Accurate numerical methods for the Boltzmann equation

  • Francis Filbet
  • Giovanni Russo
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. On the one hand, the transport is solved by a third-order accurate (in space) Positive and Flux Conservative (PFC) method. On the other hand, the collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with high-order integrators in time preserving stationary states. Several space dependent numerical tests in 2D and 3D illustrate the accuracy and robustness of the methods.


Boltzmann Equation Knudsen Number Riemann Problem Collision Operator Kernel Mode 
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  1. 1.
    Babovsky, H.: On a simulation scheme for the Boltzmann equation. Mathematical Methods in the Applied Sciences 8,223–233 (1986)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bird, G.A.: Molecular gas dynamics. Clarendon Press, Oxford (1994)Google Scholar
  3. 3.
    Bobylev, A.V. and Rjasanow, S.: Difference scheme for the Boltzmann equation based on the Fast Fourier Transform. Eur. J. Mech. B/Fluids 16,293–306 (1997)MathSciNetMATHGoogle Scholar
  4. 4.
    Buet, C: A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics. Trans. Theo. Stat. Phys. 25,33–60 (1996)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Canuto, C, Hussaim, M.Y., Quarteroni, A. and Zang, TA.: Spectral methods in fluid dynamics. Springer Verlag, New York (1988)MATHCrossRefGoogle Scholar
  6. 6.
    Cercignani, C: The Boltzmann equation and its applications. Springer-Verlag, Berlin (1988)MATHCrossRefGoogle Scholar
  7. 7.
    Filbet, F. and Pareschi, L.: A numerical method for the accurate solution of the Landau- Fokker-Planck equation in the non homogeneous case. J. Comput. Phys. 179,1–26 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Filbet, F, Sonnendrücker, E. and Bertrand, P.: Conservative Numerical schemes for the Vlasov equation. J. Comput. Phys. 172,166–187 (2001)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Filbet, F. and Sonnendrücker, E.: Comparison of Eulerian Vlasov solvers. Comput. Phys. Communications 151, 247–266 (2003)CrossRefGoogle Scholar
  10. 10.
    Filbet, F. and Russo, G.: High order numerical methods for the space non homogeneous Boltzmann equation. J. Comput. Phys. 186,457–480 (2003)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Goldstein, D., Sturtevant, B. and Broadwell, J.E.: Investigation of the motion of discrete velocity gases. Rar. Gas. Dynam., Progress in Astronautics and Aeronautics 118 AIAA, Washington (1989)Google Scholar
  12. 12.
    Inamuro, T. and Sturtevant B.: Numerical study of discrete velocity gases. Phys. Fluids A 12,2196–2203(1990)CrossRefGoogle Scholar
  13. 13.
    Jin, S.: Runge-Kutta methods for hyperbolic systems with stiff relaxation terms. J. Comput. Phys., 122,51–67(1995)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Nanbu, K.: Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent Gases. J. Phys. Soc. Japan 52, 2042–2049 (1983)Google Scholar
  15. 15.
    Nessyahu, H. and Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87,408–463 (1990)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ohwada, T: Higher Order Approximation Methods for the Boltzmann Equation. J. Comput. Phys. 139, 1–14 (1998)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Pareschi, L: Computational methods and fast algorithms for Boltzmann equations, Chapter 7, Lecture Notes on the discretization of the Boltzmann equation, ed. N.Bellomo, World Scientific, 46 pp. (to appear)Google Scholar
  18. 18.
    Pareschi, L. and Perthame, B.: A Fourier spectral method for homogeneous Boltzmann equations. Transp. Theo. Stat. Phys. 25, 369–383 (1996)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Pareschi, L. and Russo, G.: On the stability of spectral methods for the homogeneous Boltzmann equation. Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998). Transport Theory Statist. Phys. 29,431–447(2000)Google Scholar
  20. 20.
    Pareschi, L. and Russo, G.: Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37, 1217–1245 (2000)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Pareschi, L. and Russo, G.: Time Relaxed Monte Carlo methods for the Boltzmann equation. SIAM J. Sci. Comp. 23,1253–1273 (2001)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Pareschi, L.; Toscani, G. and Villani, C: Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93, no. 3, 527–548 (2003)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rogier, F. and Schneider, J.: A direct method for solving the Boltzmann equation. Trans. Theo. Stat. Phys. 23, 313–338 (1994)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, Springer 1697, 325–432 (1998)Google Scholar
  25. 25.
    Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Villani, C: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics 1,71–305, North-Holland, Amsterdam (2002)CrossRefGoogle Scholar
  27. 27.
    Whitham, G. B.: Linear and nonlinear waves. Wiley Interscience (1974)Google Scholar
  28. 28.
    Wild, E: On Boltzmann’s equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc. 47, 602–609, (1951)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Francis Filbet
    • 1
  • Giovanni Russo
    • 2
  1. 1.Mathématiques et Applications, Physique Mathématique d’Orléans (MAPMO)CNRS-Université d’OrléansOrléansFrance
  2. 2.Università di CataniaCataniaItalia

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