# On the order reduction of entropy stable DGSEM for the compressible Euler equations

## TL;DR

This paper investigates how the choice of numerical surface flux influences the convergence order of entropy stable DGSEM for the Euler equations, revealing that less dissipative fluxes achieve optimal convergence regardless of Mach number.

## Contribution

It provides numerical evidence on the impact of different surface flux functions on the convergence behavior of entropy stable DGSEM, clarifying the effects of dissipation levels.

## Key findings

- High dissipation fluxes negatively affect convergence for odd polynomial degrees.
- Less dissipative fluxes like Roe's and HLLC achieve optimal convergence rate $N+1$.
- The convergence behavior varies with Mach number and flux choice.

## Abstract

Is the experimental order of convergence lower when using the entropy stable DGSEM-LGL variant? Recently, a debate on the question of the convergence behavior of the entropy stable nodal collocation discontinuous Galerkin spectral element method (DGSEM) with Legendre-Gauss-Lobatto nodes has emerged. Whereas it is well documented that the entropy conservative variant with no additional interface dissipation shows an odd-even behavior when testing its experimental convergence order, the results in the literature are less clear regarding the entropy stable version of the DGSEM-LGL, where explicit Riemann solver type dissipation is added at the element interfaces. We contribute to the ongoing discussion and present numerical experiments for the compressible Euler equations, where we investigate the effect of the choice of the numerical surface flux function. In our experiments, it turns out that the choice of the numerical surface flux has an impact on the convergence order. Penalty type numerical fluxes with high dissipation in all waves, such as the LLF and the HLL flux, appear to affect the convergence order negatively for odd polynomial degrees $N$, in contrast to the entropy conserving variant, where even polynomial degrees $N$ are negatively affected. This behavior is more pronounced in low Mach number settings. In contrast, for numerical surface fluxes with less dissipative behavior in the contact wave such as e.g. Roe's flux, the HLLC flux and the entropy conservative flux augmented with 5-wave matrix dissipation, optimal convergence rate of $N+1$ independent of the Mach number is observed.