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
and
where
and
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 [3]
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. [161] 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. [40] and Dolezal & Wong [42] proposed the use of iterative algorithms for
and
, respectively.
In the covariant formulation of the GRHD equations presented by Papadopoulos & Font [138], 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.
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Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |