Primitive Variable Recovering in Special Relativistic Hydrodynamics Allowing Ultra- Relativistic Flows

The recovering of the primitive variables from the conserved ones is an important issue in the numerical simulation of flows at relativistic velocities. In Literature, three different techniques based on the solution of quartic equations in the Lorentz factor, in the pressure and in the modulus of the velocity. Aim of the present paper lies in comparing the main properties of such techniques: the number of roots having a physical meaning (admissible roots) for given conserved variables and the efficiency of their practical application, in terms of accuracy and computational cost. Moreover, a novel approach is proposed, which leads to a quartic in a quantity related to the modulus of the velocity: it possesses a unique physical root and leads to the best performances among all the techniques here investigated. The number of admissible roots (among which the physical one has to be selected) is investigated through a numerical application of the Sturm theorem. The conserved quantities are combined into two parameters and the roots are counted in a suitable (bounded) region of the plane of these parameters. The efficiency of the recovering techniques is numerically tested on a grid in the parameter space which allows ultra-relativistic conditions. The practical uses of both the ana- lytical solver and a numerical one (Newton method) are compared. In particular, the numerical solver exhibits better accuracy and smaller computational cost with respect to the analytical one, for all the recovering techniques here investigated.