  Active galactic nuclei, x-ray binaries, pulsars, and gamma-ray bursts are all
believed to be powered by compact objects surrounded by relativistic plasma
flows driving phenomena such as accretion, winds, and jets. These flows are
often accurately modelled by the relativistic magnetohydrodynamics (MHD)
approximation. Time-dependent numerical MHD simulations have proven to be
especially insightful, but one regime that remains difficult to simulate is
when the energy scales (kinetic, thermal, magnetic) within the plasma become
disparate. We develop a numerical scheme that significantly improves the
accuracy and robustness of the solution in this regime. We use a modified form
of the WENO method to construct a finite-volume general relativistic
hydrodynamics code called WHAM that converges at fifth order. We avoid (1)
field-by-field decomposition by adaptively reducing down to 2-point stencils
near discontinuities for a more accurate treatment of shocks, and (2) excessive
reduction to low order stencils, as in the standard WENO formalism, by
maintaining high order accuracy in smooth monotonic flows. Our scheme performs
the proper surface integral of the fluxes, converts cell averaged conserved
quantities to point conserved quantities before performing the reconstruction
step, and correctly averages all source terms. We demonstrate that the scheme
is robust in strong shocks, very accurate in smooth flows, and maintains
accuracy even when the energy scales in the flow are highly disparate.
