We start with those purely hydrodynamic codes which also consider dynamic spacetimes. Hydrodynamic codes, which only solve for the matter dynamics on fixed background metrics (quite more numerous), are listed in Table 1, where we give their main basic features and provide the relevant references for further reading. Correspondingly, Table 2 lists test-fluid GRMHD codes, which could also, in principle, deal with unmagnetized flows by simply setting to zero the magnetic field contribution. In addition, the reader is directed to the Living Review article of [240] where similar tables can be found for the particular case of special relativistic hydrodynamics.
CACTUS/WHISKY: The whisky code was the result of a collaboration among several European
institutions [246, 27]. This code solves the general relativistic hydrodynamics equations formulated as in
Section 2.1.3 on a 3D numerical grid with Cartesian coordinates. The code has been constructed within
the framework of the Cactus Computational Toolkit (see [244
] for details), developed at the
Albert Einstein Institute (Potsdam-Golm, Germany) and at Louisiana State University (Baton
Rouge, Louisiana). This public-domain code provides high-level facilities such as parallelization,
input/output, portability on different platforms and several evolution schemes to solve general systems
of partial differential equations. Special attention is dedicated to the solution of the Einstein
equations, whose matter terms in nonvacuum spacetimes are handled by the whisky code.
The initial development of whisky benefitted in part from the release of a public version of
the general-relativistic hydrodynamics code gr_astro described in [137
, 131
], and developed
mostly by the group at Washington University (St. Louis, Missouri). The main features of the
whisky code are (a) cell-reconstruction procedures with minmod and monotonized central
(MC) slope limiters, along with PPM and ENO methods; (b) full-wave decomposition and
incomplete Riemann solvers to compute the numerical fluxes such as HLLE, Roe-like, and Marquina
flux formula; (c) analytic expression for the right and left-eigenvectors of flux-vector Jacobian
matrices [176] and compact flux formulae [10] for the Roe-Riemann solver and Marquina’s
flux formula; (d) a “method of lines” (MoL) approach for the implementation of conservative,
high-order time evolution schemes; (e) the possibility to couple the GRHD equations with a
conformally-decomposed three-metric in the BSSN formulation[363
, 39
]. It is worth noting that whisky also
incorporates excision methods adapted to HRSC schemes for studying black-hole formation, as
described in [166]. In particular, a modified PPM reconstruction scheme was developed to handle
the boundary of the excised region inside apparent horizons. The accuracy of the simulations
performed with the whisky code can also benefit from the mesh refinement driver implemented in
the cactus code, called carpet, as recently demonstrated in the gravitational collapse studies
of [28
, 305
].
CoCoNuT: This code, described in detail in [97], was designed with the aim of studying general-relativistic
astrophysical scenarios such as rotational core collapse to neutron stars and black holes, as well as
pulsations and instabilities of the formed compact objects. Contrary to most 3D codes, which are based on
Cartesian coordinates, the CoCoNuT code employs spherical coordinates. Upwind HRSC methods are used
for the hydrodynamic equations in the formulation of Section 2.1.3 (Marquina’s flux formula and PPM
cell-reconstruction procedures) and spectral methods (through the LORENE library) for solving the metric
equations which are formulated in the conformal flatness (CF) approximation (an approach,
which [97
] coined Mariage des Maillages, i.e., French for grid wedding). In this approximation, and
under the maximal slicing condition, the 3+1 ADM equations reduce to a set of five coupled
elliptic (Poisson-like) nonlinear equations for the metric components (lapse function, shift vector
and conformal factor), which are optimally suited to be solved using spectral methods (see
Section 4.2.2). The CF approximation has proved to agree remarkably well with BSSN in simulations
performed with the whisky code for rotating neutron stars and stellar core collapse [98
, 305
].
Extensions of the CF approximation also implemented in the CoCoNuT code are considered
by [74
].
