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On a simulation scheme for the Boltzmann equation. (English) Zbl 0609.76084

The author investigates a Monte Carlo scheme derived by K. Nanbu [Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases, J. Phys. Soc. Japan 49, 2042-2049 (1980] which simulates solutions of the Boltzmann equation by N-point systems. This scheme may be interpreted as a method to generate paths of a stochastic process associated to a (piecewise linear) transport equation \(f_ t=J(f,f^{\Delta})\). It is shown that by rearranging certain random variables the computation effort can be reduced from \(O(N^ 2)\) to O(N). The fluctuations of the kinetic energy of the N-point system are estimated.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
65C05 Monte Carlo methods
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References:

[1] Bird, Molecular Gas Dynamics (1976)
[2] Deshpande , S. M. An Unbiased and Consistent Monte Carlo Game Simulating the Boltzmann Equation. Indian Inst. Science 1978
[3] Koura, Comment on ”Direct Simulation Scheme Derived from the Boltzmann Equation”, J. Phys. Soc. Jpn. 50 pp 3829– (1981) · doi:10.1143/JPSJ.50.3829
[4] Nanbu, Direct Simulation Scheme Derived from the Boltzmann Equation. I. Monocomponent Gases, J. Phys. Soc. Jpn. 49 pp 2042– (1980) · doi:10.1143/JPSJ.49.2042
[5] Nanbu, Reply to a Comment by Koura on ”Direct Simulation Scheme Derived from the Boltzmann Equation”, J. Phys. Soc. Jpn. 50 pp 3831– (1981) · doi:10.1143/JPSJ.50.3831
[6] Papanicolaou, Asymptotic Analysis of Transport Processes, Bull. Amer. Math. Soc. 81 pp 330– (1975) · Zbl 0361.60056 · doi:10.1090/S0002-9904-1975-13744-X
[7] Ploss , H.
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