Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

A quasi-Monte Carlo method for the Boltzmann equation


Author: Christian Lécot
Journal: Math. Comp. 56 (1991), 621-644
MSC: Primary 65C05; Secondary 76M25, 76P05, 82C40
MathSciNet review: 1068812
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Abstract: A new quasi-Monte Carlo method for solving the Boltzmann equation in a simplified case is described. The analysis is restricted to a spatially homogeneous and isotropic gas; in addition, the molecular model only involves isotropic scattering. The scheme makes use of particles and combines an Euler scheme with numerical integrations. The sequence which is used for the quadratures must possess some symmetry properties which prescribe energy conservation for colliding particles. The error of the method is estimated by means of the discrepancy of the sequence which performs the quadratures. An algorithm for generating convenient sequences is proposed. In an example, where an exact solution is known, the computation of effective errors is included.


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  • [1] H. Babovsky, F. Gropengiesser, H. Neunzert, J. Struckmeier, and B. Wiesen, Low discrepancy methods for the Boltzmann equation, 16th Internat. Sympos. on Rarefied Gas Dynamics, Pasadena, Ca., July 1988.
  • [2] G. A. Bird, Molecular gas dynamics, Clarendon Press, Oxford, 1976.
  • [3] -, Monte Carlo simulation in an engineering context, 12th Internat. Sympos. on Rarefied Gas Dynamics, Charlottesville, Va., July 1980.
  • [4] Carlo Cercignani, Theory and application of the Boltzmann equation, Elsevier, New York, 1975. MR 0406273
  • [5] Henri Faure, Discrépance de suites associées à un système de numération (en dimension 𝑠), Acta Arith. 41 (1982), no. 4, 337–351 (French). MR 677547
  • [6] J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 84–90. MR 0121961
  • [7] Edmund Hlawka, The theory of uniform distribution, A B Academic Publishers, Berkhamsted, 1984. With a foreword by S. K. Zaremba; Translated from the German by Henry Orde. MR 750652
  • [8] M. Krook and T. T. Wu, Exact solutions of the Boltzmann equation, Phys. Fluids 20 (1977), 1589-1595.
  • [9] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • [10] C. Lécot, A direct simulation Monte Carlo scheme and uniformly distributed sequences for solving the Boltzmann equation, Computing 41 (1989), no. 1-2, 41–57 (English, with German summary). MR 981669, 10.1007/BF02238728
  • [11] Christian Lécot, Low discrepancy sequences for solving the Boltzmann equation, J. Comput. Appl. Math. 25 (1989), no. 2, 237–249. MR 988058, 10.1016/0377-0427(89)90049-6
  • [12] -, An algorithm for generating low discrepancy sequences on vector computers, Parallel Comput. 11 (1989), 113-116.
  • [13] The NAG Fortran Library Manual-Mark 12, The Numerical Algorithms Group Limited, Oxford, 1987.
  • [14] K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. I, Monocomponent gases, J. Phys. Soc. Japan 49 (1980), 2042-2049.
  • [15] Heiner Niederreiter and Jörg M. Wills, Diskrepanz und Distanz von Maßen bezüglich konvexer und Jordanscher Mengen, Math. Z. 144 (1975), no. 2, 125–134 (German). MR 0376588
  • [16] Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, 10.1090/S0002-9904-1978-14532-7
  • [17] Harald Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987), no. 4, 273–337. MR 918037, 10.1007/BF01294651
  • [18] W. G. Vincenti and C. H. Kruger, Introduction to physical gas dynamics, Krieger, Malabar, 1986.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1991-1068812-4
Article copyright: © Copyright 1991 American Mathematical Society