Simple and Efficient Godunov Scheme for Computational Relativistic Gas Dynamics

The governing equations of relativistic computational fluid dynamics (CFD) are integrated numerically. The equation of state (EOS) for a gas at relativistic temperature (the thermal energy of a gas particle is on the order of its rest mass energy) is obtained as a polynomial approximation for a gas with the Maxwellian distribution function. In contrast to previous investigations by other authors, in which the polytropic index of a gas was accepted to be constant, here the relativistic dependence of the specific heat is taken into account. The use of the proposed EOS facilitates the relativistic CFD. The Riemann invariants are expressed in terms of elementary functions so that the characteristic decomposition of the governing equations is efficient and natural. The full solution of the Riemann problem (Riemann solver) is also given by elementary functions. In order to construct it numerically, a simple transcendent equation, which relates the pressure and the velocity at the contact discontinuity, should be solved using an iteration procedure, just as in nonrelativistic CFD. So the Godunov scheme based upon the exact Riemann solver becomes simple and efficient. 1D test results are presented, as well as an example of a 2D simulation.