Estimation of the statistical error of the direct statistical modeling method

M. Yu. Plotnikov   ; E. V. Shkarupa

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 2, pp. 352-361

Résumé

The statistical error of the direct simulation Monte Carlo method for numerical solution of the rarefied gas dynamics problems is investigated. Based on the central limit theorem for Markov processes, asymptotic confidence intervals for the errors connected with the number of time steps are obtained for estimates of the three main macroparameters of the flow (density, velocity, and temperature). For the quantities involved in the expressions for the confidence intervals, practical recommendations are given concerning their numerical evaluation simultaneously with the calculation of the flow macroparameters. The proposed approaches to constructing the confidence intervals are illustrated using the classical problem of heat transfer between two infinite parallel plates as an example.

Bibliographie

[1] Bird G. A., Molecular gas dynamics and the direct simulation of gas flows, Clarendon Press, Oxford, 1994 | MR

[2] Ivanov M. S., Rogazinskii S. V., Metod pryamogo statisticheskogo modelirovaniya v dinamike razrezhennogo gaza, VTs SO RAN, Novosibirsk, 1988

[3] Gallis M. A., Torczynski J. R., Rader D. J., Bird G. A., “Convergence behavior of a new DSMC algorithm”, J. Comput. Phys., 228 (2009), 4532–4548 | DOI | Zbl

[4] Rudyak V. Ya., “Kineticheskaya teoriya i sovremennaya aerogidromekhanika”, Sibirskii zhurnal industr. matem., 8:3 (23) (2009), 120–148 | MR

[5] Wagner W., “Monte Carlo methods and numerical solutions”, Proc. 24-th Internat. Symp. Rarefied Gas Dynamics, AIP Conf. Proc., 762, Melville, New York, 2005, 459–466

[6] Yanitskii V. E., “Teoretiko-veroyatnostnyi analiz pryamogo statisticheskogo modelirovaniya stolknovitelnykh protsessov v razrezhennom gaze”, Dop. 1: Berd G., Molekulyarnaya gazovaya dinamika, Mir, M., 1981, 279–302

[7] Bogomolov S. V., “Skhodimost metoda summarnoi approksimatsii dlya uravneniya Boltsmana”, Zh. vychisl. matem. i matem. fiz., 28:1 (1988), 119–126 | MR

[8] Ivanov M. S., Rogazinskii S. V., “Sravnitelnyi analiz algoritmov metoda pryamogo statisticheskogo modelirovaniya v dinamike razrezhennogo gaza”, Zh. vychisl. matem. i matem. fiz., 28:7 (1988), 1058–1070 | MR | Zbl

[9] Lukshin An. V., Smirnov S. N., “Ob odnom effektivnom stokhasticheskom algoritme resheniya uravneniya Boltsmana”, Zh. vychisl. matem. i matem. fiz., 29:1 (1989), 118–124 | MR | Zbl

[10] Nekrutkin V. V., Tur N. I., “K obosnovaniyu skhemy pryamogo modelirovaniya techenii razrezhennykh gazov”, Zh. vychisl. matem. i matem. fiz., 29:9 (1989), 1380–1392 | MR | Zbl

[11] Ivanov M. S., Rudyak V. Ya., “Osnovnoe kineticheskoe uravnenie i metod pryamogo statisticheskogo modelirovaniya”, Matem. modelirovanie, 1:7 (1989), 93–99 | Zbl

[12] Ivanov M. S., Rogazinskii S. V., “Ekonomichnye skhemy statisticheskogo modelirovaniya techenii razrezhennogo gaza”, Matem. modelirovanie, 1:7 (1989), 130–145 | MR | Zbl

[13] Lukshin A. V., “Stokhasticheskie algoritmy matematicheskoi teorii prostranstvenno-neodnorodnogo uravneniya Boltsmana”, Matem. modelirovanie, 1:7 (1989), 146–159 | MR | Zbl

[14] Lukshin A. V., Yuferov I. E., “Effektivnye algoritmy resheniya uravneniya Boltsmana dlya mnogokomponentnoi reagiruyuschei smesi”, Matem. modelirovanie, 3:6 (1991), 84–92 | MR | Zbl

[15] Korotchenko M. A., Mikhailov G. A., Rogazinskii S. V., “Modifikatsii vesovykh algoritmov metoda Monte-Karlo dlya resheniya nelineinykh kineticheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 47:12 (2007), 2110–2121 | MR

[16] Rogasinsky S. V., Levin D. A., Ivanov M. S., “Statistical errors of DSMC results for rarefied gas flow”, Proc. 25-th Internat. Symp. on Rarefied Gas Dynamics, Publish House of the Siberian Branch of Russian Academy of Sciences, Novosibirsk, 2007, 391–395

[17] Plotnikov M. Yu., Shkarupa E. V., “Application of the theory of functional Monte-Carlo algorithms to optimization of the DSMC method”, Proc. 26-th Internat. Symp. on Rarefied Gas Dynamics, Conf. Proc., 1084, Melville, New York, 2009, 377–382

[18] Ibragimov I. A., “Zamechanie o tsentralnoi predelnoi teoreme dlya zavisimykh velichin”, Teoriya veroyatnostei i ee primeneniya, 20:1 (1975), 134–139

[19] Doob J. L., Stochaztic processes, John Wiley Sons, New York; Chapmen Hall, London, 1953 | MR | MR

[20] Nagaev S. V., “Nekotorye predelnye teoremy dlya odnorodnykh tsepei Markova”, Teoriya veroyatnostei i ee primeneniya, 2:4 (1957), 389–416 | MR

[21] Ibragimov I. A., Linnik Yu. V., Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965

[22] Kramer G., Matematicheskie metody statistiki, Mir, M., 1975 | MR

[23] Ermakov S. M., Mikhailov G. A., Statisticheskoe modelirovanie, Nauka, M., 1982 | MR

[24] Rotar V. I., Teoriya veroyatnostei, Vyssh. shkola, M., 1992 | MR | Zbl

[25] Yaglom A. M., Korrelyatsionnaya teoriya statsionarnykh sluchainykh funktsii s primerami iz meteorologii, Gidrometeoizdat, L., 1981

[26] Gribanov Yu. I., Malkov V. L., Spektralnyi analiz sluchainykh protsessov, Energiya, M., 1974 | MR