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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.

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: DOI
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