The line element is written as
where
For historical reasons it is worth beginning the overview of the formulations with the pioneering numerical
work in general relativistic hydrodynamics, namely the one-dimensional gravitational collapse code of May
and White [247, 248
]. Building on theoretical work by Misner and Sharp [263
], May and White developed
a time-dependent numerical code to solve the evolution equations describing adiabatic spherical collapse in
general relativity. This code was based on a Lagrangian finite-difference scheme (see Section 4.1), in which
the coordinates are co-moving with the fluid. Artificial viscosity terms were included in the equations to
damp the spurious numerical oscillations caused by the presence of shock waves in the flow
solution. May and White’s formulation became the starting point of a large number of numerical
investigations in subsequent years and, hence, it is useful to describe its main features in some
detail.
For a spherically-symmetric spacetime, the line element can be written as
The co-moving character of the coordinates leads, for a perfect fluid, to a stress-energy tensor whose
nonvanishing components are . In these coordinates the local
conservation equation for the baryonic mass, Equation (2
), can be easily integrated to yield the metric
potential
:
The gravitational field equations, Equation (10), and the equations of motion, Equation (1
), reduce to
the following quasi-linear system of partial differential equations (see also [263]):
Hydrodynamics codes based on the original formulation of May and White and on later
versions (e.g., [406]) have been used in many nonlinear simulations of supernova and neutron-star
collapse (see, e.g., [262, 391] and references therein), as well as in perturbative computations of
spherically-symmetric gravitational collapse within the framework of the linearized Einstein
equations [346, 347]. However, the Lagrangian character of May and White’s formulation, together with
other theoretical considerations concerning the particular coordinate gauge, has prevented its
extension to multiple-dimensional calculations. In spite of this, for one-dimensional problems, the
Lagrangian approach adopted by May and White has considerable advantages with respect to an
Eulerian approach with spatially-fixed coordinates, most notably the reduction of numerical
diffusion.
The use of Eulerian coordinates in multiple-dimensional numerical-relativistic hydrodynamics started with
the pioneering work of Wilson [411, 417
]. Introducing the basic dynamic variables
,
, and
,
representing the relativistic density, momenta, and generalized internal energy, respectively, and defined
as
A direct inspection of the system shows that the equations are written as a coupled set of advection
equations. In doing so, the terms containing derivatives (in space or time) of the pressure are treated as
source terms. This approach, hence, sidesteps an important guideline for the formulation of nonlinear
hyperbolic systems of equations, namely the preservation of their conservation form. This is a necessary
condition to guarantee correct evolution in regions of sharp entropy generation (i.e., shocks).
Furthermore, some amount of numerical dissipation must be used to stabilize the solution across
discontinuities. In this spirit, the first attempt to solve the equations of general-relativistic
hydrodynamics in the original Wilson’s scheme [411] used a combination of finite-difference
upwind techniques with artificial viscosity terms. Such terms adapted the classic treatment of
shock waves introduced by von Neumann and Richtmyer [407
] to the relativistic regime (see
Section 4.1.1).
Wilson’s formulation has been (and still is) widely used in hydrodynamic codes developed by a variety of
research groups. (The latest extensions made to incorporate magnetic fields are discussed in
Section 3.1.1). Many different astrophysical scenarios were first investigated with these codes, including
axisymmetric stellar core collapse [279, 277
, 283, 38
, 386
, 319
, 118
], accretion onto compact
objects [170
, 317
], numerical cosmology [72, 73
, 17] and, more recently, the coalescence of neutron-star
binaries [416
, 418
, 242
] (see [417
] for an up-to-date summary of such studies). This formalism has also
been employed, in the special-relativistic limit, in numerical studies of heavy-ion collisions [415, 249]. We
note that in most of these investigations, the original formulation of the hydrodynamic equations
was slightly modified by redefining the dynamic variables, Equation (19
), with the addition of
a multiplicative
factor (the lapse function) and the introduction of the Lorentz factor,
:
As mentioned before, the description of the evolution of self-gravitating matter fields in general relativity
requires a joint integration of the hydrodynamic equations and the gravitational field equations (the
Einstein equations). Using Wilson’s formulation for the fluid dynamics, such coupled simulations were
first considered in [413], building on a vacuum numerical-relativity code specifically developed
to investigate the headon collision of two black holes [382]. The resulting code was axially
symmetric and aimed to integrate the coupled set of equations in the context of stellar core
collapse [120].
