Abstract
In this talk we study the relativistic Euler equations together with constitutive equations compatible with the kinetic theory, i.e., with the Synge energy for the monatomic gas and the generalized Synge energy for the diatomic Gas, see [1]. We show existence and uniqueness of self similar radially symmetric solutions to the initial-boundary problem for the system under consideration. This is the first multi–dimensional result for this system together with the Synge energy. Moreover, in contrast to previous results, due to the properties of the Synge energy, we also cover the ultra relativistic limit, as well as the classical case of the polytropic ideal gas. This is done in a unified treatment and no distinction between the cases is needed. This talk further includes new significant estimates for the Synge energy as well as numerical results.
We additionally are concerned with the ultra–relativistic Euler equations for an ideal gas. In [2] we have presented genuine multi–dimensional benchmark problems for the ultra–relativistic Euler equations. In particular, we compared full two dimensional DG simulations for radially symmetric problems with solutions computed with a specific one-dimensional scheme. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For further investigation it is of interest to study other numerical schemes. In this talk we present an entropy stable flux for this particular system. Therefore, we derive the main field (or entropy variables) and the corresponding potentials. We then present the entropy stable flux and conclude with 2D and 3D simulation results for different test cases, see [3] .
References
[1] T. Ruggeri, F. Thein, Q. Xiao: Self-Similar Radially Symmetric Solutions of the Relativistic Euler Equations with Synge Energy (2025), https://doi.org/10.48550/arXiv.2511.18971
[2] M. Kunik, A. Kolb, S. M¨uller, F. Thein. Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions. Journal of Computational Physics, 518, 113330, 2024. 10.1016/j.jcp.2024.113330
[3] H. Ranocha, F. Thein: Computing radially-symmetric solutions of the ultrarelativistic Euler equations with entropy-stable discontinuous Galerkin methods (2025), https://doi.org/10.48550/arXiv.2508.21427