Binary neutron-star mergers with Whisky and SACRA: First quantitative comparison of results from independent general-relativistic hydrodynamics codes

We present the first quantitative comparison of two independent general-relativistic hydrodynamics codes, the whisky code and the sacra code. We compare the output of simulations starting from the same initial data and carried out with the configuration (numerical methods, grid setup, resolution, gauges) which for each code has been found to give consistent and sufficiently accurate results, in particular, in terms of cleanness of gravitational waveforms. We focus on the quantities that should be conserved during the evolution (rest mass, total mass energy, and total angular momentum) and on the gravitational-wave amplitude and frequency. We find that the results produced by the two codes agree at a reasonable level, with variations in the different quantities but always at better than about 10%.

Figure 1

Comparison of the time evolution of the maximum of the rest-mass density for the two models (with different EoSs) described in Sec. 2d2. For ease of interpretation, we remind the reader that in our units $\rho =1\times {10}^{-3}$ corresponds approximately to $6.18\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$.

Figure 2

Left: Comparison of the time evolution of the rest mass (normalized to the initial value). The inset is a magnification of the higher-resolution whisky curve, in the form $M/M_0-1$. These data refer to the piecewise-polytropic EoS. As explained in the text, the larger variations in the SACRA data are due to the choice of grid structure. Right: Comparison of the time evolution of the sum (normalized to the initial value) of the ADM mass measured on the numerical grid and the energy carried away from the grid by gravitational waves. This quantity should be conserved. These data refer to the piecewise-polytropic EoS. Note that the data for the low resolution of whisky are not reliable after the formation of the AH ($t\simeq 13.4\mathrm{ms}$ for this simulation), because the volume integral with which the ADM mass is computed contained also the points inside the horizon. See text for more details.

Figure 3

Comparison of the time evolution of the angular momentum, computed as the sum of the angular momentum measured on the numerical grid and the angular momentum carried away from the grid by gravitational waves. These data refer to the piecewise-polytropic EoS. As already noted for Fig. 2, also here the data for the low resolution of whisky are not reliable after the formation of the AH ($t\simeq 13.4\mathrm{ms}$ for this simulation), because the volume integral with which the angular momentum is computed excludes the contribution of the black hole. See text for more details.

Figure 4

Left: Comparison of the time evolution of the orbital frequency, computed from ${\psi }_4$. These data refer to the ideal-fluid EoS. The inset shows the percent difference of the two curves. The curve labeled T4 is the Taylor-T4 post-Newtonian approximation [18, 69, 70]. Right: Comparison of the amplitude of the wave as a function of the frequency $M\Omega$. The inset shows the percent difference between the two curves. These data refer to the ideal-fluid EoS.

Figure 5

Comparison of the waveform $(h_{+}{)}_{22}$. These data refer to the piecewise-polytropic EoS. The upper panel refers to the “higher resolution” and the lower panel to the “lower resolution” (see text for details). As in Fig. 4, the curve labeled T4 is the Taylor-T4 post-Newtonian approximation [18, 69, 70].