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A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation. (English) Zbl 0899.76312

Summary: Bird’s direct simulation Monte Carlo method for the Boltzmann equation is considered. The limit (as the number of particles tends to infinity) of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation. A Markov jump process is introduced, which is related to Bird’s collision simulation procedure via a random time transformation. Convergence is established for the Markov process and the random time transformation. These results, together with some general properties concerning the convergence of random measures, make it possible to characterize the limiting behavior of the Bird algorithm.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
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[1] A. A. Arsen’yev, On the approximation of the solution of the Boltzmann equation by solutions of the Ito stochastic differential equations,USSR Comput. Math. Math. Phys. 27(2):51-59 (1987). · Zbl 0664.76104
[2] H. Babovsky, A convergence proof for Nanbu’s Boltzmann simulation scheme,Eur. J. Mech. B Fluids 8(1):41-55 (1989). · Zbl 0669.76096
[3] H. Babovsky and R. Illner, A convergence proof for Nanbu’s simulation method for the full Boltzmann equation,SIAM J. Numer. Anal. 26(1):45-65 (1989). · Zbl 0668.76086
[4] P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968). · Zbl 0172.21201
[5] G. A. Bird, Direct simulation and the Boltzmann equation,Phys. Fluids 13(10):2676-2681 (1970). · Zbl 0227.76111
[6] G. A. Bird,Molecular Gas Dynamics (Clarendon Press, Oxford, 1976).
[7] C. Cercignani,Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975). · Zbl 0403.76065
[8] C. Cercignani,The Boltzmann Equation and its Applications (Springer, New York, 1988). · Zbl 0646.76001
[9] M. H. A. Davis,Lectures on Stochastic Control and Nonlinear Filtering (Tata Institute of Fundamental Research, Bombay, 1984).
[10] S. M. Ermakov, V. V. Nekrutkin, and A. S. Sipin,Random Processes for Classical Equations of Mathematical Physics (Nauka, Moscow, 1984) [in Russian] (English translation, Kluwer Academic Publishers, Dordrecht, 1989). · Zbl 0604.60082
[11] S. N. Ethier and T. G. Kurtz,Markov Processes, Characterization and Convergence (Wiley, New York, 1986). · Zbl 0592.60049
[12] W. Feller,An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York, 1966). · Zbl 0138.10207
[13] W. Greenberg, J. Polewczak, and P. F. Zweifel, Global existence proofs for the Boltzmann equation, inNonlinear Phenomena. I. The Boltzmann Equation, J. L. Lebowitz and E. W. Montroll, eds. (North-Holland, Amsterdam, 1983), pp. 19-49.
[14] F. Gropengiesser, H. Neunzert, and J. Struckmeier, Computational methods for the Boltzmann equation, inApplied and Industrial Mathematics, R. Spigler, ed. (Kluwer Academic Publishers, Dordrecht, 1991), pp. 111-140.
[15] P. L. Hennequin and A. Tortra,Probability Theory and Some Applications (Nauka, Moscow, 1974) [Russian translation].
[16] R. Illner and H. Neunzert, On simulation methods for the Boltzmann equation,Transport Theory Stat. Phys. 16(2&3):141-154 (1987). · Zbl 0623.76084
[17] K. Koura, Null-collision technique in the direct-simulation Monte Carlo method,Phys. Fluids 29(11):3509-3511 (1986).
[18] J. L. Lebowitz and E. W. Montroll (eds.),Nonequilibrium Phenomena. I. The Boltzmann Equation (North-Holland, Amsterdam, 1983). · Zbl 0583.76004
[19] A. V. Lukshin and S. N. Smirnov, An efficient stochastic algorithm for solving the Boltzmann equation,USSR Comput. Math. Math. Phys. 29(1):83-87 (1989). · Zbl 0702.76091
[20] K. Nanbu, Interrelations between various direct simulation methods for solving the Boltzmann equation,J. Phys. Soc. Japan 52(10):3382-3388 (1983).
[21] H. Ploss, On simulation methods for solving the Boltzmann equation,Computing 38:101-115 (1987). · Zbl 0613.65140
[22] A. V. Skorokhod,Stochastic Equations for Complex Systems (Nauka, Moscow, 1983) [in Russian] (English translation, Reidel, Dordrecht, 1988). · Zbl 0513.60053
[23] S. N. Smirnov, Justification of a stochastic method for solving the Boltzmann equation,USSR Comput. Math. Math. Phys. 29(1):187-192 (1989). · Zbl 0702.76093
[24] A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated,J. Funct. Anal. 56:311-336 (1984). · Zbl 0547.60080
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