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 | References | Similar Articles | Additional Information

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 -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.

**[1]**H. Neunzert and J. Struckmeier,*Particle methods for the Boltzmann equation*, Acta Numerica, 1995, Cambridge University Press, Cambridge, 1995, pp. 417-457 MR**1352475****[2]**V. V. Aristov and F. G. Tcheremissine,*The conservative splitting method for solving Boltzmann's equation*, USSR Comp. Math. Phys.**21**, 208-225 (1980)**[3]**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**[4]**C. Buet,*A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics*, Transport Theory Statist. Phys.**25**, 33-60 (1996) MR**1380030****[5]**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) MR**1472201****[6]**F. Rogier and J. Schneider,*A direct method for solving the Boltzmann equation*, Transport Theory Statist. Phys.**23**, 313-338 (1994) MR**1257657****[7]**F. G. Tcheremissine,*Conservative discrete ordinates method for solving Boltzmann kinetic equation*, Comm. Appl. Math. (1997). In press.**[8]**L. Preziosi and E. Longo,*On a conservative polar discretization of the Boltzmann equation*, Japan J. Indust. Appl. Math.**14**, 399-435 (1997) MR**1475141****[9]**R. Temam,*Sur la stabilité et la convergence de la method des pas fractionaires*, Ann. Math. Pura Appl.**79**, 191-380 (1968) MR**0241838****[10]**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) MR**1428146****[11]**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) MR**1014927**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1724301

Article copyright:
© Copyright 1999
American Mathematical Society