This section collects all information concerning the characteristic structure of the general relativistic hydrodynamic equations in the Valencia formulation (Section 2.1.3). As explained in Section 2.1.3 this information is necessary in order to implement approximate Riemann solvers in HRSC finite difference schemes.
We only present the characteristic speeds and fields corresponding to the x -direction. Equivalent expressions for the two other directions can be easily obtained by symmetry considerations. The characteristic speeds (eigenvalues) of the system are given by:
where denotes the local sound speed, which can be obtained from , with and . We note that the Minkowskian limit of these expressions is recovered properly (see ) as well as the Newtonian one (, ).
A complete set of right-eigenvectors is given by (superscript T denotes transpose):
where the following auxiliary quantities are used:
Finally, a complete set of left-eigenvectors is given by:
where the following relations and auxiliary quantities have been used:
These two sets of eigenfields reduce to the corresponding ones in the Minkowskian limit .
|Numerical Hydrodynamics in General Relativity
José A. Font
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