A function of pressure, whose zero represents the
pressure in the physical state, can easily be obtained from
Equations (8, 9
, 10
, 12
, 13
):
Eulderink [83, 84] 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 entire nonlinear set of equations.
Also for ideal gases with constant , Schneider et al. [256] transform
system (8
, 9
, 10
, 12
, 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. [70] and Dolezal and
Wong [74] proposed the use of
iterative algorithms for and
, respectively.
In the covariant formulation of the GRHD equations presented by Papadopoulos and Font [220], 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.