2.3 Going further: Non-ideal hydrodynamics2 Equations of General Relativistic 2.1 Spacelike 3+1 approaches

2.2 Covariant approaches 

General (covariant) conservative formulations of the general relativistic hydrodynamic equations for ideal fluids, i.e., not restricted to spacelike foliations, have been reported in [78Jump To The Next Citation Point In The Article] and, more recently, in [221Jump To The Next Citation Point In The Article, 218Jump To The Next Citation Point In The Article]. The form invariance of these approaches with respect to the nature of the spacetime foliation implies that existing work on highly specialized techniques for fluid dynamics (i.e., HRSC schemes) can be adopted straightforwardly. In the next two sections we describe the existing covariant formulations in some detail.

2.2.1 Eulderink and Mellema 

Eulderink and Mellema [76Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article] first derived a covariant formulation of the general relativistic hydrodynamic equations. As in the formulation discussed in Section  2.1.3, these authors took special care to use the conservative form of the system, with no derivatives of the dependent fluid variables appearing in the source terms. Additionally, this formulation is strongly adapted to a particular numerical method based on a generalization of Roe's approximate Riemann solver. Such a solver was first applied to the non-relativistic Euler equations in [242Jump To The Next Citation Point In The Article] and has been widely employed since in simulating compressible flows in computational fluid dynamics. Furthermore, their procedure is specialized for a perfect fluid EOS, tex2html_wrap_inline4724, tex2html_wrap_inline4726 being the (constant) adiabatic index of the fluid.

After the appropriate choice of the state vector variables, the conservation laws, Equations (7Popup Equation) and (8Popup Equation), are re-written in flux-conservative form. The flow variables are then expressed in terms of a parameter vector tex2html_wrap_inline4878 as

  equation565

where tex2html_wrap_inline4880, tex2html_wrap_inline4882 and tex2html_wrap_inline4884 . The vector tex2html_wrap_inline4886 represents the state vector (the unknowns), and each vector tex2html_wrap_inline4888 is the corresponding flux in the coordinate direction tex2html_wrap_inline4716 .

Eulderink and Mellema computed the exact ``Roe matrix'' [242Jump To The Next Citation Point In The Article] for the vector (34Popup Equation) 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 tex2html_wrap_inline4892, 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, 78Jump To The Next Citation Point In The Article].

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 [77]. However, no astrophysical applications in strong-field general relativistic flows have yet been attempted with this formulation.

2.2.2 Papadopoulos and Font 

In this formulation [221Jump To The Next Citation Point In The Article], the spatial velocity components of the 4-velocity, tex2html_wrap_inline4894, together with the rest-frame density and internal energy, tex2html_wrap_inline4640 and tex2html_wrap_inline4690, provide a unique description of the state of the fluid at a given time and are taken as the primitive variables. They constitute a vector in a five dimensional space tex2html_wrap_inline4900 . The initial value problem for equations (7Popup Equation) and (8Popup Equation) is defined in terms of another vector in the same fluid state space, namely the conserved variables, tex2html_wrap_inline4902, individually denoted tex2html_wrap_inline4904 :

  equation606

Note that the state vector variables slightly differ from previous choices (see, e.g., Equations (19Popup Equation), (28Popup Equation), and (34Popup Equation)). With those definitions the equations of general relativistic hydrodynamics take the standard conservation law form,

  equation632

with A =(0, i,4). The flux vectors tex2html_wrap_inline4908 and the source terms tex2html_wrap_inline4910 (which depend only on the metric, its derivatives and the undifferentiated stress energy tensor), are given by

  equation647

and

  equation662

The state of the fluid is uniquely described using either vector of variables, i.e., either tex2html_wrap_inline4902 or tex2html_wrap_inline4822, and each one can be obtained from the other via the definitions (35Popup Equation) and the use of the normalization condition for the 4-velocity, tex2html_wrap_inline4916 . The local characteristic structure of the above system of equations was presented in [221Jump To The Next Citation Point In The Article], 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 [218Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline4902 . After this update the vector of primitive quantities tex2html_wrap_inline4822 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. [21Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article, 221Jump To The Next Citation Point In The Article]. 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 [78Jump To The Next Citation Point In The Article]. 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 [221Jump To The Next Citation Point In The Article], 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 [133] (see [31Jump To The Next Citation Point In The Article] for a recent implementation). This approach is geared towards smooth isentropic flows. A Lagrangian method, limited to spherical symmetry, was developed by [181Jump To The Next Citation Point In The Article]. Recent work in [71] 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 [221Jump To The Next Citation Point In The Article, 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 [221Jump To The Next Citation Point In The Article, 222Jump To The Next Citation Point In The Article]. Studies of the gravitational collapse of supermassive stars are discussed in [156Jump To The Next Citation Point In The Article], and studies of the interaction of scalar fields with relativistic stars are presented in [270Jump To The Next Citation Point In The Article]. Axisymmetric neutron star spacetimes have been considered in [269Jump To The Next Citation Point In The Article], 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 [31].



2.3 Going further: Non-ideal hydrodynamics2 Equations of General Relativistic 2.1 Spacelike 3+1 approaches

image Numerical Hydrodynamics in General Relativity
José A. Font
http://www.livingreviews.org/lrr-2003-4
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de