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Kinetic schemes for the relativistic gas dynamics. (English) Zbl 1098.76056
Summary: 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.

##### MSC:
 76M28 Particle methods and lattice-gas methods 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 76N15 Gas dynamics (general theory) 76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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