Up to now most astrophysical SRHD simulations
have assumed matter whose thermodynamic properties can be described
by an inviscid ideal equation of state with a constant adiabatic
index. This simplification may have been appropriate in the first
generation of SRHD simulations, but it clearly must be given up
when aiming at a more realistic modeling of astrophysical jets,
gamma-ray burst sources, or accretion flows onto compact objects.
For these phenomena a realistic equation of state should include
contributions from radiation ( “fluid”),
allow for the formation of electron-positron pairs at high
temperatures, and allow the ideal gas contributions to be
arbitrarily degenerate and/or relativistic.
Depending on the problem to be simulated, effects due to heat conduction, diffusion, radiation transport, cooling, nuclear reactions, and viscosity may have to be considered, too. Including any of these effects is often a non-trivial task even in Newtonian hydrodynamics, as the differential operators describing advection and convection are of hyperbolic nature, while diffusion and conduction processes give rise to parabolic differential operators, and the treatment of constraints or self-gravity involves differential operators of elliptic type (see, e.g., the contributions in the book edited by Steiner and Gautschy [268]). There has been considerable development in the coupling of Newtonian HRSC methods to the nonhyperbolic terms arising in the equations from these physical processes using semi-implicit approaches, e.g., the predictor-corrector method [18]. Another example in this context provides the recent work of Howell and Grenough [127], who have coupled an explicit Newtonian Godunov-type integrator for the hyperbolic hydrodynamic equations to an implicit multigrid solver to describe effects of radiative diffusion on the flow and vice versa. We particularly mention this work here, as it also uses a block-structured adaptive mesh refinement algorithm (see Section 8.2.2). Although such sophisticated methods have not been applied in SRHD yet, they represent an important set of ideas that could provide a starting point for more elaborate SRHD simulations.
In the context of relativistic jets, Komissarov
and Falle [148], and Scheck et
al. [255] have considered a
mixture of ideal, relativistic Boltzmann gases [272] hence allowing for jets
with general (i.e., ,
,
) composition. The
usage of such a more general ideal EOS causes no special problem
for the Riemann solvers although a higher nonlinearity is
introduced in the process of the recovery of the primitive
variables. In order to avoid this extra complexity, approximate
expressions for the relativistic ideal gas EOS have been
proposed [79, 265
]. In case of the
approximation proposed by Sokolov et al. [265], the recovery of the
primitive variables is explicit. Moreover, the authors have
developed an exact Riemann solver consistent with the approximate
EOS.
An EOS describing matter consisting of a set of ideal, non-relativistic Boltzmann gases (nuclei in nuclear statistical equilibrium), a Fermi gas of electrons and photons was used in the simulations of relativistic jets from collapsars by Aloy et al. [7].
HRSC flow simulations involving elaborate microphysics may require the extension of the presently available relativistic Riemann solvers to handle general equations of state (see Section 9.1). This is the case for the Roe-Eulderink method, which can be extended following the procedure developed in the classical case by Glaister [103]. Methods based on exact solutions of the Riemann problem, like rPPM and rGlimm, can take advantage of the solution presented in Section 2.3 to cope with a general EOS. FCT based difference schemes used in simulations of relativistic heavy ion collisions (see Section 7.3) pose no specific numerical problem in handling a general EOS.
Another interesting area that deserves further research is the application of relativistic HRSC methods in simulations of reactive multi-species flows, especially as such flows still cause problems for the Newtonian CFD community (see, e.g., [232]). The structure of the solution to the Riemann problem becomes significantly more complex with the introduction of reactions between multiple species. Riemann solvers that incorporate source terms [160], and in particular source terms due to reactions, have been proposed for classical flows [19, 132]. However, most HRSC codes still rely on operator splitting.
Peitz and Appl [221] have addressed the difficult issue of non-ideal GRHD, which is of particular importance, e.g., for the simulation of accretion discs around compact objects, rotating relativistic fluid configurations, and the evolution of density fluctuations in the early universe. They have accounted for dissipative effects by applying the theory of extended causal thermodynamics, which eliminates the causality violating infinite signal speeds arising from the conventional Navier-Stokes equation. However, Peitz, and Appl have not yet implemented their model numerically.
