Winters, Andrew R., Czernik, Christof, Schily, Moritz B. and Gassner, Gregor J. (2020). Entropy stable numerical approximations for the isothermal and polytropic Euler equations. Bit, 60 (3). S. 791 - 825. DORDRECHT: SPRINGER. ISSN 1572-9125

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Abstract

In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index gamma As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations (gamma=1 and the shallow water equations (gamma=2 We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme.

Item Type: Journal Article
Creators:
CreatorsEmailORCIDORCID Put Code
Winters, Andrew R.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Czernik, ChristofUNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Schily, Moritz B.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Gassner, Gregor J.UNSPECIFIEDUNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-322239
DOI: 10.1007/s10543-019-00789-w
Journal or Publication Title: Bit
Volume: 60
Number: 3
Page Range: S. 791 - 825
Date: 2020
Publisher: SPRINGER
Place of Publication: DORDRECHT
ISSN: 1572-9125
Language: English
Faculty: Unspecified
Divisions: Unspecified
Subjects: no entry
Uncontrolled Keywords:
KeywordsLanguage
DISCONTINUOUS GALERKIN METHOD; NONLINEAR CONSERVATION-LAWS; SHALLOW-WATER EQUATIONS; IDEAL MHD; SCHEMES; SYSTEMS; FORM; DISCRETIZATION; DISCRETE; ENERGYMultiple languages
Computer Science, Software Engineering; Mathematics, AppliedMultiple languages
URI: http://kups.ub.uni-koeln.de/id/eprint/32223

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