# zbMATH — the first resource for mathematics

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier-Stokes equations in polar coordinates. (English) Zbl 1448.65126
Summary: The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier-Stokes system involving additional 2nd order terms with a regularizing parameter $$\tau > 0$$. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier-Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced.
The main results are valid for $$\tau = 0$$ as well, i.e., for the barotropic compressible Navier-Stokes system.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76N15 Gas dynamics, general 35Q35 PDEs in connection with fluid mechanics
Full Text:
##### References:
 [1] V. Balashov, A. Zlotnik, and E. Savenkov, Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface. Russ. J. Numer. Anal. Math. Model. 32 (2017), No. 6, 347-358. · Zbl 1422.65145 [2] O. V. Bulatov and T. G. Elizarova, Regularized shallow water equations for numerical simulation of flows with a moving shoreline. Comput. Math. Math. Phys. 56 (2016), No. 4, 665-684. · Zbl 1426.76343 [3] B. N. Chetverushkin, Kinetic Schemes and Quasi-Gas Dynamic System of Equations. CIMNE, Barcelona, 2008. · Zbl 1192.76037 [4] T. G. Elizarova, Quasi-Gas Dynamic Equations. Springer, Dordrecht, 2009. · Zbl 1169.76001 [5] T. G. Elizarova, A. A. Zlotnik, and M. A. Istomina, Hydrodynamical aspects of the formation of spiral-vortical structures in rotating gaseous disks. Astron. Reports. 62 (2018), No. 1, 9-18. [6] E. Feireisl, R. Hošek, D. Maltese, and A. Novotný, Error estimates for a numerical method for the compressible Navier-Stokes system on sufficiently smooth domains. ESAIM: M2AN51 (2017), 279-319. · Zbl 1360.35144 [7] E. Feireisl, T. G. Karper, and M. Pokorný, Mathematical Theory of Compressible Viscous Fluids. Analysis and Numerics. Springer International Publishing Switzerland, Cham, 2016. · Zbl 1356.76001 [8] T. Gallouët, L. Gastaldo, R. Herbin, and J.-C. Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations. ESAIM: M2AN 42 (2008), 303-331. · Zbl 1132.35433 [9] S. Gottlieb, D. Ketcheson, and C.-W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific, Singapore, 2011. · Zbl 1241.65064 [10] R. Herbin and J.-C. Latché, Kinetic energy control in the MAC discretization of compressible Navier-Stokes equations. Int. J. Finite Volumes. 7 (2010), No. 2, 1-6. [11] R. Käppeli and S. Mishra, Well-balanced schemes for the Euler equations with gravitation. J. Comput. Phys. 259 (2014), 199-219. · Zbl 1349.76345 [12] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman and Hall/CRC, London, 2001. · Zbl 0965.35001 [13] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, 2004. [14] A. V. Popov and K. A. Zhukov, An implicit finite-difference scheme for the unsteady motion of a viscous barotropic gas. Numer. Meth. Programming14 (2013), 516-523 (in Russian). [15] J. Reiss and J. Sesterhenn, Conservative, skew-symmetric discretization of the compressible Navier-Stokes equations. In: New Results in Numerical and Experimental Fluid Mechanics VIII (Eds. A. Dillmann, G. Heller, H.-P. Kreplin et al.). Springer, Berlin, 2013, 395-402. [16] P. J. Roache, Computational Fluid Dynamics. Hermosa, Albuquerque, 1976. [17] S. B. Sorokin, A difference scheme for a conjugate-operator model of the heat conduction problem in the polar coordinates. Sib. J. Numer. Math. 20 (2017), No. 3, 297-312 (in Russian). · Zbl 1399.80006 [18] M. Svärd, Weak solutions and convergent numerical schemes of modified compressible Navier-Stokes equations. J. Comput. Phys. 288 (2015), 19-51. · Zbl 1351.76187 [19] E. Tadmor and W. Zhong, Novel entropy stable schemes for 1D and 2D fluid equations. In: Hyperbolic Problems: Theory, Numerics, Applications (Eds. S. Benzoni-Gavage, D. Serre). Springer, Berlin, 2008, 1111-1119. · Zbl 1134.76048 [20] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin, 2009. · Zbl 1227.76006 [21] A. A. Zlotnik, Energy equalities and estimates for barotropic quasi-gasdynamic and quasi-hydrodynamic systems of equations. Comput. Math. Math. Phys. 50 (2010), No. 2, 310-321. · Zbl 1224.35356 [22] A. A. Zlotnik, On construction of quasi-gasdynamic systems of equations and the barotropic system with the potential body force. Math. Modelling24 (2012), No. 4, 65-79 (in Russian). · Zbl 1289.76068 [23] 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 (2016), No. 2, 303-319. · Zbl 1382.76198 [24] A. A. Zlotnik, Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. Math. Phys. 57 (2017), No. 4, 706-725. · Zbl 06815873 [25] 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 (2008), No. 3, 420-446. · Zbl 1201.76232 [26] A. Zlotnik and V. Gavrilin, On a conservative finite-difference method for 1D shallow water flows based on regularized equations. In: Math. Problems in Meteorological Modeling (Eds. A. Batkai, P. Csomós, I. Faragó et al.). Math. Industry, Vol. 22. Springer, Berlin, 2015, 3-18.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.