For a fluid endowed with a magnetic field, the stress-energy tensor is the sum of that of the fluid and
that of the electromagnetic field, , where
is given by Equation (5
) for a
perfect fluid. On the other hand
can be obtained from the Faraday tensor as follows:
As happens for the equations of general-relativistic hydrodynamics discussed previously, available
approaches to cast the corresponding magnetohydrodynamics equations in forms suitable for numerical
work also follow to a great extent the basic distinction between conservative and nonconservative
formulations. The latter, which are covered in this section, were first laid out by Wilson in the mid
1970s, who pioneered a new emerging field in numerical astrophysics, just as he had done a
few years earlier for numerical Eulerian hydrodynamics. The original approach was followed
by [429] and more recently by [86
] and [20
], who adopted an internal energy formulation with
artificial viscosity for shock capturing and a method of characteristics for handling the magnetic
fields.
The dynamic variables used in the formulation of [86],
and
, coincide with those reported in
Equation (23
), save for the definition of the total four-momentum
, which includes the norm of the
magnetic field four-vector
, namely
, with
. With this choice, the
“conservation” equations of mass density, momentum and energy are again written as a set of advection
equations similar to Equations (20
) – (22
), except for the inclusion of the magnetic field terms,
namely:
where is again the transport velocity defined in Section 2.1.2. As in the purely hydrodynamic case,
the time component of the momentum is obtained from the corresponding normalization condition together
with the algebraic relations between the magnetic-field components. Similarly, an artificial viscosity must be
added for the numerical solution of the last two equations to increase the entropy of the fluid
at shocks. In [86
] such a viscosity prescription coincides with that used in the unmagnetized
case [171
], except for the modification of the definition of the inertia to include the magnetic
energy.
In addition, a supplementary evolution equation is needed for the constrained transport magnetic field
, the induction equation, which reads:
A similar internal-energy formulation was presented in [20]. There are, however, important subtle
differences with respect to the approach of [86
] in the particular form of the evolution equations, due
to the different definition of the Lorentz (relativistic boost) factor employed by [20
], namely
. In addition, there are a couple of points particular to the formulation of [20
],
which are worth mentioning. First, in the momentum-evolution equation, the term
is cast
into projected and divergence components
, which has significant numerical
advantages for achieving stable formulations of sheared Alfvén waves. Second, the induction equation is
rewritten in the form
We turn now to the discussion of conservative formulations for the GRMHD equations. Very recently, a
number of groups have proposed various such conservative approaches to formulating these equations, all
aimed at solving them using methods specifically designed for hyperbolic systems of conservation (or
balance) laws [149, 20
, 201
, 108
, 366
, 24
, 286
, 265
, 91
]. The existing approaches all share the key
ingredient of casting the GRMHD equations in first-order flux-conservative form, yet the choice of variables
(state-vector, fluxes, etc) differs (albeit slightly) among most of them. Therefore, for the sake of clarity, we
start here by discussing one such conservative formulation first, namely that of [24
] (which coincides
with the one obtained by [286
] and is also employed in the codes of [265
, 151
]), leaving to the
second part of this section the analysis of similarities and differences present in the additional
formulations.
Following [24] the conservation equations for the energy-momentum tensor, together with the continuity
equation and the equation for the evolution of the magnetic field, can be written as a first-order,
flux-conservative, hyperbolic system equivalent to Equation (32
). The state vector and the vector of fluxes
of the GRMHD system of equations read
The hyperbolic structure of the flux-vector Jacobian matrices corresponding to Equations (57)-(58
), the
knowledge of which is essential for building up a numerical code based on upwind HRSC schemes, is
analyzed in [24
] (see also [23
]). In the classical MHD case, the wave structure has been analyzed by [61
],
which shows that there are seven physical waves: two Alfvén waves (with eigenvalues
,
and
being the fluid and Alfvén speeds, respectively), two fast and two slow magnetosonic waves
(
,
), and one entropy wave (
). The expressions for the Alfvén and
magnetosonic speeds read
We note in passing the numerical analysis of the hyperbolic structure of the classical ideal MHD system carried out by [400], where nonuniform convergence to the true solution was found for a number of finite volume methods in coarse grid simulations.
The wave structure of the above GRMHD hyperbolic system is, understandably, more involved than
that corresponding to the GRHD case. As mentioned before, Anile [15] (see also [16, 404]) studied the
characteristic structure of these equations (eigenvalues, right and left-eigenvectors) in the (ten-dimensional)
space of covariant variables
,
being the specific entropy. It was found that only
the entropic waves and the Alfvén waves can be explicitly written in closed form. For the
formulation of [24
] discussed in this section, and along each coordinate direction, they are given
by
On the other hand, the (fast and slow) magnetosonic waves are given by the numerical solution (through
a root-finding procedure) of a quartic characteristic equation. Among the magnetosonic waves, the two
solutions with maximum and minimum speeds are called fast magnetosonic waves , and the two
solutions in between the slow magnetosonic waves
. The seven waves can be ordered as follows
We note that, as a result of working with the unconstrained, system of equations, there
appear superfluous eigenvalues associated with unphysical waves, which need to be removed (the entropy
waves as well as the Alfvén waves appear as double roots). Any attempt to develop a numerical procedure
based on the wave structure of the RMHD equations must, hence, remove these nonphysical waves (and the
corresponding eigenvectors) from the wave decomposition. In the particular case of special-relativistic MHD,
[199
] and [198
] eliminate the unphysical eigenvectors by enforcing the waves to preserve the values of the
invariants
and
, as suggested by [15]. Correspondingly, in [34
] the physical
eigenvectors are selected by comparing with the equivalent expressions in the nonrelativistic
limit.
