General-relativistic MHD is concerned with the dynamics of relativistic, electrically-conducting fluids
(plasma) in the presence of magnetic fields. Here, we concentrate on purely ideal GRMHD, neglecting the
presence of viscosity and heat conduction in the limit of infinite conductivity, i.e., the fluid is
assumed to be a perfect conductor (see Section 3.3 for a brief discussion of the validity of this
approximation). Like the GRHD equations discussed in the preceding Section 2, the GRMHD
equations can also be cast in first-order, flux-conservative, hyperbolic form that is well suited to
numerical work. Concerning this issue, the contribution of Anile [15] is remarkable (but see
also [220] for an in-depth analysis of these equations), as he carried out a comprehensive study of
the mathematical structure of the GRMHD equations. In order to analyze the hyperbolicity
of the equations Anile found it convenient to write those equations using the following set of
covariant variables
, where
is the magnetic-field four-vector in the fluid
rest frame and
is the specific entropy. With respect to these Anile variables, the system of
GRMHD equations can be written as a quasi-linear system of the form
, where
the indices
and
run from zero to nine, as the number of variables. The particular
form of the
matrices
can be found in [15
]. The eigenvalue structure of the
system can be obtained by considering a generic characteristic hypersurface of the previous
quasi-linear system,
, such that it defines a characteristic matrix
, whose
determinant must vanish, with
. If we now consider a wave propagating in an arbitrary
direction
with a speed
, the normal to the characteristic hypersurface is given by the
(spacelike) 4-vector
, which can be substituted in the determinant of the
characteristic matrix to obtain the corresponding characteristic polynomial, whose zeroes give the
characteristic speed of the waves propagating in the
-direction. Three different kinds of
waves can be obtained according to which factor in the equation resulting from the condition
vanishes, either entropic waves, Alfvén waves, or magnetosonic waves. It is
worth noting that in Anile’s study [15
] both the entropy and Alfvén waves appear as double
roots of the characteristic polynomial, a direct result of working with an augmented system of
equations. Therefore, those nonphysical waves must be removed from the wave decomposition
when building up a numerical scheme based upon such wave structure to solve the GRMHD
equations.
In recent years there has been intense research on formulating and solving numerically the MHD equations in
general-relativistic spacetimes, either background or dynamic [196, 86
, 40, 149
, 20
, 201
, 108
, 366
, 24
, 286
, 265
, 91
].
Both, artificial viscosity and HRSC schemes have been developed and most of the astrophysical applications
previously attempted with no magnetic fields included are currently being revisited, namely for studies of
gravitational collapse, neutron-star dynamics, black-hole accretion, and jet formation. We note, however,
that all HRSC numerical approaches employed, with the exception of [201
, 24
], are based on central
schemes (or incomplete Riemann solvers), for which the use of the full wave decomposition is not needed
(see below).
In terms of the (Faraday) electromagnetic tensor , Maxwell’s equations read
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