Gravitational waveforms from binary neutron star mergers with high-order weighted-essentially-nonoscillatory schemes in numerical relativity

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 waveform production, and in future large-scale investigations of the binary parameter space.

Figure 1

Central rest-mass density evolution for simulations of a single spherical star with $n=96$ and $h=0.1875$. Top: polytropic EOS; bottom: $\mathrm{\Gamma }$-law EOS.

Figure 2

Evolution of the $L_1$-distance $||\rho (t)-\rho (0)|{|}_1$ for different numerical methods and resolution. Different resolutions are shown with different colors. The dashed colored lines are the same for the corresponding solid but scaled according to second-order convergence. Top: polytropic EOS. Bottom: $\mathrm{\Gamma }$-law EOS.

Figure 3

Deviation from optimal WENOZ weights for the $\mathrm{\Gamma }=2$ EOS spherical star. The plot shows the absolute difference with respect to optimal weights $d_j^{(k)\pm }$ of (13) for various projected fluxes ${\widehat{F}}^{(k)\pm }$ and after a very short evolution time, $t\sim 0.003$. Note that, by symmetry, $d_j^{(2)\pm }=d_j^{(1)\pm }$ and $d_j^{(k)\pm }(x>0)=d_j^{(k)\mp }(x<0)$. For this test we evolve few Euler time steps at a unigrid resolution of $h=0.15$, $n=128$, and CFL of 0.01. The vertical line marks the star surface.

Figure 4

SLy135135_0060: $L_2$ norm of the Hamiltonian constraint on refinement level $l=1$ for different grid resolution and numerical schemes. Dashed lines show results scaled to second order.

Figure 5

SLy135135_0060: ADM energy Eq. (14) extracted at coordinate sphere $r=450$ for different grid resolution and numerical schemes. Dashed lines show results scaled to second order.

Figure 6

SLy135135_0060: Rest-mass conservation ${\log }_{10}(|M_b(t)/M_b(t=0)|)$ and its convergence on refinement level $l=5$ for different grid resolution and numerical schemes. Dashed lines show results scaled to fourth order, which is the order of the restriction operator (see the text).

Figure 7

Conserved quantities for SLy135135_0038 with the HO-Hyb-WENOZ scheme. Top panel: conservation of the baryonic mass measures on level $l=4$. Dashed lines correspond to an assumed fourth-order convergence, cf., Fig. 6. Middle panel: $L_2$-volume norm of the Hamiltonian constraint measured on level $l=1$. Dashed lines correspond to an assumed second-order convergence, cf., Fig. 4. Bottom panel: Self-convergence of the ADM energy, where we have not corrected the energy by the emitted GW energy. The energies are extracted at $r=1000M_{\odot }$. Dashed lines correspond to an assumed second-order convergence, cf., Fig. 5.

Figure 8

Top: $\ell =m=2$ waveforms for SLy135135_0060 from runs with grid low and varying the numerical scheme. The vertical line marks the moment of merger for the LLF simulation. Middle: Phase differences between HO-Hyb and LLF simulations employing WENOZ reconstruction and with the same grid resolutions. Bottom: Phase differences between HO-Hyb simulations employing WENOZ and MP5 reconstruction with the same grid resolution.

Figure 9

Convergence of SLy135135_006. The different panels show the phase differences in log scale for different reconstructions: MP5 (left) and WENOZ (right). The algorithms from top to bottom are LLF, HO, HO-Hyb. The vertical shaded regions represent the moment of merger for different resolutions: light gray for $[u_{\text{mrg}}^{\mathrm{L}},u_{\text{mrg}}^{\mathrm{M}}]$, gray for $[u_{\text{mrg}}^{\mathrm{M}},u_{\text{mrg}}^{\mathrm{H}}]$, and dark gray for $[u_{\text{mrg}}^{\mathrm{H}},u_{\text{mrg}}^{\mathrm{F}}]$. To show convergence we rescale the phase differences assuming second-order convergence, cf., dashed and dot-dashed lines.

Figure 10

Phase error budget of SLy135135_0060 with the HO-Hyb-WENOZ scheme. Solid lines are phase differences between runs at different resolutions; dashed lines are differences with the Richardson extrapolated data; dashed dotted lines are differences between the phase extracted at finite radii and the radius-extrapolated one.

Figure 11

Phase convergence of MS1b135135_0038 (upper panel) and SLy135135_0038 (lower panel) with the HO-Hyb-WENOZ scheme.

Figure 12

SLy135135_0060 finite radius extraction uncertainty (HO-Hyb-WENOZ scheme). The top panel shows the phase difference for different extraction radii computed to the polynomial extrapolation with $K=2$ (solid lines). The dashed lines show the difference between the extrapolated waves with $K=1$, 3 and the last radius $r_N=1100$ as well as the difference between the $K=2$ and next-to-leading order (NLO) extrapolation. The bottom shows the phase difference between the NLO extrapolated value and finite radii extracted data (solid lines). Dashed lines show the difference between the NLO extrapolation for different radii and the $K=2$ extrapolated one.

Figure 13

Phase error budget of SLy135135_0038 (left) and MS1b135135_0038 (right) similar to Fig. 10. Solid lines are phase differences between runs at different resolutions; dashed lines are differences with the Richardson extrapolated data; dashed dotted lines are differences between the phase extracted at finite radii and the radius-extrapolated one. The conservative error estimate described in the text is given by the blue shaded region.

Figure 14

Relativistic blast wave solutions at $t=0.4$ with $n=400$ points. Solid lines are the exact solutions.

Figure 15

Relativistic simple wave. Top: solution with $n=400$ points (shown 200) and CFL factor 0.125. Bottom: evolution of the experimental self-convergence factor as computed from numerical data at resolution (400, 800, 1600) points.

Figure 16

Phase differences for Sly135135_006. Solid lines refer to the phase differences of the original data (L, M, H) to the highest resolved simulation (F). Dashed lines correspond to the phase difference between $\eta$ rescaled data and the original F simulation. Dash-dotted lines are $\eta$, $\mathrm{\Phi }$-rescaled data compared to the H resolution. The solid black line marks the difference of the extrapolated data to infinite resolution according to the text and the F resolution.