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In this work we analyze the entropic properties of the Euler equations whenthe system is closed with the assumption of a polytropic gas. In this case, thepressure solely depends upon the density of the fluid and the energy equationis not necessary anymore as the mass conservation and momentum conservationthen form a closed system. Further, the total energy acts as a convexmathematical entropy function for the polytropic Euler equations. Thepolytropic equation of state gives the pressure as a scaled power law of thedensity in terms of the adiabatic index $\gamma$. As such, there are importantlimiting cases contained within the polytropic model like the isothermal Eulerequations ($\gamma=1$) and the shallow water equations ($\gamma=2$). We firstmimic the continuous entropy analysis on the discrete level in a finite volumecontext to get special numerical flux functions. Next, these numerical fluxesare incorporated into a particular discontinuous Galerkin (DG) spectral elementframework where derivatives are approximated with summation-by-parts operators.This guarantees a high-order accurate DG numerical approximation to thepolytropic Euler equations that is also consistent to its auxiliary totalenergy behavior. Numerical examples are provided to verify the theoreticalderivations, i.e., the entropic properties of the high order DG scheme.

Entropy stable numerical approximations for the isothermal and polytropic Euler equations