Geodesic
On reduction of computational cost of imitation Monte Carlo algorithms for modeling rarefied gas flows
Matematičeskoe modelirovanie, Tome 23 (2011) no. 9, pp. 65-88.

Voir la notice de l'article provenant de la source Math-Net.Ru

Publication describes Monte Carlo methods and algorithms for Boltzmann equation for rarefied gases problems in case of large-scale flow areas. We consider imitation or Continuous Time Monte Carlo methods where frequencies of interactions of particles’ pairs depend on difference of particles’ coordinates. The question about reduction computational costs of algorithms is examined using specificity of the problem. First, algorithms of an approximated method are constructed, analyzed and realized. This method is obtained using splitting (over groups of particles) of operator in master equations system. In the second place, we investigate fictitious collisions technique, where the upper bound for the number of interacting pairs is specified. Plane Poiseuille flow (in the field of external forces) problem, Heat transfer problem and Temperature discontinuity propagation problem are numerically solved using developed algorithms. Asymptotical estimates of the computational costs are confirmed with the data of the computational processes and comparative properties of the last one are fixed. Suggested algorithms of the method with splitting allow parallelization of the certain type.
Keywords: statistical modeling, Continuous Time Monte Carlo methods for Boltzmann equation, fictitious collisions technique, approximated method obtained using of splitting over groups of particles, reduction of computational cost. 
@article{MM_2011_23_9_a5,
     author = {A. I. Khisamutdinov and N. N. Velker},
     title = {On reduction of computational cost of imitation {Monte} {Carlo} algorithms for modeling rarefied gas flows},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {65--88},
     publisher = {mathdoc},
     volume = {23},
     number = {9},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2011_23_9_a5/}
}
TY  - JOUR
AU  - A. I. Khisamutdinov
AU  - N. N. Velker
TI  - On reduction of computational cost of imitation Monte Carlo algorithms for modeling rarefied gas flows
JO  - Matematičeskoe modelirovanie
PY  - 2011
SP  - 65
EP  - 88
VL  - 23
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2011_23_9_a5/
LA  - ru
ID  - MM_2011_23_9_a5
ER  - 
%0 Journal Article
%A A. I. Khisamutdinov
%A N. N. Velker
%T On reduction of computational cost of imitation Monte Carlo algorithms for modeling rarefied gas flows
%J Matematičeskoe modelirovanie
%D 2011
%P 65-88
%V 23
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2011_23_9_a5/
%G ru
%F MM_2011_23_9_a5
A. I. Khisamutdinov; N. N. Velker. On reduction of computational cost of imitation Monte Carlo algorithms for modeling rarefied gas flows. Matematičeskoe modelirovanie, Tome 23 (2011) no. 9, pp. 65-88. http://geodesic.mathdoc.fr/item/MM_2011_23_9_a5/

[1] Khisamutdinov A.I., Ob imitatsionnom metode Monte-Karlo dlya modelirovaniya dinamiki razrezhennykh gazov, preprint No 599, VTs SO AN SSSR, Novosibirsk, 1985, 17 pp. | MR

[2] Khisamutdinov A.I., “Imitatsionnoe statisticheskoe modelirovanie kineticheskogo uravneniya razrezhennykh gazov”, Dokl. AN SSSR, 302:1 (1988), 75–79 | MR | Zbl

[3] Khisamutdinov A. I., “Algoritmy s “raznovremennymi koordinatami” metodov Monte-Karlo dlya nelineinogo "sglazhennogo uravneniya Boltsmana”, Dokl. AN SSSR, 316 (1991), 289–293

[4] Khisamutdinov A. I., “On development of Continuous Time Monte Carlo methods for problems of Boltzmann equation with external forces”, Transport Theory and Statistical Physics, 33:1 (2004), 69–89 | DOI | MR | Zbl

[5] Khisamutdinov A. I., “On some properties of Markov processes and Monte Carlo methods for inhomogeneous Boltzmann equation”, Russian Journal of Numerical Analysis and Mathematical Modelling, 20:2 (2005), 131–160 | DOI | MR | Zbl

[6] Bird G. A., Molecular gas dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994 | MR

[7] Khisamutdinov A. I., “Priblizhennyi metod statisticheskogo modelirovaniya, ispolzuyuschii slabuyu zavisimost podchastei”, Voprosy korrektnosti zadach analiza, IM SO AN SSSR, Novosibirsk, 1989, 154–160 | MR

[8] Khisamutdinov A. I., “On connection between “Continuous time” and “Direct simulation” Monte Carlo methods for Boltzmann equation and on some new approximate methods”, Monte Carlo Methods and Applications, VSP, 6:4 (2000), 323–340 | DOI | MR | Zbl

[9] Ermakov S., Metod Monte-Karlo i smezhnye voprosy, Nauka, M., 1971, 327 pp. | MR | Zbl

[10] Koura K., “Null-collision technique in the direct-simulation Monte Carlo method”, Phys. Fluids, 29:11 (1986), 3509–3511 | DOI

[11] Khisamutdinov A. I., Sidorenko L. L., Ob algoritmakh metodov Monte-Karlo s nepreryvnym vremenem dlya kineticheskogo uravneniya razrezhennykh gazov, prepr. No 35, IM SO AN SSSR, Novosibirsk, 1989

[12] Khisamutdinov A. I., Sidorenko L. L. i dr., O trudoemkostyakh algoritmov metodov Monte-Karlo s nepreryvnym vremenem dlya uravneniya Boltsmana, prepr. No 21, IM SO AN SSSR, Novosibirsk, 1991 | Zbl

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

[14] Khisamutdinov A. I., “Statisticheskoe modelirovanie odnogo tipa par sluchainykh velichin s ispolzovaniem fiktivnykh skachkov”, ZhVM i MF, 47:1 (2007), 162–173 | MR

[15] Mansour Malek M., Baras F., Garcia Alejandro L., “On the validity of hydrodynamics in plane Poiseuille flows”, Physica, A240 (1997), 255–267

[16] Aoki Kazuo, Sone Yoshio, Nishino Kenji, Sugimoto Hiroshi, “Numerical Analysis of unsteady motion of a rarefied gas caused by sudden changes of wall temperature with special interest in the propagation of a discontinuity in the velocity distribution function”, Rarefied Gas Dynamics, ed. A. E. Beylich, VCH, Weinheim, 1991, 222–231