After the appropriate choice of the state vector variables, the conservation laws, Eqs. (7) and (8), are re-written in flux-conservative form. The flow variables are 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 appropriate ``Roe matrix''  for the vector (41) 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 3.2). Further technical details can be found in [62, 64].
The performance of this general relativistic Roe solver was tested in a number of one-dimensional problems for which an exact solution is 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 . No astrophysical applications in strong-field general relativistic flows have yet been attempted with this formulation.
Note that these variables slightly differ from previous choices (see, e.g., Eqs. (24), (33), (34), (35) and (41)). With those definitions the equations 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 (42, 43, 44) and the use of the normalization condition for the 4-velocity, .
The local characteristic structure of these equations has been presented in . The formulation has proved well suited for the numerical implementation of HRSC schemes. A comprehensive numerical study of this approach was also presented in , where it was applied to simulate one-dimensional relativistic flows on null spacetime foliations. The demonstrations performed include standard shock tube tests in Minkowski spacetime, perfect fluid accretion onto a Schwarzschild black hole using ingoing null Eddington-Finkelstein coordinates, and dynamical spacetime evolutions of polytropes (i.e. stellar models satisfying the Tolman-Oppenheimer-Volkoff equilibrium equations) 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 have been presented before in  (see  for a recent implementation). This approach is geared towards smooth isentropic flows. A Lagrangian method, applicable in spherical symmetry, has been presented by . Recent work in  includes a Eulerian non-conservative formulation for general fluids in null hypersurfaces and spherical symmetry, including their matching to a spacelike section.
A technical remark must be included here: In all conservative formulations reviewed in Sections 2.1.3, 2.2.1, 2.2.2, the time-update of the numerical algorithm is applied to the conserved quantities . After the update the vector of primitive quantities must be reevaluated, as those are needed in the Riemann solver (see Section 3.1.2). The relation between the two sets of variables is not in closed form and, hence, the update of the primitive variables is done using a root-finding procedure, typically a Newton-Raphson algorithm. This feature may lead to accuracy losses in regions of low density and small speeds, apart from being computationally inefficient. Specific details on this issue can be found in [17, 64, 171]. We note that the covariant formulation discussed in this section, when applied to null spacetime foliations, allows for an explicit recovery of the primitive variables, as a consequence of the particular form of the Bondi-Sachs metric. We end by pointing out that the formulation presented in this section has been developed for a perfect fluid EOS. Extensions to account for generic EOS, as well as a comprehensive analysis of general relativistic hydrodynamics in conservation form, have been recently presented in .
|Numerical Hydrodynamics in General Relativity
José A. Font
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