Numerical evolution of multiple black holes with accurate initial data

We present numerical evolutions of three equal-mass black holes using the moving puncture approach. We calculate puncture initial data for three black holes solving the constraint equations by means of a high-order multigrid elliptic solver. Using these initial data, we show the results for three black hole evolutions with sixth-order waveform convergence. We compare results obtained with the BAM and AMSS-NCKU codes with previous results. The approximate analytic solution to the Hamiltonian constraint used in previous simulations of three black holes leads to different dynamics and waveforms. We present some numerical experiments showing the evolution of four black holes and the resulting gravitational waveform.

Figure 1

2D representation of a domain divided between 2 processors and with bitant symmetry. Face (A) is internal to the global domain and handles a reflection symmetry. Face (B) is internal to the domain and manages the communication with the second processor. Faces (C) and (D) are physical boundaries, where we impose a Robin boundary condition in the normal direction $\overrightarrow{n}$ (see text).

Figure 2

Regular part $u$ of the conformal factor along the $Y$-axis of a single puncture with vanishing spin parameter and with linear momentum $P_y=0.2M$. Shown is a convergence test without scaling (left) and with scaling (right) for second-order convergence using ${\mathrm{cf}}_2=1.8409$.

Figure 3

Regular part $u$ of the conformal factor along the $Y$-axis of a single puncture with vanishing linear momentum and with spin $S_y=0.2M$. Shown is a convergence test without scaling (left) and with scaling (right) for second-order convergence using ${\mathrm{cf}}_2=1.8409$.

Figure 4

Regular part $u$ of the conformal factor along the $Y$-axis of a single puncture with vanishing spin parameter and with linear momentum $P_y=0.2M$. Eighth-order convergence of $u$ is obtained far from the puncture (${\mathrm{cf}}_8=6.4637$).

Figure 5

Regular part $u$ of the conformal factor along the $Y$-axis of a single puncture with vanishing linear momentum and with spin $S_y=0.2M$. Convergence test without scaling (left) and with scaling (right) for fourth-order convergence using ${\mathrm{cf}}_4=2.7840$.

Figure 6

Comparison between the numerical solution of the Hamiltonian constraint calculated using a single-domain spectral method and the high-order multigrid solver. The plot shows $u$ along the $X$-axis produced by the spectral code (denoted by $u_S$) and using three resolutions calculated with the eighth-order implementation of the multigrid code (labels $u^{h_i}$).

Figure 7

Absolute value of the differences between the numerical solution of $u$ for the second, fourth, sixth, and eighth-order finite difference implementation and the spectral solution. The upper part shows the result for second and fourth order, and the bottom part for fourth, sixth, and eighth order.

Figure 8

Plot of $u$ along $X$-axis for system 3BH102, comparing the approximate solution with a second-order numerical solution. The higher order numerical solutions would not be distinguishable from second order in this plot, compare Fig. 7.

Figure 9

Real part of ${\Psi }_4$ (mode $l=m=2$) calculated at $r=40M$ for system 3BH1. The lower panel shows the convergence test for 6th order (${\mathrm{cf}}_6=1.9542$).

Figure 10

Puncture tracks for system 3BH102 using approximate initial data. (see text, Sec. 2c4).

Figure 11

Puncture tracks for system 3BH102 using approximate initial data comparing results for the BAM and AMSS-NCKU codes. The difference in the trajectories is small, and the results agree in the general shape. Note that AMSS-NCKU uses fourth-order spatial discretization instead of sixth-order which is implemented in BAM.

Figure 12

Puncture tracks for system 3BH102 using the numerical solution of the Hamiltonian constraint with the 8th order multigrid method. There is a drastic change in the puncture tracks compared to the evolution of the approximate initial data, in particular, the black holes merge in a different order (compare Fig. 10).

Figure 13

Real part of $r{\Psi }_4$ (mode $l=m=2$) calculated at $r=50M$ for system 3BH102 using approximate initial data. The upper panel shows the $r{\Psi }_4$ waveform for 3 resolutions, the bottom plot shows sixth-order convergence scaling with a factor ${\mathrm{cf}}_6=1.9542$.

Figure 14

Real part of $r{\Psi }_4$ (mode $l=m=2$) calculated at $r=50M$ for system 3BH102 using the numerical solution of the Hamiltonian constraint. The upper panel shows the $r{\Psi }_4$ waveform for 3 resolutions, the bottom plot shows sixth-order convergence scaling with a factor ${\mathrm{cf}}_6=1.9542$. Near the first merger the order of convergence is closer to fourth order.

Figure 15

Symmetric configuration of four black holes. Top: Paths followed by the punctures with a rotational symmetry of 180° with respect to the $Z$-axis. Bottom: Real part of the $l=m=2$ mode of $r{\Psi }_4$ extracted at $r=50M$, the waveform shows two mergers (around $t=150M$ and $t=200M$), first between the black holes labeled BH1 and BH4, and the second merger between the resulting black hole and the black holes BH2 and BH3.

Figure 16

Triple merger of four black holes. Top: Paths followed by the punctures. Bottom: Real part of the $l=m=2$ and $l=2$, $m=0$ modes of $r{\Psi }_4$ extracted at $r=50M$. The waveform shows three mergers (around $t=55M$, $t=95M$, and $t=170M$), which are more easily identified by looking at the mode $l=2$, $m=0$.