Preprint 2002-039

Kinetic Schemes for the Ultra-relativistic Euler Equations

Matthias Kunik, Shamsul Qamar, and Gerald Warnecke

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, the spatial part of the four-velocity and the pressure. 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 equilibrium 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. In the numerical case studies the results obtained from the kinetic scheme are compared with the first order upwind and centered schemes.

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Matthias Kunik, <Matthias.Kunik@Mathematik.Uni-Magdeburg.DE>
Shamsul Qamar, <Shamsul.Qamar@Mathematik.Uni-Magdeburg.DE>
Gerald Warnecke, <Gerald.Warnecke@Mathematik.Uni-Magdeburg.DE>
Publishing information:
New version received April 16 2003.
Submitted by:
<Shamsul.Qamar@Mathematik.Uni-Magdeburg.DE> July 1 2002.

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