Advertisement

Journal of Statistical Physics

, Volume 132, Issue 1, pp 153–170 | Cite as

Construction of Discrete Kinetic Models with Given Invariants

  • A. V. Bobylev
  • M. C. Vinerean
Article

Abstract

We consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by Gatignol in connection with discrete models of the Boltzmann equation (BE) and it has been addressed in the last decade by several authors. Even though a practical criterion for the non-existence of spurious conservation laws has been devised, and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed, a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d≥2 and for any sufficiently large number N of velocities (for example, N≥6 for the planar case d=2) there exists just a finite number of distinct classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of single gases involved in the mixture, we also obtain normal DVMs).

Keywords

Boltzmann equation Discrete kinetic models Conservation laws Collision invariants 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arkeryd, L.: On the Boltzmann equation, part II: the full initial value problem. Arch. Ration. Mech. Anal. 45, 17–34 (1972) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arkeryd, L., Cercignani, C.: On a functional equation arising in the kinetic theory of gases. Rend. Mat. Acc. Lincei 1, 139–149 (1990) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bobylev, A., Palczewski, A., Schneider, J.: On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris Sér. I Math. 320, 639–644 (1995) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bobylev, A.V., Cercignani, C.: Discrete velocity models for mixtures. J. Stat. Phys. 91, 327–342 (1998) CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bobylev, A.V., Cercignani, C.: Discrete velocity models without non-physical invariants. J. Stat. Phys. 97, 677–686 (1999) CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Cabannes, H.: The discrete Boltzmann equation. Lecture notes given at the University of California at Berkeley (1980), revised jointly with R. Gatignol and L.-S. Luo, 2003 Google Scholar
  7. 7.
    Carleman, T.: Problème Mathématiques dans la Théorie Cinétique des Gaz. Almqvist-Wiksell, Uppsala (1957) Google Scholar
  8. 8.
    Cercignani, C.: Sur des critère d’existence globale en théorie cinétique discrète. C. R. Acad. Sci. Paris 301, 89–92 (1985) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cercignani, C., Cornille, H.: Shock waves for discrete velocity gas mixture. J. Stat. Phys. 99, 115–140 (2000) CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cercignani, C., Cornille, H.: Large size planar discrete velocity models for gas mixtures. J. Phys. A: Math. Gen. 34, 2985–2998 (2001) CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Cornille, H., Cercignani, C.: A class of planar discrete velocity models for gas mixtures. J. Stat. Phys. 99, 967–991 (2000) CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Cornille, H., Cercignani, C.: On a class of planar discrete velocity models for gas mixtures. In: Ciancio, V., Donato, A., Oliveri, F., Rionero, S. (eds.) Proceedings “WASCOM-99” 10th Conference on Waves and Stability in Continuous Media. World Scientific, Singapore (2001) Google Scholar
  13. 13.
    Gatignol, R.: Théorie Cinétique des Gaz à Répartition Discrète de Vitesses. Springer, New York (1975) Google Scholar
  14. 14.
    Palczewski, A., Schneider, J., Bobylev, A.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM J. Numer. Anal. 34, 1865–1883 (1997) CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Platkowski, T., Illner, R.: Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory. SIAM Rev. 30, 213–255 (1988) CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Vedenyapin, V.V.: Velocity inductive construction for mixtures. Transport. Theory Stat. Phys. 28, 727–742 (1999) CrossRefzbMATHGoogle Scholar
  17. 17.
    Vedenyapin, V.V., Orlov, Yu.N.: Conservation laws for polynomial Hamiltonians and for discrete models for Boltzmann equation. Teor. Math. Phys. 121, 1516–1523 (1999) CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Bobylev, A.V., Vinerean, M.C.: Construction and classification of discrete kinetic models without spurious invariants. Riv. Mat. Univ. Parma 7, 1–80 (2007) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsKarlstad UniversityKarlstadSweden

Personalised recommendations