Kinetic schemes for the relativistic gas dynamics.

*(English)*Zbl 1098.76056Summary: 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 \(\mathbf u\) 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 \(t_n=n\tau_M\), the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by F. Jüttner’s relativistic generalization of the classical Maxwellian phase density [Z. Phys. 47, 542–566 (1928; JFM 54.0987.01)].

(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 \(\tau_M\to 0\) 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.

(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 \(t_n=n\tau_M\), the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by F. Jüttner’s relativistic generalization of the classical Maxwellian phase density [Z. Phys. 47, 542–566 (1928; JFM 54.0987.01)].

(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 \(\tau_M\to 0\) 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.