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Despite the recent rapid progress in numerical relativity, a convergenceorder less than the second has so far plagued codes solving the Einstein-Eulersystem of equations. We report simulations of the inspiral of binary neutronstars in quasi-circular orbits computed with a new code employing high-order,high-resolution shock-capturing, finite-differencing schemes that, for thefirst time, go beyond the second-order barrier. In particular, without anytuning or alignment, we measure a convergence order above three both in thephase and in the amplitude of the gravitational waves. Because the new code isable to calculate waveforms with very small phase errors already at modestresolutions, we are able to obtain accurate estimates of tidal effects in theinspiral that are essentially free from the large numerical viscosity typicalof lower-order methods, and even for the challenging large compactness andsmall-deformability binary considered here. We find a remarkable agreementbetween our Richardson-extrapolated waveform and the one from the tidallycorrected post-Newtonian (PN) Taylor-T4 model, with a de-phasing smaller than0.2 radians during the seven orbits of the inspiral and up to the contactpoint. Because our results can be used reliably to assess the validity of thePN or other approximations at frequencies significantly larger than thoseconsidered so far in the literature, they seem to exclude at thesecompactnesses significant tidal amplifications fromnext-to-next-to-leading--order terms in the PN expansion.

Beyond second-order convergence in simulations of binary neutron stars in full general-relativity