WHAM: a WENO-based general relativistic numerical scheme, I: hydrodynamics. 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 magnetohydrodynamic (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 weighted essentially non-oscillatory (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 two-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.