A description of non-ideal hydrodynamics in
general relativity is also the aim of Richardson and Chung’s
work [242], although from a
less formal basis. The authors introduce an approach (the so-called
flow-field-dependent variation theory [54, 55] resting on the
conservative Navier-Stokes system of equations for classical fluid
dynamics) where local properties of the flow (advection,
turbulence, or gravity dominated) are captured in terms of relevant
parameters (measuring changes of the Lorentz factor, relativistic
Reynolds and Froude numbers between adjacent numerical zones,
respectively). These parameters are then used to produce a suitable
numerical model (hyperbolic, parabolic, elliptic) which is
subsequently discretized using finite difference or finite element
methods. The latter approach has been applied by Richardson and
Chung [242] for several test cases
(mildly relativistic Riemann problem and general relativistic
spherical dust infall).
Modeling astrophysical phenomena often involves
an enormous range of length and time scales to be covered in the
simulations (see, e.g., [205]). In two and definitely
in three spatial dimensions many such simulations cannot be
performed with sufficient spatial resolution on a static
equidistant or non-equidistant computational grid, but they rather
require dynamic, adaptive grids. In addition, when the flow problem
involves stiff source terms (e.g., energy generation by nuclear
reactions), very restrictive time step limitations may result. A
promising approach to overcome these complications is the coupling
of SRHD solvers with the adaptive mesh refinement (AMR)
technique [21]. AMR automatically
increases the grid resolution near flow discontinuities or in
regions of large gradients (of the flow variables) by introducing a
dynamic hierarchy of grids until a prescribed accuracy of the
difference approximation is achieved. Because each level of grids
is evolved in AMR on its own time step, time step restrictions due
to stiff source terms constrain the computational costs less than
without AMR.
For an overview of online information about AMR
visit, e.g., the AMRA home page of Plewa [229], and for public
domain AMR software, e.g., the AMRCLAW home page of LeVeque and
Berger [162], the web page of the
Lawrence Berkeley Lab dedicated to AMR [1], and the NASA
Goddard Space Flight Center web page on PARAMESH [171]. Astrophysical
applications based on PARAMESH can be found at the web site of the
ASCI / Alliances Center for Astrophysical Thermonuclear Flashes at
the University of Chicago [2]. Although, as demonstrated by these
web sites, there has been a considerable effort over the last few
years in developing frameworks for block-structured adaptive mesh
refinement, we will see that the application to SRHD is still in
its infancy.
An SRHD simulation of a relativistic jet based on
a combined HLL-AMR scheme was performed by Duncan and
Hughes [78]. Plewa et
al. [231, 230] have modeled the
deflection of highly supersonic slab jets propagating through
non-homogeneous environments using the HRSC scheme of Martí et
al. [183
] combined with the
AMR implementation AMRA of Plewa [229]. A similar study, but in
3D, was performed by Hughes et al. [128] who studied the
deflection and precession of cylindrical relativistic jets when
impinging on an oblique density gradient using the SRHD code of
Duncan and Hughes [78] extended to 3D and their
own implementation of the AMR technique of Berger and
Colella [21]. Komissarov and
Falle [147] have combined their
numerical scheme with the adaptive grid code Cobra, which has been
developed by Mantis Numerics Ltd. for industrial
applications [88], and which uses a
hierarchy of grids with a constant refinement factor of two between
subsequent grid levels.
Up to now only very few attempts have been made
to extend HRSC methods to GRHD (for a comprehensive review see
Font [91]). All these
attempts are based on the usage of linearized Riemann
solvers [179
, 84
, 249, 15, 93
]. In the most recent
of these approaches, Font et al. [93] have developed a 3D
general relativistic HRSC hydrodynamic code where the matter
equations are integrated in conservation form and fluxes are
calculated with Marquina’s formula.
A very interesting and powerful procedure was proposed by Balsara [13] and has been implemented by Pons et al. [233]. This procedure allows one to exploit all the developments in the field of special relativistic Riemann solvers in general relativistic hydrodynamics. The procedure relies on a local change of coordinates at each zone interface such that the spacetime metric is locally flat. In that locally flat spacetime any special relativistic Riemann solver can be used to calculate the numerical fluxes, which are then transformed back. The transformation to an orthonormal basis is valid only at a single point in spacetime. Since the use of Riemann solvers requires the knowledge of the behavior of the characteristics over a finite volume, the use of the local Lorentz basis is only an approximation. The effects of this approximation will only become known through the study of the performance of these methods in situations where the structure of the spacetime varies rapidly in space and perhaps time as well. In such a situation finer grids and improved time advancing methods will definitely be required. The implementation is simple and computationally inexpensive.