Shibata’s GRHD code: In this code the hydrodynamics equations are formulated both in a
nonconservative way (following Wilson’s approach of Section 2.1.2; see [353]) and in a conservative way
similar to the approach of Section 2.1.3 (see [355
]). In the former case, an important difference with respect
to the original system given by Equations (20
) – (22
), is that an equation for the entropy is
solved instead of the energy equation. The hydrodynamic equations are integrated using van
Leer’s [403
] second-order finite difference scheme with artificial viscosity, following the approach of a
precursor code developed by [303]. For the conservative system, both upwind HRSC schemes
and high-order central schemes are implemented in the code, along with third-order parabolic
cell-reconstruction procedures. As mentioned before, the choice of conserved variables in the conservative
formulation is, however, slightly different to that of Section 2.1.3, mainly due to the presence of
a common factor
in all quantities,
being a spacetime conformal factor, motivated
by the formulation used in the code for solving Einstein’s equations. Those are written using
the BSSN formulation and solved with finite differences (see [355
] for details particular to the
formulation of the spacetime variables and their solution). In addition, [355
] uses the transport
velocity throughout (instead of
) and in the energy equation the continuity equation is
not subtracted (contrary to the approach of [36
]). The choice of coordinates in Shibata’s code
depends on the dimensionality of the simulation under study. In the full 3D case, Cartesian
coordinates
are employed. While this is also the case in axisymmetry there is an
important subtle difference worth mentioning; the hydrodynamics equations are first written using
cylindrical coordinates
, with
and
. However, the
Einstein’s equations are solved in the
plane using Cartesian coordinates. To be able to
compute
derivatives in this plane the code implements the “cartoon” method [6
], which makes
possible axisymmetric computations with a Cartesian grid. Next, the hydrodynamics equations are
rewritten in Cartesian coordinates using appropriate relations at the
plane (see [355
]
for details). Comprehensive tests of the code are described in [353
, 355
, 361
]. Shibata’s code
has allowed important breakthroughs in the study of the morphology and dynamics of various
general-relativistic astrophysical problems, such as black-hole formation resulting from both the
gravitational collapse of rotating neutron stars and rotating supermassive stars, dynamic instabilities of
rotating neutron stars, and, perhaps most importantly, the coalescence of neutron-star binaries, a
long-standing problem in numerical relativistic hydrodynamics. These applications are discussed in
Section 5.
Duez et al. [109]: The code of [109
] shares many features with that of Shibata just discussed. As in the
case of [355
] the Einstein equations are formulated in the BSSN approach and solved using an iterative
Crank–Nicholson scheme for the time update and second-order centered differencing for the spatial
derivatives. Correspondingly, the hydrodynamics equations are formulated in the same conservative
approach followed by [355
], yet they are not solved using HRSC schemes, but through an artificial
viscosity (either quadratic or linear), which allows it to handle shock waves. A noteworthy feature
of this code is the incorporation of an algorithm, which makes unnecessary the addition of
a low-density atmosphere required in most Eulerian hydrodynamic codes, either Newtonian
or relativist, when studying the dynamics of matter sources of compact support (as, e.g., the
pulsations of stars). Technical details of boundary conditions and gauge choices, along with a
comprehensive list of tests passed by the code, are available in [109
]. Extensions of this code to account
for viscous fluids through the solution of the relativistic Navier–Stokes equations are reported
in [107].
|
Oechslin et al. [299]: As in the CoCoNuT code described above, in the code of [299
] the relativistic
hydrodynamics equations are solved together with the Einstein equations in the conformally-flat
approximation. The code has evolved from an early Newtonian version, which was designed with the
definite aim of studying the merger of neutron star binaries. It has gradually been improved to provide
ever more accurate descriptions of such mergers. In its latest version the code incorporates
microphysical treatment of neutron-star matter through two finite-temperature nuclear EOSs
along with a modern treatment to extract gravitational waveform information. Compared to the
other codes mentioned before, the most noticeable feature of the code of [299
] is the use of
SPH techniques to solve for the relativistic hydrodynamics equations, which are formulated in
conservation form. The code implements the new SPH artificial viscosity formalism of [79] in which
the artificial viscosity interaction is determined by approximately solving a Riemann problem
between particle neighbors. Binary merger simulations performed with this code are discussed in
Section 5.
