  We report simulations of the inspiral and merger of binary neutron stars
performed with \texttt{WhiskyTHC}, the first of a new generation of numerical
relativity codes employing higher than second-order methods for both the
spacetime and the hydrodynamic evolution. We find that the use of higher-order
schemes improves substantially the quality of the gravitational waveforms
extracted from the simulations when compared to those computed using
traditional second-order schemes. The reduced de-phasing and the faster
convergence rate allow us to estimate the phase evolution of the gravitational
waves emitted, as well as the magnitude of finite-resolution effects, without
the need of phase- or time-alignments or rescalings of the waves, as sometimes
done in other works. Furthermore, by using an additional unpublished simulation
at very high resolution, we confirm the robustness of our high convergence
order of $3.2$.
