Semi-Lagrangian Approximation of BGK Models for Inert and Reactive Gas Mixtures

  • M. GroppiEmail author
  • G. Russo
  • G. Stracquadanio
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 258)


Recent relaxation time-approximation models of BGK-type for the kinetic description of both inert and reacting gas mixtures are reviewed and their main properties are recalled. The models are characterized by only one Maxwellian attractor for each species; such attractors are defined in terms of auxiliary parameters. For their numerical approximation, semi-Lagrangian schemes are proposed. Numerical simulations are presented with the aim of showing the peculiarities of the different BGK models and the performance of the numerical method.



This work was supported by MIUR, by the National Group of Mathematical Physics (GNFM-INdAM), by the National Group sof Scientific Computing (GNCS-INdAM), and by the Universities of Catania and Parma (Italy).


  1. 1.
    Aimi, A., Diligenti, M., Groppi, M., Guardasoni, C.: On the numerical solution of a BGK-type model for chemical reactions. Eur. J. Mech. B/Fluids 26, 455–472 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106, 993–1018 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhatnagar, P.L., Gross, E.P., Krook, K.: A model for collision processes in gases. Phys. Rev. 94, 511–525 (1954)CrossRefGoogle Scholar
  4. 4.
    Bisi, M., Groppi, M., Spiga, G.: Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit. Phys. Rev. E 81(036327), 1–9 (2010)Google Scholar
  5. 5.
    Bobylev, A.V., Bisi, M., Groppi, M., Spiga, G., Potapenko, I.F.: A general consistent BGK model for gas mixtures. Kinet. Relat. Models 11, 1377–1393 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Carlini, E., Ferretti, R., Russo, G.: A weighted essentially nonoscillatory, large time-step scheme for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 27, 1071–1091 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cercignani, C.: The Boltzmann Equation and its Applications. Springer, New York (1988)CrossRefGoogle Scholar
  8. 8.
    Chu, C.K.: Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8, 12–21 (1965)CrossRefGoogle Scholar
  9. 9.
    Dimarco, G., Pareschi, L.: Implicit-explicit linear multistep methods for stiff kinetic equations. SIAM J. Numer. Anal. 55(2), 664–690 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garzó, V., Santos, A., Brey, J.J.: A kinetic model for a multicomponent gas. Phys. Fluids A 1(2), 380–383 (1989)CrossRefGoogle Scholar
  11. 11.
    Gross, E.P., Krook, M.: Model for collision processes in gases: small-amplitude oscillations of charged two-component systems. Phys. Rev 102, 593 (1956)CrossRefGoogle Scholar
  12. 12.
    Groppi, M., Lichtenberger, P., Schürrer, F., Spiga, G.: Conservative approximation schemes of kinetic equations for chemical reactions. Eur. J. Mech. B Fluids 27(2), 202–217 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Groppi, M., Rjasanow, S., Spiga, G.: A kinetic relaxation approach to fast reactive mixtures: shock wave structure. J. Stat. Mech. Theory Exp. 10(P10010), 1–15 (2009)Google Scholar
  14. 14.
    Groppi, M., Russo, G., Stracquadanio, G.: High order semilagrangian methods for BGK models. Commun. Math. Sci. 14(2), 389–414 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Groppi, M., Russo, G., Stracquadanio, G.: Boundary conditions for semi-Lagrangian methods for the BGK model. Commun. Appl. Ind. Math. 7(3), 135–161 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Groppi, M., Spiga, G.: A Bhatnagar-Gross-Krook type approach for chemically reacting gas mixtures. Phys. Fluids 16, 4273–4284 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hairer, E., Warner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1996)Google Scholar
  18. 18.
    Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. Math. Mod. Numer. Anal. 33(3), 547–571 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mieussens, L.: Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Meth. Appl. Sci. 10(8), 1121–1149 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Pareschi, L., Russo, G.: Efficient Asymptotic Preserving Deterministic Methods for the Boltzmann Equation, AVT-194 RTO AVT/VKI. Models and Computational Methods for Rarefied Flows, Lecture Series held at the von Karman Institute, pp. 24–28. Rhode St. Genèse, Belgium (2011)Google Scholar
  22. 22.
    Pieraccini, S., Puppo, G.: Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput. 32, 1–28 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rossani, A., Spiga, G.: A note on the kinetic theory of chemically reacting gases. Phys. A 272, 563 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Russo, G., Santagati, P., Yun, S.-B.: Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation. SIAM J. Numer. Anal. 50, 1111–1135 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sivorich, L.: Kinetic modeling of gas mixtures. Phys. Fluids 5, 908–918 (1962)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Welander, P.: On the temperature jump in a rarefied gas. Ark. Fys. 7, 507–553 (1954)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical, Physical and Computer SciencesUniversity of ParmaParmaItaly
  2. 2.Department of Mathematics and Computer SciencesUniversity of CataniaCataniaItaly

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