Towards high-precision inspiral gravitational waveforms from binary neutron star mergers in numerical relativity

We report the performance of a newly implemented fourth-order accurate finite-volume HLLC Riemann solver in the adaptive-mesh-refinement numerical relativity code {\tt SACRA-MPI}. First, we validate our implementation in one-dimensional special relativistic hydrodynamics tests, i.e., a simple wave and shock tube test, which have analytic solutions. We demonstrate that the fourth-order convergence is achieved for the smooth flow, which cannot be achieved in our original second-order accurate finite-volume Riemann solver. We also show that our new solver is robust for the strong shock wave emergence problem. Second, we validate the implementation in a dynamical spacetime by demonstrating that {\tt SACRA-MPI} perfectly preserves the $\pi$-symmetry without imposing the $\pi$-symmetry in a short-term ($\sim 20~{\rm ms}$ in the inspiral and subsequent post-merger phase) non-spinning equal-mass binary neutron star merger simulations. Finally, we quantify the accuracy of $\approx 28$ cycles inspiral gravitational waveforms from binary neutron star mergers by conducting a resolution study with $\approx 78, 94$, $118$, and $135$ m. We find that the fourth-order accurate Riemann solver achieves the convergence order $\approx 2.1\pm{0.05}$--$2.4\pm{0.27}$, i.e., slightly evolving with time, in the inspiral gravitational wave phase, while the second-order accurate Riemann solver achieves the convergence order $\approx 2.0\pm{0.5}$. The residual phase error towards the continuum limit at the merger is $0.27\pm 0.07$ rad and $0.58\pm 0.22$ rad out of a total phase of $\approx 176$ rad, respectively, for the fourth- and second-order accurate Riemann solver.