Methods and Applications of Analysis

Volume 28 (2021)

Number 4

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part III

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Radially symmetric solutions of the ultra-relativistic Euler equations

Pages: 401 – 422



Matthias Kunik (Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Germany)

Hailiang Liu (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)

Gerald Warnecke (Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Germany)


The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$, the spatial part $\underline{u} \in \mathbb{R}^3$ of the dimensionless four-velocity and the particle density $n$. Radially symmetric solutions of these equations are studied. Analytical solutions are presented for the linearized system. For the original nonlinear equations we design and analyze a numerical scheme for simulating radially symmetric solutions in three space dimensions. The good performance of the scheme is demonstrated by numerical examples. In particular, it was observed that the method has the capability to capture accurately the pressure singularity formation caused by shock wave reflections at the origin.


relativistic Euler equations, conservation laws, hyperbolic systems, Lorentz transformations, shock waves, entropy conditions, rarefaction waves

2010 Mathematics Subject Classification

35L45, 35L60, 35L65, 35L67

Received 10 February 2020

Accepted 10 August 2020

Published 10 June 2022