2.3 Going further2 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 derived in [64Jump To The Next Citation Point In The Article] and, more recently, in [171Jump To The Next Citation Point In The Article, 168Jump 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 them in some detail.

2.2.1 Eulderink and Mellema 

Eulderink and Mellema [62Jump To The Next Citation Point In The Article, 64Jump To The Next Citation Point In The Article] first derived a covariant formulation of the general relativistic hydrodynamic equations taking special care, as in the Valencia formulation, of the conservative form of the system, with no derivatives of the dependent fluid variables appearing in the source terms. Additionally, this formulation is strongly tailored towards the use of a numerical method based on a generalization of Roe's approximate Riemann solver for the non-relativistic Euler equations in Cartesian coordinates [183Jump To The Next Citation Point In The Article]. Their procedure is specialized for a perfect fluid EOS, tex2html_wrap_inline3729, tex2html_wrap_inline3731 being the (constant) adiabatic index of the fluid.

After the appropriate choice of the state vector variables, the conservation laws, Eqs. (7Popup Equation) and (8Popup Equation), are re-written in flux-conservative form. The flow variables are expressed in terms of a parameter vector tex2html_wrap_inline3733 as

  equation536

where tex2html_wrap_inline3735, tex2html_wrap_inline3737 and tex2html_wrap_inline3739 . The vector tex2html_wrap_inline3741 represents the state vector (the unknowns), and each vector tex2html_wrap_inline3743 is the corresponding flux in the coordinate direction tex2html_wrap_inline3603 .

Eulderink and Mellema computed the appropriate ``Roe matrix'' [183Jump To The Next Citation Point In The Article] for the vector (41Popup 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_inline3747, 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, 64Jump 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 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 [63]. 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 [171Jump To The Next Citation Point In The Article] the spatial velocity components of the 4-velocity, tex2html_wrap_inline3749, together with the rest-frame density and internal energy, tex2html_wrap_inline3575 and tex2html_wrap_inline3577, provide a unique description of the state of the fluid and are taken as the primitive variables. They constitute a vector in a five dimensional space, tex2html_wrap_inline3755 . 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_inline3757, individually denoted tex2html_wrap_inline3759 :

  equation577

  equation584

  equation593

Note that these variables slightly differ from previous choices (see, e.g., Eqs. (24Popup Equation), (33Popup Equation), (34Popup Equation), (35Popup Equation) and (41Popup Equation)). With those definitions the equations take the standard conservation law form,

  equation607

with A =(0, i,4). The flux vectors tex2html_wrap_inline3763 and the source terms tex2html_wrap_inline3765 (which depend only on the metric, its derivatives and the undifferentiated stress energy tensor), are given by

  equation622

  equation637

The state of the fluid is uniquely described using either vector of variables, i.e. either tex2html_wrap_inline3757 or tex2html_wrap_inline3677, and each one can be obtained from the other via the definitions (42Popup Equation, 43Popup Equation, 44Popup Equation) and the use of the normalization condition for the 4-velocity, tex2html_wrap_inline3771 .

The local characteristic structure of these equations has been presented in [171Jump To The Next Citation Point In The Article]. The formulation has proved well suited for the numerical implementation of HRSC schemes. A comprehensive numerical study of this approach was also presented in [171Jump To The Next Citation Point In The Article], 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 [103] (see [28] for a recent implementation). This approach is geared towards smooth isentropic flows. A Lagrangian method, applicable in spherical symmetry, has been presented by [139Jump To The Next Citation Point In The Article]. Recent work in [58] 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 tex2html_wrap_inline3757 . 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 [17Jump To The Next Citation Point In The Article, 64Jump To The Next Citation Point In The Article, 171Jump To The Next Citation Point In The Article]. 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 [168].



2.3 Going further2 Equations of General Relativistic 2.1 Spacelike (3+1) approaches

image Numerical Hydrodynamics in General Relativity
José A. Font
http://www.livingreviews.org/lrr-2000-2
© Max-Planck-Gesellschaft. ISSN 1433-8351
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