Preprint 2002-040

Kinetic Schemes for the Relativistic Gas Dynamics

Matthias Kunik, Shamsul Qamar, and Gerald Warnecke

Abstract: A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density, the spatial part of the four-velocity and the inverse temperature. In this paper we present a general frame work for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now.

(i) There are periods of free flight where the gas particles move according to the free kinetic transport equation.

(ii) At the maximization times which is the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density.

(iii) At each new maximization time we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at that time.

(iv) If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem.

In the Eulerian limit we obtain the weak solutions of Euler's equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach.

Available as PostScript (19.5 Mbytes) or gzipped PostScript (360 Kbytes; uncompress using gunzip).
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:
The paper is an extension of Preprint 2002-039.
Submitted by:
<Shamsul.Qamar@Mathematik.Uni-Magdeburg.DE> July 2 2002.

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