On Gas Dynamic Hierarchy

  • S. V. BogomolovEmail author
  • N. B. Esikova
  • A. E. Kuvshinnikov
  • P. N. Smirnov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


On the example of a simple and clear, but far from being trivial, model of hard sphere gas, we will try to show the main stages in constructing the mathematical formalization of a complex physical system.

We are considering a set of about \({10}^{25}\) solid balls that just fly and collide. A mathematical description of the evolution of such a system inevitably leads to the necessity of using the apparatus of the theory of random processes. To identify the mathematical and computational features of the problem under study it is important to write it in a dimensionless form. This procedure leads to the appearance of the Knudsen number, the physical meaning of which is the ratio of the mean free path of molecules to the characteristic size of the problem. The hierarchy of micro-macro models is constructed in accordance with the change in this parameter from values of the order of unity (micro) to magnitudes of the order of 0.1 (meso) and further to 0.01 (macro). Accurate movement along this path leads to more accurate mathematical models, in comparison with traditional ones, which affects their greater computational fitness - nature pays for a careful attitude towards it. In particular, obtained macroscopic equations are softer for simulations than the classical Navier-Stokes equations.

This hierarchy of mathematical statements generates a corresponding chain of computational methods. Microscopic problems are most often solved using Monte Carlo methods, although there are research groups that are committed to nonrandom methods for solving the Boltzmann equation. Recently, much attention has been paid to mesomodels based on modeling the Brownian motion or solving the deterministic Fokker - Planck - Kolmogorov equations. To solve the problems of a continuous medium, difference methods, finite element methods, and particle methods are used. The latter ones, in our opinion, are particularly promising for the entire hierarchy, uniting different statements with a single computational ideology. A discontinuous particle method is particularly effective.


Boltzmann equation Kolmogorov – Fokker – Planck equation Navier – Stokes equation Random processes Stochastic differential equations with respect to Poisson and Wiener measures Discontinuous particle method 


  1. 1.
    Boltzmann, L.: Weitere Studien über das Wärme gleichgenicht unfer Gasmoläkuler. Sitzungsberichte der Akademie der Wissenschaften 66, 275–370 (1872)Google Scholar
  2. 2.
    Jun, E., Hossein Gorji, M., Grabe, M., Hannemann, K.: Assessment of the cubic Fokker-Planck-DSMC hybrid method for hypersonic rarefied flows past a cylinder. Comput. Fluids 168, 1–13 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zhang, J., Zeng, D., Fan, J.: Analysis of transport properties determined by Langevin dynamics using Green-Kubo formulae. Physica A: Stat. Mech. Appl. 411, 104–112 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gupta, V.K., Torrilhon, M.: Comparison of relaxation phenomena in binary gas-mixtures of Maxwell molecules and hard spheres. Comput. Math. Appl. 70, 73–88 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogomolov, S.V.: An approach to deriving stochastic gas dynamics models. Doklady Math. 78, 929–931 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Arsen’yev, A.A.: On the approximation of the solution of the Boltzmann equation by solutions of the ito stochastic differential equations. USSR Comput. Math. Math. Phys. 27, 51–59 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bogomolov, S.V., Dorodnitsyn, L.V.: Equations of stochastic quasi-gas dynamics: viscous gas case. Math. Models Comput. Simul. 3, 457–467 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Skorokhod, A.V.: Stochastic Equations for Complex Systems. Kluwer Academic, Dordrecht (1987)CrossRefGoogle Scholar
  9. 9.
    Bogomolov, S.V., Gudich, I.G.: Diffusion model of gas in a phase space for moderate Knudsen numbers. Math. Models Comput. Simul. 5, 130–144 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bogomolov, S.V., Gudich, I.G.: Verification of a stochastic diffusion gas model. Math. Models Comput. Simul. 6, 305–316 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chetverushkin, B.N.: Resolution limits of continuous media mode and their mathematical formulations. Math. Models Comput. Simul. 5, 266–279 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Elizarova, T.G.: Quasi-Gas Dynamic Equations. Springer, Heidelberg (2009). Scholar
  13. 13.
    Bogomolov, S.V., Esikova, N.B., Kuvshinnikov, A.E.: Micro-macro Kolmogorov-Fokker-Planck models for a rigid-sphere gas. Math. Models Comput. Simul. 8(5), 533–547 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bogomolov, S.V., Esikova, N.B., Kuvshinnikov, A.E.: Meso - Macro models for a hard sphere gas. In: Proceedings of the ECCOMAS Congress, Crete Island, Greece (2016)Google Scholar
  15. 15.
    Mathiaud, J., Mieussens, L.: A Fokker-Planck model of the Boltzmann equation with correct Prandtl number. J. Stat. Phys. 162, 397–414 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Morinishi, K.: A continuum/kinetic hybrid approach for multi-scale flow. In: Proceedings of the ECCOMAS CFD, Egmond aan Zee, Netherlands (2006)Google Scholar
  17. 17.
    Aringazin, A.K., Mazhintov, M.I.: Stochastic models of Lagrangian acceleration of fluid particle in developed turbulence. Int. J. Mod. Phys. B 18, 3095–3168 (2004)CrossRefGoogle Scholar
  18. 18.
    Oksendal, B.: Stochastic Differental Equations, 6th edn. Springer, Heidelberg (2000). Scholar
  19. 19.
    Stepanov, S.S.: Stochastic World. Springer, Switzerland (2013). Scholar
  20. 20.
    Dadzie, S.K., Reese, J.M.: Spatial stochasticity and non-continuum effects in gas flows. Phys. Lett. A 376, 967–972 (2012)CrossRefGoogle Scholar
  21. 21.
    Bayev, A.Z., Bogomolov, S.V.: On the stability of the discontinuous particle method for the transfer equation. Math. Models Comput. Simul. 10(2), 186–197 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • S. V. Bogomolov
    • 1
    Email author
  • N. B. Esikova
    • 1
  • A. E. Kuvshinnikov
    • 1
  • P. N. Smirnov
    • 1
  1. 1.M.V. Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations