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2.2 Covariant approaches

General (covariant) conservative formulations of the general-relativistic hydrodynamics equations for ideal fluids, i.e., not restricted to spacelike foliations, have been reported in [117Jump To The Next Citation Point311Jump To The Next Citation Point309Jump To The Next Citation Point]. 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 2.2.1 and 2.2.2 we describe the existing covariant formulations in some detail.

2.2.1 Eulderink and Mellema

Eulderink and Mellema [115Jump To The Next Citation Point117Jump To The Next Citation Point] were the first to derive a covariant formulation of the GRHD 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 upon a generalization of Roe’s approximate Riemann solver. Such a solver was first applied to the nonrelativistic Euler equations in [337Jump To The Next Citation Point] and has been widely employed since in simulating compressible flows in computational fluid dynamics. Furthermore, their procedure is specialized for a perfect-fluid EOS, p = (Γ − 1)ρɛ, Γ being the (constant) adiabatic index of the fluid.

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

α ( [ --Γ--- 4] α α β 4 αβ ) F = K − Γ − 1 ω ω ,ω ω + K ω g , (38 )
where α α ω ≡ Ku, 4 p ω ≡ K ρh and 2 √ --- α β K ≡ − gρh = − g αβω ω. The vector 0 F represents the state vector (the unknowns), and each vector Fi is the corresponding flux in the coordinate direction xi.

Eulderink and Mellema computed the exact “Roe matrix” [337Jump To The Next Citation Point] for the vector (38View 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

ωL + ωR &tidle;ω = ----------, (39 ) KL + KR
where indices L and R denote the left and right states in a Riemann problem (see Section 4.1.2). Further technical details can be found in [115117Jump To The Next Citation Point].

The performance of this general-relativistic Roe solver was tested in a number of one-dimensional problems for which exact solutions are known, including nonrelativistic shock tubes, special-relativistic shock tubes, and the 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 [116]. 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 [311Jump To The Next Citation Point], the spatial velocity components of the 4-velocity, ui, together with the rest-frame density and internal energy, ρ and ɛ, 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 w = (ρ,ui,ɛ). The initial-value problem for Equations (7View Equation) and (8View Equation) is defined in terms of another vector in the same fluid-state space, namely the conserved variables, U, individually denoted i (D, S ,E ):

0 0 D = J = ρu , (40 ) Si = T0i = ρhu0ui + pg0i, (41 ) 00 0 0 00 E = T = ρhu u + pg . (42 )
Note that the state vector variables differ slightly from previous choices (see, e.g., Equations (19View Equation), (28View Equation), (29View Equation), (30View Equation), and (38View Equation)). With those definitions the GRHD equations take also the standard conservation law form,
√ --- √ --- ∂(--− gU-) ∂(--−-gFj)- ∂x0 + ∂xj = S . (43 )

The flux vectors j F and the source terms S (which depend only on the metric, its derivatives, and the undifferentiated stress-energy tensor), are given by

j j ji j0 j i j ij 0 j 0j F = (J ,T ,T ) = (ρu ,ρhu u + pg ,ρhu u + pg ), (44 )
√ --- √ --- S = (0,− − gΓ iμλTμλ,− − gΓ 0μλT μλ ). (45 )

The state of the fluid is uniquely described using either vector of variables, i.e., either U or w, and each one can be obtained from the other via the definitions (40View Equation) – (42View Equation) and the use of the normalization condition for the 4-velocity, g uμuν = − 1 μν. The local characteristic structure of the above system of equations was presented in [311Jump To The Next Citation Point], where the formulation proved well suited for the numerical implementation of HRSC schemes. The formulation of [311Jump To The Next Citation Point] was developed for a perfect fluid EOS. Extensions to account for generic EOS are given in [309Jump To The Next Citation Point], where a comprehensive analysis of general-relativistic hydrodynamics in conservation form is also provided.

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 U. After this update the vector of primitive quantities w must be reevaluated, as those are needed in the Riemann solver (see Section 4.1.2). 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. The specific details of this issue for each formulation of the equations can be found in [36Jump To The Next Citation Point117Jump To The Next Citation Point311Jump To The Next Citation Point]. 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 the solving of a quartic polynomial. Therefore, iterative methods are still preferred [117Jump To The Next Citation Point]. 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 [311Jump To The Next Citation Point], 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, dynamic spacetime evolutions of relativistic polytropes (i.e., stellar models satisfying the 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 are presented in [182] (see [45Jump To The Next Citation Point46Jump To The Next Citation Point] for more recent implementations). A Lagrangian method, limited to spherical symmetry, is developed in [258]. Recent work in [103] includes an Eulerian, nonconservative, formulation for general fluids in null hypersurfaces and spherical symmetry, including their matching to a spacelike section.

The general formalism laid out in [311Jump To The Next Citation Point309Jump To The Next Citation Point] has been applied to astrophysical problems of increasing complexity. Applications in spherical symmetry include the investigation of accreting dynamic black holes, which can be found in [311Jump To The Next Citation Point312Jump To The Next Citation Point]. Studies of the gravitational collapse of supermassive stars are discussed in [225] and studies of the interaction of scalar fields with relativistic stars are presented in [380]. Axisymmetric neutron-star spacetimes are considered in [378Jump To The Next Citation Point], and results for axisymmetric gravitational-core collapse using characteristic numerical relativity can be found in [379Jump To The Next Citation Point]. A proof-of-principle demonstration of the inclusion of matter fields in three dimensions was first given in[45] and short-time evolutions of a self-gravitating star in close orbit around a black hole have only been accomplished recently [46Jump To The Next Citation Point].

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