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Kinetic Schemes for the Ultra-Relativistic Euler Equations

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

Series: 2001-21, Preprints

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

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

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