Characteristic formulations of the Einstein field
equations are able to handle the long term numerical description of
single black hole spacetimes in vacuum [24]. In order to include
matter in such an scenario, Papadopoulos and Font [220] have generalized
the HRSC approach to cope with the hydrodynamic equations in such a
null foliation of spacetime. Actually, they have presented a
complete (covariant) reformulation of the equations in GR, which is
also valid for spacelike foliations in SR. They have extensively
tested their method, calculating, among other tests, shock tube
problem 1 (see Section 6.2.1), but posed on a light cone
and using the appropriate transformations of the exact
solution [180] to account for advanced
and retarded times.
Other developments in GRHD in the past included finite element methods for simulating spherically symmetric collapse in general relativity [173], general relativistic pseudo-spectral codes based on the (3+1) ADM formalism [11] for computing radial perturbations [112] and 3D gravitational collapse of neutron stars [32], general relativistic [172, 204] and post-Newtonian [12] SPH. The potential of these methods for the future is unclear, as none of them is specifically appropriate for ultra-relativistic speeds and strong shock waves which are characteristic of most astrophysical applications.
Let us remark that, in the case of dynamic spacetimes, the equations of relativistic hydrodynamics are solved on the local (in space and time) background solution provided by the Einstein equations at every time step [91]. The solution of the Einstein gravitational field equations and its coupling with the hydrodynamic equations is the realm of Numerical Relativity, which is beyond the scope of this article (see, e.g., Lehner [157] for a recent review).
The inclusion of magnetic effects is of great importance for many astrophysical phenomena. The formation and collimation process of (relativistic) jets (powering powerful extragalactic radio sources, galactic microquasars, and GRBs) most likely involves dynamically important magnetic fields and occurs in strong gravitational fields. The same is likely to be true for accretion discs around black holes. Magneto-relativistic effects even play a non-negligible role in the formation of proto-stellar jets in regions close to the light cylinder [41]. Thus, relativistic MHD codes are a very desirable tool in astrophysics. The non-trivial task of developing such a kind of code is considerably simplified by the fact that because of the high conductivity of astrophysical plasmas one must only consider ideal RMHD in most applications.
The aim of any (Newtonian or relativistic) MHD
code is to evolve the induction equation to obtain the magnetic
fields from which to calculate the Lorentz force. Magnetic fields
are divergence free, i.e., . Hence,
numerical schemes are required to maintain this constraint (if
fulfilled for the initial data) during the evolution. A first step
towards the development of a relativistic (in fact, general
relativistic) MHD code was made by Evans and Hawley [86] who incorporated a
numerical scheme for the induction equation (constrained transport), which maintained
zero divergence of the magnetic field up to machine round-off
error, into the axisymmetric, two-dimensional finite difference
code of Hawley et al. [123]. In Evans and Hawley’s
method the magnetic flux through cell interfaces is the fundamental
“magnetic” variable. Their method is also based on the use of a
staggered mesh (some quantities including the magnetic field
components are defined at grid interfaces). Thus, even in classical
MHD, the extension of the constrained transport method to
Riemann-solver-based schemes (that rely on fluxes at cell
interfaces derived from cell averaged quantities) is
non-trivial [65, 252]. Tóth [280] reviews and compares
strategies (namely the eight-wave
formulation, several versions of the constrained transport,
and the projection scheme) used in
HRSC schemes in classical MHD to maintain the constraint
numerically. His conclusions also apply to RMHD.
Special relativistic 2D MHD test problems with
Lorentz factors up to have been investigated by
Dubal [77] with a code based on FCT
techniques (see Section 4).
Van Putten [286, 287, 290] has proposed a method for accurate and stable numerical simulations of RMHD in the presence of dynamically significant magnetic fields in two dimensions and up to moderate Lorentz factors. The method is based on MHD in divergence form using a 2D shock-capturing method in terms of a pseudo-spectral smoothing operator (see Section 4). He applied the method to 2D blast waves [289] and astrophysical jets [288, 291].
