Calibration of moving puncture simulations

We present single and binary black-hole simulations that follow the “moving-puncture” paradigm of simulating black-hole spacetimes without excision, and use “moving boxes” mesh refinement. Focusing on binary black-hole configurations where the simulations cover roughly two orbits, we address five major issues determining the quality of our results: numerical discretization error, finite extraction radius of the radiation signal, physical appropriateness of initial data, gauge choice, and computational performance. We also compare results we have obtained with the BAM code described here with the independent LEAN code.

Figure 1

The BSSN variables ${\tilde{g}}_{xx}$, $\chi$, $\alpha$, and ${\beta }^y$ after $50M$ of evolution of a Schwarzschild puncture using the $\chi$ method. A small pulse due to the initial adjustment of the gauge can be seen at about $y=60M$ in ${\tilde{g}}_{xx}$. The main features of the other variables are confined to $y<10M$.

Figure 2

Fourth-order convergence of a Schwarzschild puncture after $50M$ of evolution, using the $\varphi$ method. Results were taken from runs with $N={64}^3$, ${96}^3$ and ${128}^3$ points with octant symmetry. The plots show the differences between the three runs, scaled to be consistent with fourth-order accuracy. For the Hamiltonian constraint and $y$ component of the momentum constraint, which should converge to zero, we show the logarithm of the scaled values.

Figure 3

Fourth-order convergence of a Schwarzschild puncture after $50M$ of evolution, using the $\chi$ method. The parameters match those used for the runs discussed in Fig. 2.

Figure 4

Left: Coordinate location of $R=2M$ after $50M$ of evolution, as a function of the damping parameter, $\eta$, with initial lapse $\alpha =1$ and $\alpha ={\psi }^{-2}$. Right: Coordinate location $R=2M$ as a function of time, for $M\eta =0,2.0,3.5$, using initial $\alpha =1$.

Figure 5

Real part of the $l=2$, $m=0$ mode of $r{\Psi }_4$, extracted at $r=30m_1$, for a simulation of a single spinning Bowen-York puncture with $S=0.2$.

Figure 6

Errors in the real part of the $l=2$, $m=0$ mode of $r{\Psi }_4$, extracted at $r=30m_1$ for the same simulations as in Fig. 5. The upper plot shows results from runs with the $\varphi$ method, while the lower plot shows results with the $\chi$ method. See text for the grid details and discussion.

Figure 7

Convergence of the $\mathrm{BAM}{\chi }_{\eta =2}[5\times i:4\times 2i:6][i=32,40,48,56,72]$ series. Top: Scaled differences between results for different resolutions demonstrate fourth-order convergence. Bottom: The time $t_{\max }$ (in units of $M$) of the maximal amplitude in ${\Psi }_4$ is plotted versus resolution, and Richardson-extrapolated values are shown. The three most accurate runs ($i=48$, 56, and 72) show fourth-order convergence (solid line), while the runs with ($i=32$, 40, 48, and 56) show second-order convergence (dashed line). The points on the far left are the Richardson-extrapolated values.

Figure 8

Convergence of the $\mathrm{BAM}{\chi }_{\eta =2}[5\times i:4\times 2i:6][i=48,56,72]$ series. Top: The early part of the waveform does not show fourth-order convergence until approximately $t=125M$. Bottom: Clean fourth-order convergence is lost at approximately $t=310M$.

Figure 9

Overlay of $\mathrm{Re}(r{\Psi }_4)$, extracted at $r=30$ for highest resolution LEAN ${\varphi }_{\eta =1}$ (as published in 13) and $\mathrm{BAM}{\chi }_{\eta =2}[5\times 72:4\times 144:6][44.8,5.714]$ runs. The results have been shifted in time so $\max (|\Psi 4|)$ is aligned with $t=0$.

Figure 10

The Richardson-extrapolated “main” part of the waveform is shown for the $\mathrm{BAM}{\chi }_{\eta =2}$ runs, error bars are obtained from the difference between Richardson-extrapolated result and the highest resolution run. The BAM result is overlaid with the highest resolution LEAN result. The results have been shifted in time so $\max (|\Psi 4|)$ is aligned with $t=0$.

