Conservative energy discretization of Boltzmann collision operator
Authors:
L. Preziosi and L. Rondoni
Journal:
Quart. Appl. Math. 57 (1999), 699-721
MSC:
Primary 76P05; Secondary 82C40
DOI:
https://doi.org/10.1090/qam/1724301
MathSciNet review:
MR1724301
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Abstract: The paper deduces a kinetic model obtained introducing a discretization of the Boltzmann equation based on an equispaced distribution of allowed particle energies. The model obtained is a system of integro-differential equations with integration over suitable angular variables: one over the portion of the unit sphere between two parallels symmetric with respect to the equatorial plane perpendicular to the velocity of the field particle, and one over a unit circle. The model preserves mass, momentum and energy. Furthermore, there exists an $H$-functional describing trend toward an equilibrium state described by a Gaussian distribution. Particular attention is paid to the identification of a criterion which indicates the values of the discretization parameters.
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H. Neunzert and J. Struckmeier, Particle methods for the Boltzmann equation, Acta Numerica, 1995, Cambridge University Press, Cambridge, 1995, pp. 417–457
V. V. Aristov and F. G. Tcheremissine, The conservative splitting method for solving Boltzmann’s equation, USSR Comp. Math. Phys. 21, 208–225 (1980)
C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas-dynamics, in Proceedings of 19th Rarefied Gas Dynamics Symposium, J. Harvey and G. Lord, eds., Oxford University Press, 1994, pp. 878–884
C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics, Transport Theory Statist. Phys. 25, 33–60 (1996)
A. V. Bobylev, A. Palczewski, and J. Schneider, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal. 34, 1865–1883 (1997)
F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation, Transport Theory Statist. Phys. 23, 313–338 (1994)
F. G. Tcheremissine, Conservative discrete ordinates method for solving Boltzmann kinetic equation, Comm. Appl. Math. (1997). In press.
L. Preziosi and E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. Indust. Appl. Math. 14, 399–435 (1997)
R. Temam, Sur la stabilité et la convergence de la method des pas fractionaires, Ann. Math. Pura Appl. 79, 191–380 (1968)
L. Desvillettes and S. Mischler, About the splitting algorithm for Boltzmann and B.G.K. equations, Math. Models Methods Appl. Sci. 6, 1079–1101 (1996)
R. J. Di Perna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. 130, 321–366 (1989)
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© Copyright 1999
American Mathematical Society