Skip to main content
SpringerLink
Log in

Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations

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

Abstract

The multidimensional quasi-gasdynamic system written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization of these equations on a nonuniform rectangular grid is constructed (with the basic unknown functions—density, velocity, and temperature—defined on a common grid and with fluxes and viscous stresses defined on staggered grids). Primary attention is given to the analysis of entropy behavior: the discretization is specially constructed so that the total entropy does not decrease. This is achieved via a substantial revision of the standard discretization and applying numerous original features. A simplification of the constructed discretization serves as a conservative discretization with nondecreasing total entropy for the simpler quasi-hydrodynamic system of equations. In the absence of regularizing terms, the results also hold for the Navier–Stokes equations of a viscous compressible heat-conducting gas.

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. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 2009).

    Book  MATH  Google Scholar 

  2. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Chapman and Hall/CRC, London, 2001; Fizmatlit, Moscow, 2012).

    MATH  Google Scholar 

  3. B. N. Chetverushkin, Kinetic Schemes and Quasi-Gas Dynamic System of Equations (MAKS, Moscow, 2004; CIMNE, Barcelona, 2008).

    MATH  Google Scholar 

  4. T. G. Elizarova, Quasi-Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer-Verlag, Berlin, 2009).

    MATH  Google Scholar 

  5. Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (NITs Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2009) [in Russian].

    Google Scholar 

  6. A. A. Zlotnik and B. N. Chetverushkin, “Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them,” Comput. Math. Math. Phys. 48 (3), 420–446 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. A. Zlotnik, “Quasi-gasdynamic system of equations with general equations of state,” Dokl. Math. 81 (2), 312–316 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. A. Zlotnik, “On the quasi-gasdynamic system of equations with general equations of state and a heat source,” Mat. Model. 22 (7), 53–64 (2010).

    MathSciNet  MATH  Google Scholar 

  9. A. A. Zlotnik, “Linearized stability of equilibrium solutions to the quasi-gasdynamic system of equations,” Dokl. Math. 82 (2), 811–815 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  10. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).

    MATH  Google Scholar 

  11. S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauchnaya Kniga, Novosibirsk, 1998; Kluwer, New York, 2003).

    Book  MATH  Google Scholar 

  12. S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids (Nauka, Novosibirsk, 1983; North Holland, Amsterdam, 1990).

    MATH  Google Scholar 

  13. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer, Berlin, 2010).

    Book  MATH  Google Scholar 

  14. E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford Univ. Press, Oxford, 2004).

    MATH  Google Scholar 

  15. E. Tadmor, “Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems,” Acta Numerica 12, 451–512 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  16. G. P. Prokopov, “Necessity of entropy control in gasdynamic computations,” Comput. Math. Math. Phys. 47 (9), 1528–1537 (2007).

    Article  MathSciNet  Google Scholar 

  17. U. S. Fjordholm, S. Mishra, and E. Tadmor, “Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws,” SIAM J. Numer. Anal. 50, 544–573 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  18. S. K. Godunov and I. M. Kulikov, “Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee,” Comput. Math. Math. Phys. 54 (6), 1012–1024 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Svärd and H. Özcan, “Entropy stable schemes for the Euler equations with far-field and wall boundary conditions,” J. Sci. Comput. 58 (1), 61–89 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Lv and M. Ihme, “Entropy-bounded discontinuous Galerkin scheme for Euler equations,” J. Comput. Phys. 295, 715–739 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. N. Mohammed and F. Ismail, “Entropy consistent methods for the Navier–Stokes equations,” J. Sci. Comput. 63, 612–631 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. R. Winters and G. J. Gassner, “Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations,” Comput. Phys. 304, 72–108 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. A. Amosov and A. A. Zlotnik, “A study of finite-difference method for the one-dimensional viscous heat conductive gas flow equation: Part I. A priori estimates and stability,” Sov. J. Numer. Anal. Math. Model. 2 (3), 159–178 (1987).

    MATH  Google Scholar 

  24. A. A. Amosov and A. A. Zlotnik, “A difference scheme on a nonuniform mesh for the equations of one-dimensional magnetic gas dynamics,” Comput. Math. Math. Phys. 29 (2), 129–139 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  25. A. A. Amosov and A. A. Zlotnik, “A finite-difference scheme for quasi-averaged equations of one-dimensional viscous heat-conducting gas flow with nonsmooth data,” Comput. Math. Math. Phys. 39 (4), 564–583 (1999).

    MathSciNet  MATH  Google Scholar 

  26. A. A. Zlotnik, “Spatial discretization of one-dimensional quasi-gasdynamic systems of equations and the entropy and energy balance equations,” Dokl. Math. 86 (1), 464–468 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  27. A. A. Zlotnik, “Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation,” Comput. Math. Math. Phys. 52 (7), 1060–1071 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. A. Zlotnik, “Spatial discretization of the one-dimensional barotropic quasi-gasdynamic system of equations and the energy balance equation,” Mat. Model. 24 (10), 51–64 (2012).

    MathSciNet  MATH  Google Scholar 

  29. V. A. Gavrilin and A. A. Zlotnik, “On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance,” Comput. Math. Math. Phys. 55 (2), 264–281 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  30. A. A. Zlotnik and V. A. Gavrilin, “On discretization of the one-dimensional quasi-hydrodynamic system of equations for real gas,” Vestn. Mosk. Energ. Inst., No. 1 (2016).

    Google Scholar 

  31. A. A. Zlotnik, “On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force,” Comput. Math. Math. Phys. 56 (2), 303–319 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  32. A. A. Amosov and A. A. Zlotnik, “Two-level finite-difference schemes for one-dimensional equations of magnetic gas dynamics (viscous heat-conducting case),” Sov. J. Numer. Anal. Math. Model. 4 (3), 179–197 (1989).

    MathSciNet  MATH  Google Scholar 

  33. A. A. Zlotnik, “Parabolicity of a quasi-hydrodynamic system of equations and the stability of its small perturbations,” Math. Notes 83 (5), 610–623 (2008).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, 101000, Russia

    A. A. Zlotnik

Authors
  1. A. A. Zlotnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Zlotnik.

Additional information

Original Russian Text © A.A. Zlotnik, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 4, pp. 710–729.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zlotnik, A.A. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. and Math. Phys. 57, 706–725 (2017). https://doi.org/10.1134/S0965542517020166

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542517020166

Keywords

Search

Navigation