9.2 Spectral decomposition of the 9 Additional Information9 Additional Information

9.1 Algorithms to recover primitive quantities 

The expressions relating the primitive variables tex2html_wrap_inline5739 to the conserved quantities tex2html_wrap_inline5741 depend explicitly on the equation of state tex2html_wrap_inline6623 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 Eqs. (8Popup Equation, 9Popup Equation, 10Popup Equation, 12Popup Equation), and (13Popup Equation):

  equation2801

with tex2html_wrap_inline6625 and tex2html_wrap_inline6627 given by

  equation2810

and

  equation2816

where

  equation2824

and

equation2832

The root of (62Popup 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 [107Jump To The Next Citation Point In The Article, 109, 111]. The derivative of f with respect to tex2html_wrap_inline6631, tex2html_wrap_inline6633, can be approximated by [3]

  equation2841

where tex2html_wrap_inline6635 is the sound speed which can efficiently be computed for any EOS. Moreover, approximation (67Popup Equation) tends towards the exact derivative as the solution is approached.

Eulderink [49Jump To The Next Citation Point In The Article, 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 tex2html_wrap_inline5745, Schneider et al. [161] transform the system (8Popup Equation, 9Popup Equation, 10Popup Equation), (12Popup Equation), and (13Popup 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. [40] and Dolezal & Wong [42] proposed the use of iterative algorithms for tex2html_wrap_inline6639 and tex2html_wrap_inline5637, 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.



9.2 Spectral decomposition of the 9 Additional Information9 Additional Information

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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