by Mahmoud A.E. Abdelrahman; Matthias Kunik.

**Series:** 2013-18, Preprints

- MSC:
- 35L45 Initial value problems for first-order hyperbolic systems
- 35L60 Nonlinear first-order hyperbolic equations
- 35L65 Conservation laws
- 35L67 Shocks and singularities
- 76Y05 Quantum hydrodynamics and relativistic hydrodynamics
- 65M99 None of the above, but in this section

**Abstract:**

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$,

the spatial part $textbf{u} in mathbb{R}^3$ of the dimensionless four-velocity

and the particle density $n$.

Two schemes for these equations are presented in one space dimension, namely

a front tracking and a cone-grid scheme.

A new front tracking technique for the ultra-relativistic

Euler equations is introduced, which gives weak solutions.

The front tracking method is based on piecewise constant

approximations to Riemann solutions, called front tracking Riemann

solutions, where continuous rarefaction waves are approximated by

finite collections of discontinuities, so called non-entropy shocks.

This method can be used for analytical as well as for numerical purposes.

A new unconditionally stable cone-grid scheme is also derived in this paper,

which is based on the Riemann solution for the ultra-relativistic

Euler equations. Both schemes are compared by two numerical examples, where

explicit solutions are known.

**Keywords:**

relativistic Euler equations, conservation laws, hyperbolic systems, entropy, front tracking

**This paper was published in:**

Journal of Computational Physics,

Volume 275, 15 October 2014, Pages 213-235,

DOI: 10.1016/j.jcp.2014.06.051