  We present a new numerical tool for solving the special relativistic ideal
MHD equations that is based on the combination of the following three key
features: (i) a one-step ADER discontinuous Galerkin (DG) scheme that allows
for an arbitrary order of accuracy in both space and time, (ii) an a posteriori
subcell finite volume limiter that is activated to avoid spurious oscillations
at discontinuities without destroying the natural subcell resolution
capabilities of the DG finite element framework and finally (iii) a space-time
adaptive mesh refinement (AMR) framework with time-accurate local
time-stepping. The divergence-free character of the magnetic field is instead
taken into account through the so-called "divergence-cleaning" approach. The
convergence of the new scheme is verified up to 5th order in space and time and
the results for a set of significant numerical tests including shock tube
problems, the RMHD rotor and blast wave problems, as well as the Orszag-Tang
vortex system are shown. We also consider a simple case of the relativistic
Kelvin-Helmholtz instability with a magnetic field, emphasizing the potential
of the new method for studying turbulent RMHD flows. We discuss the advantages
of our new approach when the equations of relativistic MHD need to be solved
with high accuracy within various astrophysical systems.
