Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

• Published in: Journal of computational physics 2024-12, Vol.518, p.113330, Article 113330
• Main Authors: Kunik, Matthias, Kolb, Adrian, Müller, Siegfried, Thein, Ferdinand
• Format: Article
• Language: English
• Subjects: Conservation laws; Entropy conditions; Hyperbolic systems; Lorentz transformations; Rarefaction waves; Relativistic Euler equations; Shock waves
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2024.113330

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers. •A novel scheme is presented, able to efficiently compute solutions to the ultra-relativistic Euler equations in multi-d.•For self-similar problems the one-dimensional scheme is compared to the solution of a corresponding ODE system.•Five tests are presented. All solutions computed with the novel scheme are compared with the results of a multi-d DG solver.•All presented test cases may serve as benchmark problems.•We give an improved estimate for the shock speed in the self-similar case.