by Kunik, M., Qamar, S., Warnecke, G..

**Series:** 2001-21, Preprints

- MSC:
- 65M99 None of the above, but in this section
- 76Y05 Quantum hydrodynamics and relativistic hydrodynamics

**Abstract:**

We present a kinetic numerical scheme for the

relativistic Euler equations, which describe the flow of a perfect

fluid in terms of the particle density $n$, the spatial part of the

four-velocity $\bu$ and the pressure $p$. The kinetic approach is very

simple in the ultra-relativistic limit, but may also be applied to

more general cases. The basic ingredients of the kinetic scheme

are the phase-density in equiblerium and the free flight. The

phase-density generalizes the non-relativistic Maxwellian for

a gas in local equilibrium. The free flight is given by solutions

of a collision free kinetic transport equation. We

establish that the conservation laws of mass, momentum and energy

as well as the entropy inequality are everywhere exactly satisfied

by the solution of the kinetic scheme. For that reason we obtain

weak admissible Euler solutions including arbitrarily complicated shock

interactions. We computed test cases with explicitly

known shock solutions, which will also be presented in this

paper

**Keywords:**

Relativistic Euler equations, kinetic schemes, conservation laws, hyperbolic systems, entropy conditions, shock solutions