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3.2 Recovery of primitive variables

On the other hand, as for the case of the GRHD equations discussed before, iterative (root-finding) algorithms are also required for the GRMHD equations to recover the primitive variables w from the state vector U. Not surprisingly, the recovery procedure is in the GRMHD case more involved than for unmagnetized flows. A number of approaches are available in the literature. In particular, we direct the interested reader to [289Jump To The Next Citation Point] for a comparison of six different methods for performing such inversions from conserved to primitives variables for the case in which the matter thermodynamics is described by an ideal gas EOS. In [289] not only the accuracy of each of the methods is assessed but also its run-time.

For illustrative purposes, we discuss next with some detail the specific approach followed by [24Jump To The Next Citation Point], which is an extension to full general relativity of that developed by [199Jump To The Next Citation Point] in the special relativistic case. The procedure relies on the fact that it is not necessary to solve the system given by the definition of the conserved variables in terms of the primitives (see Equation (57View Equation)) for the three components of the momentum, but instead for its modulus 2 i S = S Si. By eliminating the components of α b through Equations (59View Equation) and after some algebra, it is possible to write 2 S as

2 2 2W--2 −-1 2 (BiSi)2- S = (Z + B ) W 2 − (2Z + B ) Z2 , (68 )
where 2 Z ≡ ρhW. In addition, the equation for the total energy can be worked out in a similar way
B2 (BiS )2 τ = Z + B2 − p − ----2 − ----i2---− D. (69 ) 2W 2Z
Equations (68View Equation) and (69View Equation), together with the definitions of D and Z, form a system for the unknowns ρ, p and W, assuming the function h = h(ρ,p) is provided. The roots of this system can be found using a two-dimensional scheme (e.g. bisection or Newton–Raphson).
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