6 Acknowledgments5 Additional Information5.1 Riemann problems in locally

5.2 Characteristic fields in the 3+1 conservative Eulerian formulation of Ibáñez and coworkers

 

This section collects all information concerning the characteristic structure of the general relativistic hydrodynamic equations in the Eulerian formulation of Section  2.1.3 . As explained in Section  3.1.2, this information is necessary in order to implement approximate Riemann solvers in HRSC finite difference schemes.

We present only the characteristic speeds and fields corresponding to the x -direction. Equivalent expressions for the other two directions can be obtained easily from symmetry considerations. The characteristic speeds (eigenvalues) of the system are given by:

eqnarray2043

where tex2html_wrap_inline5646 denotes the local sound speed, which can be obtained from

displaymath2065

We note that the Minkowskian limit of these expressions is recovered properly (see [69Jump To The Next Citation Point In The Article]) as well as the Newtonian one (tex2html_wrap_inline5648, tex2html_wrap_inline5650).

A complete set of right-eigenvectors is given by

equation2079

equation2090

equation2100

equation2110

where the following auxiliary quantities are used:

displaymath2121

displaymath2132

displaymath2145

Finally, a complete set of left-eigenvectors is given by:

equation2153

equation2162

equation2181

equation2200

where the following relations and auxiliary quantities have been used:

displaymath2246

displaymath2260

displaymath2274

displaymath2282

displaymath2291

displaymath2307

These two sets of eigenfields reduce to the corresponding ones in the Minkowskian limit [69].



6 Acknowledgments5 Additional Information5.1 Riemann problems in locally

image Numerical Hydrodynamics in General Relativity
José A. Font
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