  The theoretical modeling of gravitational waveforms from binary neutron star
mergers requires precise numerical relativity simulations. Assessing
convergence of the numerical data and building the error budget is currently
challenging due to the low accuracy of general-relativistic hydrodynamics
schemes and to the grid resolutions that can be employed in $(3+1)$-dimensional
simulations. In this work, we explore the use of high-order
weighted-essentially-nonoscillatory (WENO) schemes in neutron star merger
simulations and investigate the accuracy of the waveforms obtained with such
methods. We find that high-order WENO schemes can be robustly employed for
simulating the inspiral-merger phase and they significantly improve the
assessment of the waveform's error budget with respect to finite-volume
methods. High-order WENO schemes can be thus efficiently used for high-quality
waveforms production, also in future large-scale investigations of the binary
parameter space.