We also note that, as in the classical case, the relativistic MHD equations have degenerate states in
which two or more wavespeeds coincide. This breaks the strict hyperbolicity of the system and may affect
numerical schemes for its solution. These degeneracies are analyzed in [199], which finds that,
with respect to the fluid rest frame, the degeneracies in both classical and relativistic MHD
are the same; either the slow and Alfvén waves have the same speed as the entropy wave
when propagating perpendicularly to the magnetic field (Degeneracy I), or the slow or the
fast wave (or both) have the same speed as the Alfvén wave when propagating in a direction
aligned with the magnetic field (Degeneracy II). On the other hand, [24
] derives a covariant
characterization of such degenerate states, which can be checked in any reference frame. In
addition, [23
] also works out a single set of right and left eigenvectors, which are regular and
span a complete basis in any physical state, including degenerate states. Such renormalization
procedure can be seen as a relativistic generalization of the work performed in [61] in classical
MHD.
Finally, it is worth pointing out that major advances in the physical understanding of the wave structure
of the GRMHD equations have been possible in recent years owing to the remarkable derivation of the exact
solution of the Riemann problem in special relativistic MHD [340, 150
].
First, in the formulations of Gammie et al. [149] and Komissarov [201
], the choice of conserved variables
corresponding to the momentum and energy equations follows directly from the conservation of energy
momentum, which is written with the free index down, i.e.,
. The continuity equation,
however, remains identical to the expression used in [24
]. Therefore, the vector of conserved variables
in [201
, 149
] reads
The choice of for the energy-momentum equations is not capricious, but rather motivated by the
reduction on the number of source terms in the corresponding evolution equations for the case of spacetime
metrics with symmetries (ignorable coordinates
; see also [309] for a similar approach in the purely
hydrodynamic case). This is not the situation in the formulation of [24
] in which the energy-momentum
equations are obtained, in the spirit of the 3+1 splitting, from projections of the conservation law, namely
and
.
Regarding the induction equation, both [149] and [201
] use the following expression
On the other hand, in the approach followed by Shibata and Sekiguchi [366], the evolution equations for
the energy-momentum equations are also derived as in [24
], from projections of the conservation law.
However, the choice of conserved variables is slightly different, mainly due to the presence in all equations of
a common factor
,
being a spacetime conformal factor, motivated by the formulation used in the
numerical code for solving Einstein’s equations, namely BSSN [363
, 39
]. (We note that such factors are also
present in the formulation used by Shibata in the purely hydrodynamic case; see, e.g., [355
].)
Another important difference with [24
] is the choice of the transport velocity
throughout (instead of
) and the fact that in the final form of the energy equation the
continuity equation has not been subtracted. Additionally, the induction equation also adopts a
different form as it is written for a three-magnetic field given by
, with
.
Correspondingly, in the conservative formulation of Anninos et al. [20] the main differences appear in
the energy-momentum equations, leaving the continuity equation unaltered with respect to
the nonconservative formulation discussed in Section 3.1.1. Regarding the momentum and
energy-conservation equations the choice of variables in [20
] is different to those of [24
] and [366
], as they
are components of the energy-momentum tensor and not projections of the conservation law. But
the choice is also different to that in [149
] and [201
], as the indices of the energy-momentum
tensor are both chosen to be contravariant (instead of
and
). More precisely, the
corresponding conserved variables are
and
. Another important point
to mention regarding the set of equations of this conservative formulation (and also in their
nonconservative internal-energy formulation) is the presence of factors of
in some terms, which
have not been absorbed by the definition of the electromagnetic tensor (as in most of the other
approaches). Finally, regarding the induction equation, we note that it is also cast in fully conservative
form
Next, the formulation of Duez et al. [108] also shares some similarities and has some differences
with all previous approaches discussed so far. The transport velocity is used throughout. The
choice of conserved variable for the continuity equation, which the authors name
, is the
same as in [20
], being the equation also cast as a source-free advection equation. As for the
momentum equation the choice of conserved variable coincides with that of [149
, 201
], i.e., direct
components of the energy-momentum tensor, namely
. However, for the remaining part of
the energy-momentum conservation law, which leads to the energy equation, Duez et al. [108
]
do not follow [149
, 201
] and, instead of using
, they choose
, the same variable
as [24
], also subtracting the continuity equation, namely
. Finally, the
induction equation is written as in Equation (67
), but without the divergence cleanser coefficient
(
).
Finally, in the formulation of Del Zanna et al. [91] an effort is made in the choice of the state vector so
that the GRMHD equations keep a somewhat close relation to classical MHD and to simplify the expression
of the source terms. As in Antón et al. [24
] the same definition for the three-velocity is used, along with
the same variable for the relativistic density
(densitized with a factor
). For the Euler equation,
components of the stress-energy tensor
are used, as in the approaches of [149
, 201
, 108
], which
reduces the complexity of the corresponding source term. The most important difference with respect to
other formulations is in the choice of conserved variable for the energy equation, for which [91
] take
. The continuity equation is not subtracted from the energy equation. As far as
the induction equation is concerned, the same conservative expression used by [149
, 201
] is
employed.
http://www.livingreviews.org/lrr-2008-7 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |