Numerical integration of the Teukolsky equation in the time domain

We present a fourth-order convergent, $(2+1)$-dimensional, numerical formalism to solve the Teukolsky equation in the time domain. Our approach is first to rewrite the Teukolsky equation as a system of first-order differential equations. In this way we get a system that has the form of an advection equation. This is then used in combination with a series expansion of the solution in powers of time. To obtain a fourth-order scheme we kept terms up to fourth derivative in time and use the advectionlike system of differential equations to substitute the temporal derivatives by spatial derivatives. This scheme is applied to evolve gravitational perturbations in the Schwarzschild and Kerr backgrounds. Our numerical method proved to be stable and fourth-order convergent in $r^{*}$ and $\theta$ directions. The correct power-law tail, $\sim 1/t^{2\ell +3}$, for general initial data, and $\sim 1/t^{2\ell +4}$, for time-symmetric data, was found in our runs. We noted that it is crucial to resolve accurately the angular dependence of the mode at late times in order to obtain these values of the exponents in the power-law decay. In other cases, when the decay was too fast and round-off error was reached before a tail was developed, then the quasinormal modes frequencies provided a test to determine the validity of our code.

Figure 1

White circles: ghost zones, black circles: normal grid points

Figure 2

Convergence test in $r^{*}$ at $\theta =\pi /2$ and $t=200M$.

Figure 3

Convergence test in $r^{*}$ at $\theta =\pi /2$ and $t=900M$.

Figure 4

Convergence test in $r^{*}$ at $\theta =\pi /2$ at time intervals of $100M$.

Figure 5

Distribution of the grid points ${\theta }_k$ for different resolutions (Only one ghost point shown here.)

Figure 6

Convergence test in $\theta$ at $r^{*}=20M$ at time intervals of $200M$. The continuous lines represents the difference (res2-res1), the dashed one represents (res3 - res1)/$5.23$ and the dotted is (res4-res1)/$31.7$. The resolutions res1, res2, res3 and res4 are $\pi /27,\pi /18,\pi /12,\pi /8$; respectively

Figure 7

Snapshots of an outgoing Gaussian pulse. Numerical dispersion effects appear at late times (space and time are in units of $M$).

Figure 8

Local power index ${\mu }_N$ for $m=0,a=0$. Part A, B and C correspond to $\ell$ values of 2, 3 and 4;, respectively, with their corresponding power-law tails of 7, 9 and 11.

Figure 9

Power-law tails for $m=0$, $a=0$ and different values of $\ell$. The evolution time is $1600M$ with a resolution $\delta r^{*}=0.25M$ and $\delta \theta =0.098$.

Figure 10

Power-law tails, scalar case, $s=0,\ell =0,m=0$. Observer’s position is in $M$ units.

Figure 11

Power-law tail $\ell =2$, $m=0,1,2$, $a=0$.

Figure 12

Power-law tail $\ell =3$, $m=0,1,2,3$, $a=0$.

Figure 13

Power-law tail $\ell =4$, $m=0,1,3,4$, $a=0$.

Figure 14

Power-law tails for $a=0.5$.

Figure 15

Power-law tail $\ell =3$, $m=0$, $a=0,0.5,0.9$.

Figure 16

Power-law tail $\ell =4$, $m=0$, $a=0,0.5,0.9$.

Figure 17

$|\xi {|}^2$ vs $k\delta x$ for several values of $\alpha$.