After the appropriate choice of the state vector variables, the conservation laws, Equations (7) and (8), are re-written in flux-conservative form. The flow variables are then expressed in terms of a parameter vector as
where , and . The vector represents the state vector (the unknowns), and each vector is the corresponding flux in the coordinate direction .
Eulderink and Mellema computed the exact ``Roe matrix''  for the vector (34) and obtained the corresponding spectral decomposition. The characteristic information is used to solve the system numerically using Roe's generalized approximate Riemann solver. Roe's linearization can be expressed in terms of the average state , where L and R denote the left and right states in a Riemann problem (see Section 3.1.2). Further technical details can be found in [76, 78].
The performance of this general relativistic Roe solver was tested in a number of one-dimensional problems for which exact solutions are known, including non-relativistic shock tubes, special relativistic shock tubes, and spherical accretion of dust and a perfect fluid onto a (static) Schwarzschild black hole. In its special relativistic version it has been used in the study of the confinement properties of relativistic jets . However, no astrophysical applications in strong-field general relativistic flows have yet been attempted with this formulation.
Note that the state vector variables slightly differ from previous choices (see, e.g., Equations (19), (28), and (34)). With those definitions the equations of general relativistic hydrodynamics take the standard conservation law form,
with A =(0, i,4). The flux vectors and the source terms (which depend only on the metric, its derivatives and the undifferentiated stress energy tensor), are given by
The state of the fluid is uniquely described using either vector of variables, i.e., either or , and each one can be obtained from the other via the definitions (35) and the use of the normalization condition for the 4-velocity, . The local characteristic structure of the above system of equations was presented in , where the formulation proved well suited for the numerical implementation of HRSC schemes. The formulation presented in this section has been developed for a perfect fluid EOS. Extensions to account for generic EOS are given in . This reference further contains a comprehensive analysis of general relativistic hydrodynamics in conservation form.
A technical remark must be included here: In all conservative formulations discussed in Sections 2.1.3, 2.2.1, and 2.2.2, the time update of a given numerical algorithm is applied to the conserved quantities . After this update the vector of primitive quantities must be re-evaluated, as those are needed in the Riemann solver (see Section 3.1.2). The relation between the two sets of variables is, in general, not in closed form and, hence, the recovery of the primitive variables is done using a root-finding procedure, typically a Newton-Raphson algorithm. This feature, distinctive of the equations of (special and) general relativistic hydrodynamics - it does not exist in the Newtonian limit - may lead in some cases to accuracy losses in regions of low density and small speeds, apart from being computationally inefficient. Specific details on this issue for each formulation of the equations can be found in Refs. [21, 78, 221]. In particular, for the covariant formulation discussed in Section 2.2.1, there exists an analytic method to determine the primitive variables, which is, however, computationally very expensive since it involves many extra variables and solving a quartic polynomial. Therefore, iterative methods are still preferred . On the other hand, we note that the covariant formulation discussed in this section, when applied to null spacetime foliations, allows for a simple and explicit recovery of the primitive variables, as a consequence of the particular form of the Bondi-Sachs metric.
Lightcone hydrodynamics: A comprehensive numerical study of the formulation of the hydrodynamic equations discussed in this section was presented in , where it was applied to simulate one-dimensional relativistic flows on null (lightlike) spacetime foliations. The various demonstrations performed include standard shock tube tests in Minkowski spacetime, perfect fluid accretion onto a Schwarzschild black hole using ingoing null Eddington-Finkelstein coordinates, dynamical spacetime evolutions of relativistic polytropes (i.e., stellar models satisfying the so-called Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium) sliced along the radial null cones, and accretion of self-gravitating matter onto a central black hole.
Procedures for integrating various forms of the hydrodynamic equations on null hypersurfaces are much less common than on spacelike (3+1) hypersurfaces. They have been presented before in  (see  for a recent implementation). This approach is geared towards smooth isentropic flows. A Lagrangian method, limited to spherical symmetry, was developed by . Recent work in  includes an Eulerian, non-conservative, formulation for general fluids in null hypersurfaces and spherical symmetry, including their matching to a spacelike section.
The general formalism laid out in [221, 218] is currently being systematically applied to astrophysical problems of increasing complexity. Applications in spherical symmetry include the investigation of accreting dynamic black holes, which can be found in [221, 222]. Studies of the gravitational collapse of supermassive stars are discussed in , and studies of the interaction of scalar fields with relativistic stars are presented in . Axisymmetric neutron star spacetimes have been considered in , as part of a broader program aimed at the study of relativistic stellar dynamics and gravitational collapse using characteristic numerical relativity. We note that there has been already a proof-of-principle demonstration of the inclusion of matter fields in three dimensions .
|Numerical Hydrodynamics in General Relativity
José A. Font
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