A function of pressure, whose zero represents the pressure in the physical state, can easily be obtained from Eqs. (8, 9, 10, 12), and (13):
with and given by
The root of (62) can be obtained by means of a nonlinear root-finder (e.g., a one-dimensional Newton-Raphson iteration). For an ideal gas with a constant adiabatic exponent such a procedure has proven to be very successful in a large number of tests and applications [107, 109, 111]. The derivative of f with respect to , , can be approximated by 
where is the sound speed which can efficiently be computed for any EOS. Moreover, approximation (67) tends towards the exact derivative as the solution is approached.
Eulderink [49, 50] has also developed several procedures to calculate the primitive variables for an ideal EOS with a constant adiabatic index. One procedure is based on finding the physically admissible root of a fourth-order polynomial of a function of the specific enthalpy. This quartic equation can be solved analytically by the exact algebraic quartic root formula although this computation is rather expensive. The root of the quartic can be found much more efficiently using a one-dimensional Newton-Raphson iteration. Another procedure is based on the use of a six-dimensional Newton-Kantorovich method to solve the whole nonlinear set of equations.
Also for ideal gases with constant , Schneider et al.  transform the system (8, 9, 10), (12), and (13) algebraically into a fourth-order polynomial in the modulus of the flow speed, which can be solved analytically or by means of iterative procedures.
For a general EOS, Dean et al.  and Dolezal & Wong  proposed the use of iterative algorithms for and , respectively.
In the covariant formulation of the GRHD equations presented by Papadopoulos & Font , which also holds in the Minkowski limit, there exists a closed form relationship between conserved and primitive variables in the particular case of a null foliation and an ideal EOS. However, in the spacelike case their formulation also requires some type of root-finding procedure.
|Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
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