  We present a discontinuous Galerkin-finite-difference hybrid scheme that
allows high-order shock capturing with the discontinuous Galerkin method for
general relativistic magnetohydrodynamics. The hybrid method is conceptually
quite simple. An unlimited discontinuous Galerkin candidate solution is
computed for the next time step. If the candidate solution is inadmissible, the
time step is retaken using robust finite-difference methods. Because of its a
posteriori nature, the hybrid scheme inherits the best properties of both
methods. It is high-order with exponential convergence in smooth regions, while
robustly handling discontinuities. We give a detailed description of how we
transfer the solution between the discontinuous Galerkin and finite-difference
solvers, and the troubled-cell indicators necessary to robustly handle
slow-moving discontinuities and simulate magnetized neutron stars. We
demonstrate the efficacy of the proposed method using a suite of standard and
very challenging 1d, 2d, and 3d relativistic magnetohydrodynamics test
problems. The hybrid scheme is designed from the ground up to efficiently
simulate astrophysical problems such as the inspiral, coalescence, and merger
of two neutron stars.
