  We present a new approach for achieving high-order convergence in fully
general-relativistic hydrodynamic simulations. The approach is implemented in
WhiskyTHC, a new code that makes use of state-of-the-art numerical schemes and
was key in achieving, for the first time, higher than second-order convergence
in the calculation of the gravitational radiation from inspiraling binary
neutron stars Radice et al. (2013). Here, we give a detailed description of the
algorithms employed and present results obtained for a series of classical
tests involving isolated neutron stars. In addition, using the
gravitational-wave emission from the late inspiral and merger of binary neutron
stars, we make a detailed comparison between the results obtained with the new
code and those obtained when using standard second-order schemes commonly
employed for matter simulations in numerical relativity. We find that even at
moderate resolutions and for binaries with large compactness, the phase
accuracy is improved by a factor 50 or more.
