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Entropy stable numerical approximations for the isothermal and polytropic Euler equations | | Andrew R. Winters
; Christof Czernik
; Moritz B. Schily
; Gregor J. Gassner
; | | Date: |
7 Jul 2019 | | 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. | | Source: | arXiv, 1907.3287 | | Services: | Forum | Review | PDF | Favorites | |
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