Figure 11

Radiated energies plotted versus time for the runs ${\chi }_{\eta =2}[5\times i:4\times 2i:6]$ ($i=56,72$) and extraction radii $r=25,30,35,40$. The dashed lines are from the $i=56$ run, the full lines from the $i=72$ run. Larger extraction radii yield smaller values for the radiated energy. The left image shows the result for the complete run, the right image zooms in on the late stage of the run. Results have been shifted in time by Eq. (61) to minimize the phase difference. Clearly, the error from the variation of extraction radius is not smaller than numerical error.

Figure 12

Shape of the apparent horizon in the $x\mathrm{\text{−}}y$ plane plotted each $10M$ for a simulation with initial separation of $r=3.5M$. The dotted line represents the trajectory of the puncture. The behavior of the second horizon can be obtained by the symmetry of the problem. The common apparent horizon (peanut shaped in the figure) appears when the black holes have merged.

Figure 13

Coordinate radius $r_{AH}$ of the apparent horizons as function of time for $\eta =0$ and $\eta =2$. The choice of $\eta$ has a strong influence in this gauge dependent quantity. The apparent-horizon mass $M_{AH}$ does not show any such gauge dependence—the respective curves are essentially on top of each other.

Figure 14

Angular momentum computed at $r=30$ for gauge parameters $\eta =1,2$ with the $\mathrm{BAM}\chi$ series—no significant dependency on the gauge is seen.

Figure 15

Orbital motion with numerical errors obtained from difference of Richardson-extrapolated value to highest resolution result [top: $r(t)$; middle: $\omega (t)$]. The dashed curves represent ${\chi }_{\eta =2}$ and the solid curves ${\varphi }_{\eta =1}$. The errors are so small that not all of the six lines in the plot are distinguishable; the main point is that the differences between the ${\chi }_{\eta =2}$ and ${\varphi }_{\eta =1}$ results can be distinguished in $r(t)$, but not in $\omega (t)$. Bottom: The logarithm of the puncture radial position shows an exponential decay at late times, with a damping time of $\tau =3.48\pm 0.01$ ($M_{\mathrm{final}}$).

Figure 16

Fourth-order convergence for $r(t)$ demonstrated for ${\chi }_{\eta =2}$ (top) and ${\varphi }_{\eta =1}$ (bottom) series.

Figure 17

Relative errors obtained from difference of Richardson-extrapolated value to highest resolution result (top: ${\chi }_{\eta =2}$; bottom: ${\varphi }_{\eta =1}$).

Figure 18

Fourth-order convergence for $\omega (t)$ demonstrated for ${\chi }_{\eta =2}$ series, at late times only first-order convergence is seen.

Figure 19

Fourth-order convergence for $\omega (t)$ demonstrated for ${\varphi }_{\eta =1}$ series, at late times only first-order convergence is seen.

Figure 20

Relative errors obtained from difference of Richardson-extrapolated value to highest resolution result (top: ${\chi }_{\eta =2}$; bottom: ${\varphi }_{\eta =1}$).

Figure 21

Results from evolution of Bowen-York puncture data using three choices of initial parameters, described in the text. Top: Real part of the $l=2$, $m=2$ mode of $r{\Psi }_4$, extracted at $r=40M$. Lower left: Paths followed by the punctures during evolution. Lower right: Energy extracted at $r=40M$ after $300M$ of evolution.

Figure 22

Results from evolution of Bowen-York puncture data using three choices of initial parameters, described in the text. Top: Absolute value of the $l=2$, $m=2$ mode of $r{\Psi }_4$, extracted at $r=40M$. Bottom: Phase angle (in units of $2\pi$), shifted in time to align the maxima of $\mathrm{abs}(r{\Psi }_4)$ at $t=0$, and also aligning the phase at $t=0$.