9.2 Algorithms to recover primitive quantities

The expressions relating the primitive variables i (ρ,v ,p) to the conserved quantities i (D, S ,τ) depend explicitly on the equation of state p(ρ,𝜀), and simple expressions are only obtained for simple equations of state (i.e., ideal gas).

A function of pressure, whose zero represents the pressure in the physical state, can easily be obtained from Equations (8View Equation, 9View Equation, 10View Equation, 12View Equation, 13View Equation):

f(¯p) = p (ρ∗(p¯),𝜀 ∗(¯p)) − ¯p, (73 )
with ρ (¯p) ∗ and 𝜀 (p¯) ∗ given by
D ρ∗(¯p) = W--(¯p) (74 ) ∗
τ + D [1 − W (¯p)] + p¯[1 − W (¯p)2]) 𝜀∗(p¯) = ------------∗--------------∗-----, (75 ) DW ∗(p¯)
1 W ∗(¯p) = ∘---------------- (76 ) 1 − vi∗(¯p)v∗i(¯p)
i ----Si---- v∗(¯p) = τ + D + p¯. (77 )
The root of Equation (73View Equation) 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 [179Jump To The Next Citation Point181183]. The derivative of f with respect to ¯p, ′ f, can be approximated by [6]
′ i 2 f = v∗(¯p)v∗i(¯p)cs∗(p¯) − 1, (78 )
where cs∗ is the sound speed which can efficiently be computed for any EOS. Moreover, approximation (78View Equation) tends to the exact derivative as the solution is approached.

Eulderink [83Jump To The Next Citation Point84] 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 γ, Schneider et al. [257] transform system (8View Equation, 9View Equation, 10View Equation, 12View Equation, 13View Equation) 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 2 v and ρ, respectively.

In the covariant formulation of the GRHD equations presented by Papadopoulos and Font [221], 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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