A stochastic particle numerical method for 3D Boltzmann equations without cutoff
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- by Nicolas Fournier and Sylvie Méléard PDF
- Math. Comp. 71 (2002), 583-604 Request permission
Abstract:
Using the main ideas of Tanaka, the measure-solution $\{P_t\}_t$ of a $3$-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to $\{P_t\}_t$ of solutions $\{P^l_t\}_t$ of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate $\{P^l_t\}_t$ with the empirical measure $\{\mu ^{l,n}_t\}_t$ of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.References
- Hans Babovsky and Reinhard Illner, A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal. 26 (1989), no. 1, 45–65. MR 977948, DOI 10.1137/0726004
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620, DOI 10.1007/978-1-4419-8524-8
- Laurent Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys. 168 (1995), no. 2, 417–440. MR 1324404
- Laurent Desvillettes, Carl Graham, and Sylvie Méléard, Probabilistic interpretation and numerical approximation of a Kac equation without cutoff, Stochastic Process. Appl. 84 (1999), no. 1, 115–135. MR 1720101, DOI 10.1016/S0304-4149(99)00056-3
- Fournier, N.: Calcul des variations stochastiques sur l’espace de Poisson, applications à des E.D.P.S. paraboliques avec sauts et à certaines équations de Boltzmann. Thèse de l’Université Paris 6, Chapitre 4, (1999).
- Carl Graham and Sylvie Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab. 25 (1997), no. 1, 115–132. MR 1428502, DOI 10.1214/aop/1024404281
- Carl Graham and Sylvie Méléard, Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations, Comm. Math. Phys. 205 (1999), no. 3, 551–569. MR 1711273, DOI 10.1007/s002200050689
- J. Horowitz and R. L. Karandikar, Martingale problems associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989) Progr. Probab., vol. 18, Birkhäuser Boston, Boston, MA, 1990, pp. 75–122. MR 1042343
- Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133, DOI 10.1007/978-3-662-02514-7
- Sylvie Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1627, Springer, Berlin, 1996, pp. 42–95. MR 1431299, DOI 10.1007/BFb0093177
- Sylvie Meleard, Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations, Stochastics Stochastics Rep. 63 (1998), no. 3-4, 195–225. MR 1658082, DOI 10.1080/17442509808834148
- Nanbu, K.: Interrelations between various direct simulation methods for solving the Boltzmann equation, J. Phys. Soc. Japan 52, 3382-3388 (1983).
- Tokuzo Shiga and Hiroshi Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 3, 439–459. MR 787607, DOI 10.1007/BF00532743
- Alain-Sol Sznitman, Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. Verw. Gebiete 66 (1984), no. 4, 559–592 (French, with English summary). MR 753814, DOI 10.1007/BF00531891
- Hiroshi Tanaka, On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) Wiley, New York-Chichester-Brisbane, 1978, pp. 409–425. MR 536022
- Hiroshi Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 1, 67–105. MR 512334, DOI 10.1007/BF00535689
- G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys. 94 (1999), no. 3-4, 619–637. MR 1675367, DOI 10.1023/A:1004589506756
- Wolfgang Wagner, A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation, J. Statist. Phys. 66 (1992), no. 3-4, 1011–1044. MR 1151989, DOI 10.1007/BF01055714
Additional Information
- Nicolas Fournier
- Affiliation: Institut Elie Cartan, Faculté de Sciences, B.P. 239, 54506 Vandoeune-les-Nancy Cedex, France
- Email: Nicolas.Fournier@antares.iecn.u-nancy.fr
- Sylvie Méléard
- Affiliation: Laboratoire de Probabilités, Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France; and MODALX, UFR SEGMI, Université Paris 10, 92000 Nanterre, France
- Email: sylm@ccr.jussieu.fr
- Received by editor(s): March 14, 2000
- Published electronically: October 25, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Math. Comp. 71 (2002), 583-604
- MSC (2000): Primary 60J75, 60H10, 60K35, 82C40
- DOI: https://doi.org/10.1090/S0025-5718-01-01339-4
- MathSciNet review: 1885616