# Computing Radially-Symmetric Solutions of the Ultra-Relativistic Euler Equations with Entropy-Stable Discontinuous Galerkin Methods

## TL;DR

This paper develops an entropy-stable discontinuous Galerkin method for solving ultra-relativistic Euler equations, enabling accurate simulation of shock waves and pressure blow-up in radially symmetric relativistic gases.

## Contribution

It introduces an entropy-stable flux and corresponding variables for the ultra-relativistic Euler equations, advancing numerical stability and accuracy in relativistic fluid simulations.

## Key findings

- Successful 2D and 3D simulations of shock formation
- Stable solutions with pressure blow-up behavior
- Validation against benchmark problems

## Abstract

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.