# Entropy stable numerical approximations for the isothermal and polytropic Euler equations

## TL;DR

This paper develops high-order entropy stable numerical methods for the polytropic Euler equations, including special cases like isothermal and shallow water equations, ensuring physical consistency and accuracy.

## Contribution

It introduces entropy stable finite volume fluxes integrated into a high-order DG spectral element framework for the polytropic Euler equations.

## Key findings

- Numerical fluxes preserve entropy properties.
- High-order DG scheme verified for entropic consistency.
- Method applicable to special cases like isothermal and shallow water equations.

## 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.