More recently, Wilson’s formulation has been applied to the numerical study of the coalescence of
neutron-star binaries in general relativity [416, 418
, 242
] (see Section 5.3.2). These studies adopt an
approximation scheme for the gravitational field, by imposing the simplifying condition that the
three-geometry (the 3-metric
) is conformally flat. The curvature of the three metric is then described
by a position-dependent conformal factor
times a flat-space Kronecker delta, and the line element,
Equation (11
), reads
Therefore, in this approximation scheme all radiation degrees of freedom are removed. Moreover, under
the maximal-slicing condition (), the field equations reduce to a set of five Poisson-like elliptic
equations in flat spacetime for the lapse, the shift vector, and the conformal factor. While in spherical
symmetry this approach is no longer an approximation, being identical to Einstein’s theory,
beyond spherical symmetry its quality degrades. In [193] it was shown by means of numerical
simulations of extremely relativistic disks of dust that it has the same accuracy as the first
post-Newtonian approximation. In less extreme situations, however, as in rotational stellar-core
collapse, the approximation yields results comparable to those obtained using full general relativity
(see [74
, 92
, 365
]).
Wilson’s formulation showed some limitations in handling situations involving ultrarelativistic flows
(), as first pointed out by Centrella and Wilson [73
]. Norman and Winkler [291
] performed a
comprehensive numerical assessment of such formulation by means of special-relativistic hydrodynamics
simulations. Figure 1
reproduces a plot from [291
] in which the relative error of the density compression
ratio in the relativistic shock reflection problem – the heating of a cold gas, which impacts at relativistic
speeds with a solid wall and bounces back – is displayed as a function of the Lorentz factor
of
the incoming gas. The source of the data is [73
]. This figure shows that for Lorentz factors of
about 2 (
), the threshold of the ultrarelativistic limit, the relative errors are between
5% and 7% (depending on the adiabatic exponent of the gas), showing a linear growth with
.
|
Norman and Winkler [291] conclude that those large errors were mainly due to the way in which the
artificial viscosity terms are included in the numerical scheme in Wilson’s formulation. These terms,
commonly called
in the literature (see Section 4.1.1), are only added to the pressure terms in some
cases, namely at the pressure gradient in the source of the momentum equation, Equation (21
), and at the
divergence of the velocity in the source of the energy equation, Equation (22
). However, [291
] proposes that
one add the
terms in a consistent way, in order to consider the artificial viscosity as a real viscosity.
Hence, the hydrodynamic equations should be rewritten for a modified stress-energy tensor of the following
form:
Recently, Anninos and Fragile [19] have compared state-of-the-art artificial viscosity schemes and
high-order nonoscillatory central schemes (see Section 4.1.3) using Wilson’s formulation for the former class
of schemes and a conservative formulation for the latter (similar to the one considered in [311
, 309
]; see
Section 2.2.2). Using the three-dimensional Cartesian code cosmos, these authors found that
earlier results for artificial viscosity schemes in shock tube tests or shock reflection tests are
not improved, i.e. the numerical solution becomes increasingly unstable for shock velocities
greater than
. On the other hand, results for the shock-reflection problem with a
second-order finite-difference central scheme show the suitability of such a scheme to handle
ultrarelativistic flows (see Figure 8 of [19
]), the underlying reason being the use of a conservative
formulation of the hydrodynamic equations rather than the particular scheme employed (see
Section 4.1.3).
In 1991, Martí, Ibáñez, and Miralles [237] presented a new formulation of the (Eulerian)
general-relativistic hydrodynamics equations. This formulation was designed to take fundamental advantage
of the hyperbolic and conservative character of the equations, contrary to the formulation discussed in the
previous Section 2.1.2. From the numerical point of view, the hyperbolic and conservative nature of the
relativistic Euler equations allows for the use of schemes based on the characteristic fields of the system,
bringing to relativistic hydrodynamics existing tools of computational classical fluid dynamics. This
procedure departs from earlier approaches, most notably in avoiding the need for artificial dissipation
terms to handle discontinuous solutions [411
, 413
], as well as implicit schemes as proposed
in [291
].
