Entropy stable numerical approximations for the isothermal and polytropic Euler 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 γ. As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations (γ = 1) and the shallow water equations (γ = 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.the internal energy e ∈ R, and the pressure p ∈ R + . Thus, in order to close the system, an equation of state is necessary to relate thermodynamic state variables like pressure, density, and internal energy. Depending on the fluid and physical processes we wish to model the equation of state changes.

Kinetic-Energy- and Pressure-Equilibrium-Preserving Schemes for Real-Gas Turbulence in the Transcritical Regime

“…3 Working directly in a finite-volume framework, other classes of schemes can be obtained that may or may not be recast as a linear combination of finite-difference split forms. For instance, numerical flux functions based on the logarithmic average, or on its generalizations, have been developed to achieve entropy conservation for ideal-gases [40,41], polytropic models [42], or thermally-perfect gases and multi-component flows [43][44][45][46]. An approach based on a square-root density splitting was recently proposed by Edoh [29], which induces a geometric average in the numerical fluxes.…”