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

TL;DR

This work analyzes how the numerical surface flux affects the convergence order of entropy-stable DGSEM-LGL for the compressible Euler equations. By comparing standard DGSEM (Gauss and LGL) with Split-DGSEM EC/ES variants across fluxes that range from dissipation-free to Roe-like, the authors show that full $N+1$ convergence is achieved when the dissipation accounts for all waves (e.g., ECKEP-Roe), while dissipation based on maximum wave speeds (LLF/HLL) yields order reductions at low Mach numbers, with an observed odd–even pattern dependent on flux type. The results are demonstrated via a 2D density-wave test across Mach numbers $ ext{Ma} o 0.2$, 1.0, and 3.5, using the $L_2$ density error as the metric. The findings clarify the critical role of surface flux design in convergence behavior and caution that overly simplified fluxes can mask the scheme’s true order, a nuance not captured by manufactured-solution tests. Overall, the paper provides practical guidance for achieving reliable high-order accuracy in entropy-stable DGSEM formulations on LGL nodes.

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.