In a series of papers, Koide and
coworkers [138, 136
, 209
, 210
, 139
] have investigated
relativistic magnetized jets using a symmetric TVD scheme (see
Section 3). Koide, Nishikawa, and
Mutel [138
] simulated a 2D RMHD
slab jet, whereas Koide [136
] investigated the
effect of an oblique magnetic field on the propagation of a
relativistic slab jet. Nishikawa et al. [209
, 210] extended these
simulations to 3D and considered the propagation of a relativistic
jet with a Lorentz factor
along an aligned and an
oblique external magnetic field. The 2D and 3D simulations
published up to now only cover the very early propagation of the
jet (up to 20 jet radii) and are performed with moderate spatial
resolution on an equidistant Cartesian grid (up to 101 zones
per dimension, i.e., 5 zones per beam radius). Concerning higher
order symmetric non-oscillatory schemes, the very recent work by
Del Zanna et al. [72
] has to be
mentioned. Their third order scheme produces results which are
competitive with those obtained by Riemann-solver based methods
(see next paragraph) but avoiding all the complexity associated
with the spectral decomposition into characteristic fields
(particularly the degeneracies). Its high order and its simplicity
make this approach very appealing.
Steps towards the extension of linearized Riemann
solvers to ideal RMHD have already been taken. All theoretical
aspects (RMHD as a quasi-linear hyperbolic system, spectral
decomposition of the Jacobian of the flux vector in covariant form,
study of simple waves and shock waves) are compiled in the book by
Anile [8], augmenting previous work of
Lichnerowicz [163]. Romero [250] derived an analytic
expression for the spectral decomposition of the Jacobian matrix of
the flux vector in the case of a planar relativistic flow field
permeated by a transversal magnetic field (nonzero field component
only orthogonal to flow direction). Anile and Pennisi [9] and Van Putten [292] studied the
characteristic structure of the RMHD equations in (constraint free)
covariant form. Finally, Balsara [14] and
Komissarov [143] have developed robust,
second-order accurate (in space and time), Godunov-type schemes for
1D and 2D RMHD, respectively. Both start from the spectral
decomposition of the RMHD system of (ten) equations in covariant
form, extract the relevant information (wave speeds, jumps in the
characteristic variables) for the (seven) physical waves, and
analyze the cases of degeneracy (i.e., cases where several wave
speeds corresponding to different waves become degenerate).
Komissarov’s RMHD scheme is an extension of the scheme developed
for RHD [89] described in
Section 3.5, which avoids the use of the
left eigenvectors (in [14] they are computed
numerically). In its multi-dimensional version, Komissarov’s code
enforces
by employing the integral form of the
induction equation. This code has been used to study the
propagation of light, highly relativistic jets carrying toroidal
magnetic fields [144].
Koide, Shibata, and Kudoh [139] performed simulations of magnetically driven axisymmetric jets from black hole accretion disks. Their GRMHD code [140] is an extension of the special relativistic MHD code developed by Koide et al. [138, 136, 209]. The necessary modifications of the code were quite simple, because in the (nonrotating) black hole’s Schwarzschild spacetime the GRMHD equations are identical to the SRMHD equations in general coordinates, except for the gravitational force terms and the geometric factors of the lapse function. The authors have recently extended their code to Kerr spacetimes [141] and performed simulations of axisymmetric jets formed by extracting rotational energy from a black hole [137, 142]. Finally, using a 3D GRMHD code, Nishikawa et al. [211] have investigated the dynamics of a freely falling corona and of a Keplerian accretion disk around a Schwarzschild black hole to form bipolar relativistic jets assuming axisymmetry as in previous simulations.
With the pioneering work of Koide and collaborators, numerical simulations have entered into the realm of GRMHD. However, despite their success, present simulations only cover a tiny fraction of dynamical time scales (about 2 rotational periods of the accretion disk) and jets are formed with very small terminal speeds (Lorentz factors less than 2). Hence, the quest for robust codes able to follow the formation of steady relativistic jets is still open. Given their success in SRHD, the extension of Riemann-solver based HRSC methods is an obvious option to bear in mind. Again, the third-order symmetric HRSC algorithms developed recently [72] represent a very interesting alternative.