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  is remarkable (but see also  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 . 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  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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