01-21
Kinetic Schemes for the Ultra-Relativistic Euler Equations
by Kunik, M.; Qamar, S.; Warnecke, G.
Preprint series: 01-21, Preprints
- MSC:
- 65M99 None of the above but in this section
- 76Y05 Quantum hydrodynamics and relativistic hydrodynamics, See also {83C55, 85A30}
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
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