If a numerical scheme written in conservation form converges, it automatically guarantees the correct
Rankine–Hugoniot (jump) conditions across discontinuities - the shock-capturing property (see, e.g., [218]).
Writing the relativistic hydrodynamic equations as a system of conservation laws, identifying the suitable
vector of unknowns, and building up an approximate Riemann solver permitted the extension of
state-of-the-art high-resolution shock-capturing schemes (HRSC in the following) from classical fluid
dynamics into the realm of relativity [237
].
Theoretical advances in the mathematical character of the relativistic hydrodynamic equations were first
achieved studying the special relativistic limit. In Minkowski spacetime, the hyperbolic character of
relativistic hydrodynamics and MHD is exhaustively studied by Anile and collaborators (see [15] and
references therein) by applying Friedrichs’ definition of hyperbolicity [143] to a quasi-linear form of the
system of hydrodynamic and MHD equations,
The approach followed in [134] for the equations of special-relativistic hydrodynamics was
extended to general relativity in [36]. The choice of state vector (conserved quantities) in the 3+1
Eulerian formulation developed by Banyuls et al. [36
] differs slightly from that of Wilson’s
formulation [411
]. It comprises the rest-mass density (
), the momentum density in the
-direction (
), and the total energy density (
), measured by a family of observers, which
is the natural extension (for a generic spacetime) of the Eulerian observers in classical fluid
dynamics. Interested readers are directed to [36
] for more complete definitions and geometrical
foundations.
In terms of the primitive variables , the conserved quantities are written
as:
With this choice of variables the equations can be written in conservation form. Strict conservation is only possible in flat spacetime. For curved spacetimes there exist source terms, arising from the spacetime geometry. However, these terms do not contain derivatives of stress-energy–tensor components. More precisely, the resulting first-order flux-conservative hyperbolic system, well suited for numerical applications, reads:
with The local characteristic structure of the previous system of equations was presented in [36]. The
eigenvalues (characteristic speeds) of the corresponding Jacobian matrices are all real (but not distinct, one
showing a threefold degeneracy as a result of the assumed directional splitting approach) and a complete set
of right eigenvectors exists. More precisely, for a fluid moving along the
-direction, the eigenvalues
read:
Three-dimensional, Eulerian codes, which evolve the coupled system of Einstein and hydrodynamics
equations following the conservative Eulerian approach discussed in this section have been developed by
Font et al. [137] and by Baiotti et al. [27
] (see Section 4.4 for further details). In particular, in [137
] the
spectral decomposition (eigenvalues and right-eigenvectors) of the general-relativistic hydrodynamics
equations, valid for general spatial metrics, was derived, extending earlier results of [36
] for nondiagonal
metrics. A complete set of left-eigenvectors was presented by Ibáñez et al. [176
]. Due to the paramount
importance of the characteristic structure of the equations in the design of upwind HRSC schemes
based upon Riemann solvers, we summarize all necessary information in Section 6.2 of the
article.
The range of applications considered so far in relativistic astrophysics employing the above formulation
of the hydrodynamics equations, Equations (32) – (35
), includes the study of gravitational collapse (both
stellar-core collapse to neutron stars and black-hole formation), accretion flows on to black holes, as well as
neutron-star pulsations and neutron-star–binary mergers (see Section 5). The first applications in general
relativity were performed, in one spatial dimension, in [237
], using a slightly different form
of the equations. Preliminary investigations of gravitational stellar collapse were attempted
in [235, 53] by coupling the above formulation of the hydrodynamic equations to a hyperbolic
formulation of the Einstein equations developed by [54]. Evolutions of fully-dynamic spacetimes in
the context of stellar-core collapse, both in spherical symmetry and in axisymmetry, have also
been done [178
, 339
, 96
, 97
, 74
]. These investigations are discussed in Section 5.1.1. In the
special-relativistic limit this formulation has successfully been applied to simulate the evolution of
(ultra-) relativistic extragalactic jets, using numerical models of increasing complexity (see,
e.g., [241, 7
]).
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