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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 2009).
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).
B. N. Chetverushkin, Kinetic Schemes and Quasi-Gas Dynamic System of Equations (MAKS, Moscow, 2004; CIMNE, Barcelona, 2008).
T. G. Elizarova, Quasi-Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer-Verlag, Berlin, 2009).
Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (NITs Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2009) [in Russian].
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).
A. A. Zlotnik, “Quasi-gasdynamic system of equations with general equations of state,” Dokl. Math. 81 (2), 312–316 (2010).
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).
A. A. Zlotnik, “Linearized stability of equilibrium solutions to the quasi-gasdynamic system of equations,” Dokl. Math. 82 (2), 811–815 (2010).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).
S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauchnaya Kniga, Novosibirsk, 1998; Kluwer, New York, 2003).
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).
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer, Berlin, 2010).
E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford Univ. Press, Oxford, 2004).
E. Tadmor, “Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems,” Acta Numerica 12, 451–512 (2003).
G. P. Prokopov, “Necessity of entropy control in gasdynamic computations,” Comput. Math. Math. Phys. 47 (9), 1528–1537 (2007).
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).
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).
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).
Y. Lv and M. Ihme, “Entropy-bounded discontinuous Galerkin scheme for Euler equations,” J. Comput. Phys. 295, 715–739 (2015).
A. N. Mohammed and F. Ismail, “Entropy consistent methods for the Navier–Stokes equations,” J. Sci. Comput. 63, 612–631 (2015).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
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).
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517020166