Characteristic numerical relativity codes: Although most numerical relativity codes are based on the
Cauchy problem, there also exist a couple of characteristic numerical relativity codes, which can handle
hydrodynamics. On the one hand, there is the axisymmetric code of [378, 379
], which has been used to
perform dynamic evolutions of neutron stars, both pulsating and those formed after a core collapse. In
this code the hydrodynamics equations are implemented following the conservative approach
outlined in Section 2.2.2 and solved using upwind HRSC schemes, high-order cell-reconstruction
procedures, and conservative Runge–Kutta schemes for the time update. Regarding full 3D, the
Pittsburgh characteristic vacuum code has recently been upgraded to account for perfect fluid matter
sources [46
], using the same conservative formulation of Section 2.2.2. However, instead of
relying on Riemann-solver-based methods for their solution, the hydrodynamics equations are
solved in the Pitt code with a dissipative central scheme designed by [83
]. Using this code,
short time evolutions of a self-gravitating star in close orbit around a black hole are discussed
in [46].
As an important note we point out that all codes based on the Cauchy approach place the outer
boundary of the grid at a finite distance, which may potentially lead to numerical problems caused by
unwanted reflections of outgoing waves back into the computational domain. Further work on the
development of sophisticated boundary conditions is needed to solve (or at least ameliorate) this type of
problem. Alternative solutions, which allow for compactified spacetimes that include future null infinity on
the computational grid, are provided by the light-cone approach developed by Winicour et al. [419] or the
conformal formalism of Friedrich [142].
As in the previous Section 4.4.1, we focus on magnetohydrodynamic codes, which also consider dynamic
spacetimes. GRMHD codes which “only” solve for the matter dynamics on fixed background
metrics are listed in Table 2, where we give their main basic features and provide references for
further reading. As mentioned before, the development of such test-fluid GRMHD codes has
witnessed spectacular progress in the last few years. This has been possible not only thanks to the
various conservative formulations, which have been put forward but also, and perhaps most
importantly, because of the use of accurate, characteristic-information-free HRSC schemes. Such
schemes have rendered possible systematic investigations of strongly magnetized scenarios of
relativistic astrophysics, which had thus far remained out of numerical reach. We note that in the
characterization followed in Table 2 to classify the various existing codes we have made the
explicit distinction between full-wave-decomposition approximate Riemann solvers (such as
those used in the codes of [201, 24
]) and incomplete Riemann solvers in which only a subset of
wave speeds are used. In this regard, we place the widely used HLL (and HLLE) scheme in
the latter group, together with symmetric schemes, despite the fact that HLL was originally
developed as an approximate Riemann solver scheme [164] with a particularly simple numerical
flux.
Shibata’s GRMHD code [366]: As described in Section 3.1.2 the GRMHD equations are formulated in
the code of [366
] in a conservative way. For their solution the semi-discrete high-resolution central scheme
of Kurganov–Tadmor [211] is used (see also [361]), together with a piecewise parabolic interpolation of the
primitive variables for the cell-reconstruction scheme. In order to obtain the maximum characteristic
speeds needed for the central scheme, instead of solving the quartic equation in the characteristic
polynomial, Shibata and Sekiguchi [366
] follow the convenient prescription proposed by [149
] and
approximate the fourth-order equation by a quadratic equation for which there exists an analytic
solution. The magnetic field divergence-free constraint is enforced with a constrained transport
scheme. Correspondingly, and as in the purely hydrodynamic code, Einstein’s equations are also
written in the GRMHD code using the BSSN formulation and solved with finite differences
(see [355
] for further details). The code is axisymmetric and, as in the hydrodynamic case, the
“cartoon” method is employed [6
] for evolving the BSSN equations, which makes possible the use of
Cartesian coordinates in axisymmetric simulations, and a cylindrical grid is used for the GRMHD
equations. It has recently been applied in the study of the collapse of magnetized hypermassive
neutron stars to black holes in the context of gamma-ray burst mechanisms, as we discuss in
Section 5.
