6 Acknowledgments5 Additional Information5.1 Riemann problems in locally

5.2 Characteristic fields in the Valencia general relativistic hydrodynamics formulation

 

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:

eqnarray1568

where tex2html_wrap_inline4247 denotes the local sound speed, which can be obtained from tex2html_wrap_inline4249, with tex2html_wrap_inline4251 and tex2html_wrap_inline4253 . We note that the Minkowskian limit of these expressions is recovered properly (see [56Jump To The Next Citation Point In The Article]) as well as the Newtonian one (tex2html_wrap_inline4255, tex2html_wrap_inline4257).

A complete set of right-eigenvectors is given by (superscript T denotes transpose):

eqnarray1590

where the following auxiliary quantities are used:

equation1611

equation1621

equation1634

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

eqnarray1642

where the following relations and auxiliary quantities have been used:

equation1722

equation1736

equation1750

equation1758

equation1771

These two sets of eigenfields reduce to the corresponding ones in the Minkowskian limit [56].



6 Acknowledgments5 Additional Information5.1 Riemann problems in locally

image Numerical Hydrodynamics in General Relativity
José A. Font
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