Skip to main content
SpringerLink
Log in

Solution to the Boltzmann kinetic equation for high-speed flows

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The Boltzmann kinetic equation is solved by a finite-difference method on a fixed coordinate-velocity grid. The projection method is applied that was developed previously by the author for evaluating the Boltzmann collision integral. The method ensures that the mass, momentum, and energy conservation laws are strictly satisfied and that the collision integral vanishes in thermodynamic equilibrium. The last property prevents the emergence of the numerical error when the collision integral of the principal part of the solution is evaluated outside Knudsen layers or shock waves, which considerably improves the accuracy and efficiency of the method. The differential part is approximated by a second-order accurate explicit conservative scheme. The resulting system of difference equations is solved by applying symmetric splitting into collision relaxation and free molecular flow. The steady-state solution is found by the relaxation method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. G. Cheremisin, “Conservative Method for Evaluating the Boltzmann Collision Integral,” Dokl. Akad. Nauk 357, 53–56 (1997) [Dokl. Phys. 42, 607–610 (1997)].

    MATH  MathSciNet  Google Scholar 

  2. F. G. Cheremisin, “Solving the Boltzmann Equation in the Case of Passing to the Hydrodynamic Flow Regime,” Dokl. Akad. Nauk 373, 483–486 (2000) [Dokl. Phys. 45, 401–404 (2000)].

    MathSciNet  Google Scholar 

  3. F. G. Tcheremissine, “Conservative Evaluation of Boltzmann Collision Integral in Discrete Ordinates Approximation,” Comput. Math. Appl. 35(1/2), 215–221 (1998).

    MATH  MathSciNet  Google Scholar 

  4. S. P. Popov and F. G. Cheremisin, “A Conservative Method for Solving the Boltzmann Equation with Centrally Symmetric Interaction Potentials,” Zh. Vychisl. Mat. Mat. Fiz. 39, 163–176 (1999) [Comput. Math. Math. Phys. 39, 156–169 (1999)].

    MathSciNet  Google Scholar 

  5. S. P. Popov and F. G. Cheremisin, “Example of Simultaneous Numerical Solution of the Boltzmann and Navier-Stokes Equations,” Zh. Vychisl. Mat. Mat. Fiz. 41, 489–500 (2001) [Comput. Math. Math. Phys. 41, 457–468 (2001)].

    MathSciNet  Google Scholar 

  6. S. P. Popov and F. G. Tcheremissine, “Supersonic Rarefied Gas Flow past a Lattice of Plane Transverse Plates,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 167–176 (2002).

  7. S. P. Popov and F. G. Tcheremissine, “Dynamics of the Interaction of a Shock Wave with a Lattice in Rarefied Gas,” Aerodin. Gaz. Din., No. 3, 31–38 (2003).

  8. S. P. Popov and F. G. Tcheremissine, “A Method of Joint Solution of the Boltzmann and Navier-Stokes Equations,” 24th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. 762 (Melville, New York, 2005), pp. 82–87.

    Google Scholar 

  9. F. Tcheremissine, “Direct Numerical Solution of the Boltzmann Equation,” 24th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. 762 (Melville, New York, 2005), pp. 667–685.

    Google Scholar 

  10. F. G. Cheremisin, “Solution of the Wang-Chang-Uhlenbeck Master Equation,” Dokl. Akad. Nauk 387, 1–4 (2002) [Dokl. Phys. 47, 872–875 (2002)].

    Google Scholar 

  11. V. V. Aristov and F. G. Tcheremissine, Direct Numerical Solution of the Boltzmann Equation (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 1992) [in Russian].

    Google Scholar 

  12. G. N. Patterson, Molecular Flow of Gases (Willey, New York, 1956; Fizmatlit, Moscow, 1960).

    Google Scholar 

  13. F. Rogier and J. Schnider, “A Direct Method for Solving the Boltzmann Equation,” Transport Theory Stat. Phys. 23(1–3) (1994).

  14. N. M. Korobov, Trigonometric Sums and Applications (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

  15. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge Univ. Press, Cambridge, 1952; Inostrannaya Literatura, Moscow, 1960).

    Google Scholar 

  16. J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972; Mir, Moscow, 1976).

    Google Scholar 

  17. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Willey, New York, 1954; Inostrannaya Literatura, Moscow, 1961).

    Google Scholar 

  18. J. P. Boris and D. L. Book, “Flux-Corrected Transport, 1.SHASTA: A Fluid Transport Algorithm That Works,” J. Comput. Phys. 11(1), 38–69 (1973).

    Article  Google Scholar 

  19. G. Strang, “On the Construction and Comparison of Difference Schemes,” SIAM J. Numer. Anal., No. 5, 506–517 (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

    F. G. Tcheremissine

Authors
  1. F. G. Tcheremissine
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © F.G. Tcheremissine, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 2, pp. 329–343.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tcheremissine, F.G. Solution to the Boltzmann kinetic equation for high-speed flows. Comput. Math. and Math. Phys. 46, 315–329 (2006). https://doi.org/10.1134/S0965542506020138

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542506020138

Keywords

Search

Navigation