Abstract
Two convergence results related to the approximation of the Boltzmann equation by discrete velocity models are presented. First we construct a sequence of deterministic discrete velocity models and prove convergence (as the number of discrete velocities tends to infinity) of their solutions to the solution of a spatially homogeneous Boltzmann equation. Second we introduce a sequence of Markov jump processes (interpreted as random discrete velocity models) and prove convergence (as the intensity of jumps tends to infinity) of these processes to the solution of a deterministic discrete velocity model.
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References
R. O. Barrachina, Wild's solution of the nonlinear boltzmann equation,J. Stat. Phys. 52(1/2):357–368 (1988).
J.-M. Bony, Existence globale à données de Cauchy petites pour les modèles discrets de l'équation de Boltzmann,Commun. Partial Differential Equations 16(4&5):533–545 (1991).
C. Cercignani,The Boltzmann Equation and its Applications (Springer, New York, 1988).
S. N. Ethier and T. G. Kurtz,Markov Processes, Characterization and Convergence (Wiley, New York, 1986).
R. Illner and W. Wagner, A random discrete velocity model and approximation of the Boltzmann equation,J. Stat. Phys. 70(3/4):773–792 (1993).
J. L. Lebowitz and E. W. Montroll, eds.,Nonequilibrium Phenomena. I. The Boltzmann Equation (North-Holland, Amsterdam, 1983).
Y. L. Martin, F. Rogier, and J. Schneider, A deterministic method for solving the nonhomogeneous Boltzmann equation,C. R. Acad. Sci. Paris I Math. 314(6):483–487 (1992).
H. P. McKean, Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas,Arch. Rat. Mech. Anal. 21:343–367 (1966).
N. A. Nurlybaev, Discrete velocity method in the theory of kinetic equations,Transport Theory Stat. Phys. 22(1):109–119 (1993).
T. Płatkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory,SIAM Rev. 30(2):213–255 (1988).
A. S. Sznitman, Équations de type de Boltzmann, spatialement homogènes,Z. Wahrsch. Verw. Geb. 66(4):559–592 (1984).
H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,Z. Wahrsch. Verw. Geb. 46(1):67–105 (1978).
W. Wagner, A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation,J. Stat. Phys. 66(3/4):1011–1044 (1992).
E. Wild, On Boltzmann's equation in the kinetic theory of gases,Proc. Camb. Phil. Soc. 47:602–609 (1951).
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Wagner, W. Approximation of the Boltzmann equation by discrete velocity models. J Stat Phys 78, 1555–1570 (1995). https://doi.org/10.1007/BF02180142
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DOI: https://doi.org/10.1007/BF02180142