Computing Radially-Symmetric Solutions of the Ultra-Relativistic Euler Equations with Entropy-Stable Discontinuous Galerkin Methods
This work provides a numerical method for accurately simulating ultra-relativistic gases, which is important for astrophysics and high-energy physics.
The authors derived an entropy-stable flux for the ultra-relativistic Euler equations and demonstrated its effectiveness through 2D and 3D simulations, capturing shock waves and pressure blow-up.
The ultra--relativistic Euler equations describe gases in the relativistic case when the thermal energy dominates. These equations for an ideal gas are given in terms of the pressure, the spatial part of the dimensionless four-velocity, and the particle density. Kunik et al.\ (2024, https://doi.org/10.1016/j.jcp.2024.113330) proposed genuine multi--dimensional benchmark problems for the ultra--relativistic Euler equations. In particular, they compared full two-dimensional discontinuous Galerkin simulations for radially symmetric problems with solutions computed using a specific one-dimensional scheme. Of particular interest in the solutions are the formation of shock waves and a pressure blow-up. In the present work we derive an entropy-stable flux for the ultra--relativistic Euler equations. Therefore, we derive the main field (or entropy variables) and the corresponding potentials. We then present the entropy-stable flux and conclude with simulation results for different test cases both in 2D and in 3D.