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

Matthias Kunik, Adrian Kolb, Siegfried Müller, Ferdinand Thein

Abstract

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.

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

Abstract

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.
Paper Structure (10 sections, 4 theorems, 92 equations, 7 figures, 2 tables)

This paper contains 10 sections, 4 theorems, 92 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conservative Formulations of the Equations
  3. Formulation of a numerical scheme for the radially symmetric solutions
  4. Numerical Results
  5. Numerical methods
  6. Benchmark Tests
  7. Discussion of results
  8. Conclusion
  9. Application of Geng Lai's analysis
  10. Eigensystem for ultra-relativistic Euler equations

Key Result

Lemma 3.1

Assume that $|b_{\pm}|< a_{\pm}$ and put $c_{\pm}=c(a_{\pm},b_{\pm})$ according to cdefinition. We recall that $\lambda\geq 1$. Then

Figures (7)

  • Figure 1: The computational domain $\mathcal{D}$
  • Figure 2: The balance region $\Omega$
  • Figure 3: Example 1: Comparison for MultiWave (green $L = 8$, blue $L = 9$) and RadSymS (red) in radial direction with the solution of ODE system \ref{['GL_ODE_sys']} (black) at final time $t_{end} = 1$ for the pressure $p$, see (a), and velocity $v$, see (b).
  • Figure 4: Example 2: Comparison for MultiWave (blue) and RadSymS (red) in radial direction with the solution of ODE system \ref{['GL_ODE_sys']} (green) at final time $t_{end} = 1$ for the pressure $p$, see (a), and velocity $v$, see (b).
  • Figure 5: Example 3: Comparison for MultiWave (left) and RadSymS (right) in the $t$--$x$ plane, see (a), (b). Further RadSymS (red) and MultiWave (blue) are compared in radial direction at final time $t_{end} = 6$, see (c), (d), for pressure $p$ and velocity $v$.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3: Numerical solution $(a',b')$ for the balance region $\Omega$
  • Definition 3.4: The function Euler
  • Theorem 3.5