Riemann Solvers and Alfven Waves in Black Hole Magnetospheres
  In the magnetosphere of a rotating black hole, an inner Alfven critical
surface (IACS) must be crossed by inflowing plasma. Inside the IACS, Alfven
waves are inward directed toward the black hole. The majority of the proper
volume of the active region of spacetime (the ergosphere) is inside of the
IACS. The charge and the totally transverse momentum flux (the momentum flux
transverse to both the wave normal and the unperturbed magnetic field) are both
determined exclusively by the Alfven polarization. Thus, it is important for
numerical simulations of black hole magnetospheres to minimize the dissipation
of Alfven waves. Elements of the dissipated wave emerge in adjacent cells
regardless of the IACS, there is no mechanism to prevent Alfvenic information
from crossing outward. Thus, numerical dissipation can affect how simulated
magnetospheres attain the substantial Goldreich-Julian charge density
associated with the rotating magnetic field. In order to help minimize
dissipation of Alfven waves in relativistic numerical simulations we have
formulated a one-dimensional Riemann solver, called HLLI, which incorporates
the Alfven discontinuity and the contact discontinuity. We have also formulated
a multidimensional Riemann solver, called MuSIC, that enables low dissipation
propagation of Alfven waves in multiple dimensions. The importance of higher
order schemes in lowering the numerical dissipation of Alfven waves is also
catalogued.
