Kinetic schemes for the relativistic gas dynamics

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 n, the spatial part of the four-velocity u and the inverse temperature beta. In this paper we present a general framework 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 of duration tau(M), where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times t(n)=ntau(M), the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Juttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time t(n)>0 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 t(n). (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 limit tau(M)-->0 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.