Duez et al. [108]: The development of this code has proceeded simultaneously to that of [366], to the
point that they share some important features and both have been used to study similar astrophysical
problems, whose results have been presented in joint publications [105
, 106]. This code solves the GRMHD
equations in both two dimensions (axisymmetry) and three dimensions, using a Cartesian grid for the latter
case and the “cartoon” method in the former (although a cylindrical grid is used for the induction and
MHD evolution equations). As in the purely hydrodynamic code of the same group [109
] the Einstein
equations are formulated in the BSSN approach and solved using a three-step iterative Crank–Nicholson
scheme for the time update, which yields second-order accuracy in time. The main modification of the
GRMHD code with respect to its predecessor is the complete reformulation of the hydrodynamic
sector, whose accuracy has been improved through the use of shock-capturing capabilities. More
precisely, the GRMHD evolution equations are solved with the HLL approximate Riemann
solver, together with different possibilities for the cell-reconstruction step (either MC, CENO, or
PPM). It is worth noting that, contrary to the no-atmosphere approach used in the purely
hydrodynamic code [109
] to handle vacuum regions outside stars, the GRMHD code of [108
] is not
suitable for such an approach and a very small positive density must be maintained outside the
stars.
|
WHISKY-MHD: As the name indicates, this code is the extension of the hydrodynamic code whisky
discussed above to solve for the full set of GRMHD equations in a dynamic spacetime. The code is
described in [151]. It is a fully three-dimensional code in Cartesian coordinates, based on HRSC techniques
on domains with adaptive mesh refinement, accomplished as in the whisky code through the AMR driver
implemented in the cactus code, called carpet. The whisky-MHD code uses, hence, the cactus
computational toolkit which provides the infrastructure for the parallelization and the I/O of the code,
along with methods for the solution of the Einstein equations formulated in the BSSN approach [363, 39].
The GRMHD equations are implemented using the conservative formulation of [24] discussed in
Section 3.1.2 and integrated in time employing the method of lines. The code uses an HLLE approximate
Riemann solver, with second-order TVD slope limiters for the cell-reconstruction procedure, and the
magnetic-field divergence-free constraint is enforced using the flux-CT scheme briefly discussed in
Section 4.3.
Anderson et al. [12]: The most distinctive feature of this code, which is mainly described in [12
] (see
also [286, 14] for further details and [13
] for the assessment of the purely hydrodynamic module of the
code), is the fact that it uses a sophisticated computational infrastructure, which provides distributed
AMR. This has allowed the investigation of both unmagnetized and magnetized neutron-star binary
mergers and the computation of the gravitational radiation in the wave zone far from the merger [13
, 12
].
As mentioned in Section 3.1.2 the code uses a conservative formulation of the GRMHD equations. These
are solved by means of the HLLE approximate Riemann solver with PPM cell reconstruction, in conjunction
with divergence cleaning to control the divergence-free constraint. Correspondingly, the Einstein
equations, cast in first-order symmetric hyperbolic form, are solved in the generalized harmonic
decomposition.
Cerdá-Durán et al. [75]: The GRMHD code developed by [75
] has been designed to study
magneto-rotational, relativistic, stellar core collapse [76
]. It is an extension of the axisymmetric
hydrodynamics code developed by [95
] (whose 3D extension constitutes the CoCoNuT code described above),
in which magnetic fields are included following the approach laid out in [24
]. Einstein’s equations are
formulated using the conformally-flat condition, which has proved very accurate for studying rotational core
collapse [96
, 98
] and are solved using spectral methods. In the code, both the HLLE approximate Riemann
solver and the Kurganov–Tadmor central scheme are implemented to solve for the MHD evolution
equations, along with the flux-CT scheme for the magnetic-field divergence-free constraint. As a first step
towards magneto-rotational core collapse simulations, an early version of the code assumed a passive
(test) magnetic field, a justifiable assumption since weakly-magnetized fluids are present in such
astrophysical scenarios. An extension of the code to relax this assumption has recently been
finished.
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