Simulations of binary black hole mergers using spectral methods

Several improvements in numerical methods and gauge choice are presented that make it possible now to perform simulations of the merger and ringdown phases of “generic” binary black hole evolutions using the pseudospectral evolution code SpEC. These improvements include the use of a new damped-wave gauge condition, a new grid structure with appropriate filtering that improves stability, and better adaptivity in conforming the grid structures to the shapes and sizes of the black holes. Simulations illustrating the success of these new methods are presented for a variety of binary black hole systems. These include fairly generic systems with unequal masses (up to $2:1$ mass ratios), and spins (with magnitudes up to $0.4M^2$) pointing in various directions.

Figure 1

Illustrates a basic element of the grid structure used to fill the 3D volume between a sphere and a concentric cube.

Figure 2

Grid structure around each black hole. A series of spherical shells (two shells shown) is surrounded by a series of cubed-sphere layers (three shown) to fill up a cube. This figure illustrates one quarter of the cube structure surrounding one of the black holes.

Figure 3

Illustrates the intermediate grid structure surrounding the two black holes, consisting of a set of rectangular blocks placed around the cubic block containing each hole. This collection of blocks fills up a cube which is surrounded by a series of cubed-sphere layers (two shown) all surrounded by a series of spherical shells (one shell shown) that are centered on the center of mass of the binary system.

Figure 4

Two-dimensional illustration of the grid-frame coordinate domain on which the function $f_A({\overline{r}}_A,{\overline{\theta }}_A,{\overline{\varphi }}_A)$ is defined in Eq. (17).

Figure 5

Illustrates the inertial frame representation of the grid structure around one black hole, just at the time of merger. This grid structure has been distorted in relation to the inertial frame by the shape-control maps ${\mathcal{M}}_{\mathrm{S}}$ described here.

Figure 6

The minimum of the outgoing (into the hole) characteristic speeds on the nine innermost spherical grid boundaries (labeled $A_0\dots A_8$) around the larger black hole, for part of run F.

Figure 7

The minimum of the outgoing (into the hole) characteristic speeds on the remaining four spherical-shell boundaries (labeled $A_5\dots A_8$) around the larger black hole, for a portion of run F. The dashed curves correspond to runs without shape control, and the solid curves correspond to runs with shape control turned on at $t=62M$. The run without shape control terminates at proper separation $5.2M$ (time $t=106M$) because the excision algorithm fails on surface $A_7$.

Figure 8

Constraint norm $||{\mathcal{C}}_{\mathrm{GH}}||$ as a function of time for a portion of case F. The dashed curve corresponds to a run with spherical shells around each hole, and the solid curve corresponds to an identical run with these inner spherical shells replaced by cubed-sphere shells at $t=80M$. The constraint norm grows larger and begins to grow earlier when spherical shells are used.

Figure 9

The minimum of the outgoing (into the hole) characteristic speeds on the two innermost cubed-sphere boundaries (labeled $B_0$ and $B_1$) around the smaller black hole, for a portion of run F. The dashed curves correspond to runs without size control, and the solid curves correspond to runs with size control turned on at $t=120.2M$. Without size control, the innermost cubed sphere is dropped at $t=124.5M$ and $B_1$ becomes the excision boundary. However, the characteristic speeds become negative on $B_1$ at $t=127M$ and excision fails. With size control, the characteristic speeds on the excision boundary $B_0$ remain positive until proper separation $0.075M$, well after merger.

Figure 10

Three snapshots of the apparent horizons of the nonspinning equal-mass binary black hole merger (case A). Solid curves are the orbital plane cross section when a common horizon is first detected; dotted curves represent the time of transition between the binary merger and the single-hole ringdown evolutions; dashed curve shows the final equilibrium horizon.

Figure 11

Apparent horizon snapshots as in Fig. 10, except for a $2:1$ mass ratio nonspinning binary black hole merger (case B).

Figure 12

Apparent horizon snapshots as in Fig. 10, except for an antialigned-spin equal-mass binary black hole merger (case C).

Figure 13

Apparent horizon snapshots as in Fig. 10, except for an aligned-spin equal-mass binary black hole merger (case D).

Figure 14

Three snapshots of the apparent horizons of a generic binary black hole merger (case E). The plane of the cross sections is a coordinate plane perpendicular to the instantaneous orbital axis at the time the common horizon is first detected. The additional degree of freedom is fixed by having this plane go through the coordinate center of the shape of the common horizon at its first detection. Solid curves are the cross section when a common horizon is first detected; dotted curves represent the time of transition between the binary merger and the single-hole ringdown evolutions; dashed curve shows the horizon at a time well after merger. Note that the individual apparent horizons intersect; this was also observed in Ref. [79].

Figure 15

Apparent horizon snapshots as in Fig. 14, except for the generic binary black hole merger case F in Table .

Figure 16

Constraint norm $||{\mathcal{C}}_{\mathrm{GH}}||$ as a function of time for the generic (case F) binary black hole merger and ringdown, computed with four different numerical resolutions. The small inset graph shows that numerical convergence is maintained (at a reduced rate) even during the most dynamical part of the merger.