, M., 1977, 101–183

[14] Belotserkovskii O. M., Yanitskii V. E., “Statisticheskii metod chastits v yacheikakh dlya resheniya zadach dinamiki razrezhennogo gaza. Ch. II. Vychislitelnye aspekty metoda”, Zh. vychisl. matem. i matem. fiz., 15:6 (1975), 1553–1567 | MR

[15] Belotserkovskii O. M., Yanitskii V. E., “Problemy chislennogo modelirovaniya techenii razrezhennogo gaza”, Uspekhi mekhan., 1:1/2 (1978), 69–112 | MR

[16] Belotserkovskii O. M., Chislennoe modelirovanie v mekhanike sploshnykh sred, Nauka, M., 1984 | MR

[17] Vlasov V. I., “Raschet obtekaniya plastiny pod uglom ataki potokom razrezhennogo gaza”, Uch. zap. TsAGI, 4:1 (1973), 17–24 | MR

[18] Gorelov S. L., Erofeev A. I., “Vliyanie vnutrennikh stepenei svobody na obtekanie plastiny giperzvukovym potokom razrezhennogo gaza”, Izv. AN SSSR. Mekhan. zhidkosti i gaza, 1979, no. 6, 151–156

[19] Belotserkovskii O. M., Erofeev A. I., Yanitskii V. E., “O nestatsionarnom metode pryamogo statisticheskogo modelirovaniya techenii razrezhennogo gaza”, Zh. vychisl. matem. i matem. fiz., 20:5 (1975), 1174–1204 | MR

[20] Belotserkovskii O. M., Erofeev A. I., Yanitskii V. E., Pryamoe statisticheskoe modelirovanie zadach aerodinamiki, VTs AN SSSR, M., 1983

[21] Belotserkovsky O. M., Yerofeev A. I., Yanitsky V. E., “Direct Monte-Carlo simulation of aerohydrodynamics problems”, Proc. XIII Internat. Symp. on RGD. V. 1, Plenum Press, New York, 1985, 313–332

[22] Khlopkov Yu. I., “Reshenie linearizovannogo uravneniya Boltsmana”, Zh. vychisl. matem. i matem. fiz., 13:5 (1973), 1307–1314

[23] Khlopkov Yu. L., Kravchuk A. S., “Simulation of rarefied gas flow and of a continuum”, Proc. XVII Internat. Symp. on RGD. V. 1, RWTH, Aachen, 1990, 332

[24] Kogan M. N., Kravchuk A. C., Khlopkov Yu. M., “Metod “relaksatsiya-perenos” dlya resheniya zadach dinamiki gaza v shirokom diapazone razrezhennosti sredy”, Uch. zap. TsAGI, 19:2 (1988), 106–109

[25] Gorelov S. L., Zharov V. A., Khlopkov Yu. L., “The Kinetic approaches to the turbulence description”, Proc. XX Internat. Symp. on RGD, Beijing, China, 1997, 192–199

[26] Rutgers M. A., “Forced 2D turbulence: experimental evidence of simultaneous inverse energy and forward enstophy cascades”, Phys. Rev. Letts, 81 (1998), 2244–2247 | DOI

[27] Sakurai A., Takayama F., “Molecular kinetic approach to the problem of compressible turbulence”, Phys. Fluids, 15 (2003), 1282–1294 | DOI

[28] Khlopkov Yu. I., Voronich I. V., Rovenskaya O. I., “Chislennoe modelirovanie nelineinykh effektov v szhimaemom gaze na osnove uravnenii Nave–Stoksa i kineticheskogo uravneniya”, Tezisy. II Mezhdunar. nauchno-tekhn. konf. “Aviadvigateli XXI veka”, TsIAM, M., 2005, 146–149

[29] Chetverushkin B. N., Kineticheski-soglasovannye raznostnye skhemy v gazovoi dinamike, Izd-vo MGU, M., 1999