BGK Polyatomic Model for Rarefied Flows

  • Florian BernardEmail author
  • Angelo Iollo
  • Gabriella Puppo


In this work we present a new model of BGK type for polyatomic gases. The model incorporates the different relaxation rates of translational, rotational and/or vibrational modes characterizing polyatomic molecules using a BGK-type equation, and additional relaxation equations for the temperatures associated to each internal energy mode. We construct an efficient numerical scheme which is implicit in the relaxation terms, and test the model and the scheme on several problems, confirming the Asymptotic Preserving properties of the scheme, and comparing the results provided by the model with experimental and DSMC simulations, carried out on the full Boltzmann polyatomic equation.


Kinetic models Polyatomic gas Multi-temperature BGK model AP schemes 



Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil Régional d’Aquitaine, FeDER, Université de Bordeaux and CNRS (see This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the “Investments for the future” Programme IdEx Bordeaux (ANR-10-IDEX-03-02), Cluster of excellence CPU, and partly by the Italian State under GNCS-INDAM 2015 project “Metodi numerici per la quantificazione dell’incertezza in equazioni iperboliche e cinetiche”.


  1. 1.
    Alsmeyer, H.: Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74, 497–513 (1976)CrossRefGoogle Scholar
  2. 2.
    Andries, P., Bourgat, J.-F., Le Tallec, P., Perthame, B.: Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Comput. Methods Appl. Mech. Eng. 191(31), 3369–3390 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andries, P., Le Tallec, P., Perlat, J.-P., Perthame, B.: The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B. Fluids 19(6), 813–830 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Andries, P., Perthame, B.: The ES-BGK model equation with correct Prandtl number. In: 22nd International Symposium on Rarefied Gas Dynamics. AIP Conference Proceeding vol. 585, p. 30, (2001)Google Scholar
  5. 5.
    Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. IMACS J. 25(2–3), 151–167 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bernard, F.: Efficient asymptotic preserving schemes for BGK and ES-BGK models on cartesian grids. Politecnico di Torino, Université de Bordeaux, Theses (2015)Google Scholar
  7. 7.
    Bernard, F., Iollo, A., Puppo, G.: A local velocity grid approach for BGK equation. Commun. Comput. Phys. 16(4), 956–982 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bernard, F., Iollo, A., Puppo, G.: Accurate asymptotic preserving boundary conditions for kinetic equations on cartesian grids. J. Sci. Comput. 65, 735–766 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)CrossRefGoogle Scholar
  10. 10.
    Borgnakke, C., Larsen, P.S.: Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. J. Comput. Phys. 18, 405–420 (1975)CrossRefGoogle Scholar
  11. 11.
    Brull, S., Schneider, J.: On the ellipsoidal statistical model for polyatomic gases. Contin. Mech. Thermodyn. 20(8), 489–508 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cabannes, H., Gatignol, R., Luol, L.-S.: The discrete Boltzmann equation. In: Lecture Notes at University of California, Berkley, pp. 1–65 (1980)Google Scholar
  13. 13.
    Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, Berlin (1988)CrossRefGoogle Scholar
  14. 14.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. No. v. 106 in Applied Mathematical Sciences Series. Springer, Berlin (1994)CrossRefGoogle Scholar
  15. 15.
    Chu, C.K.: Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8, 12–22 (1965)CrossRefGoogle Scholar
  16. 16.
    Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Riv. di Mat. della Univ. di Parma Ser. 73(2), 177–216 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Liu, C., Xu, K., Sun, Q., Cai, Q.: A unified gas-kinetic scheme for continuum and rarefied flows IV: full Boltzmann and model equations. J. Comput. Phys. 314, 305–340 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, S., Yu, P., Xu, K., Zhong, C.: Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes. J. Comput. Phys. 259(C), 96–113 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mieussens, L.: Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys. 162(2), 429–466 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25(1), 129–155 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pieraccini, S., Puppo, G.: Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput. 32(1), 1–28 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pieraccini, S., Puppo, G.: Microscopically implicit-macroscopically explicit schemes for the BGK equation. J. Comput. Phys. 231, 299–327 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rykov, V.: A model kinetic equation for a gas with rotational degrees of freedom. Fluid Dyn. 6(10), 956–966 (1975)Google Scholar
  24. 24.
    Semyonov, Y., Borisov, P., Suetin, P.: Investigation of heat transfer in rarefied gases over a wide range of Knudsen number. Int. J. Heat Mass Transf. 10, 1789–1799 (1984)CrossRefGoogle Scholar
  25. 25.
    Shapiro, A.: The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronald Press, New York (1953)Google Scholar
  26. 26.
    Tantos, C., Frezzotti, A., Valougeorgis, D., Morini, G.L.: Conductive heat transfer in a rarefied polyatomic gas confined between coaxial cylinders. Int. J. Heat and Mass Transf. 79, 378–389 (1997)CrossRefGoogle Scholar
  27. 27.
    Xu, K., Xin, H., Chunpei, C.: Multiple temperature kinetic model and gas-kinetic method for hypersonic non equilibrium flow computations. J. Comput. Phys. 227, 6779–6794 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Florian Bernard
    • 1
    • 2
    Email author
  • Angelo Iollo
    • 1
    • 2
  • Gabriella Puppo
    • 3
  1. 1.University of Bordeaux, IMBTalenceFrance
  2. 2.INRIA, Team MEMPHISTalenceFrance
  3. 3.Università dell’InsubriaComoItaly

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