Strategies for the characteristic extraction of gravitational waveforms

We develop, test, and compare new numerical and geometrical methods for improving the accuracy of extracting waveforms using characteristic evolution. The new numerical method involves use of circular boundaries to the stereographic grid patches which cover the spherical cross sections of the outgoing null cones. We show how an angular version of numerical dissipation can be introduced into the characteristic code to damp the high frequency error arising form the irregular way the circular patch boundary cuts through the grid. The new geometric method involves use of the Weyl tensor component ${\Psi }_4$ to extract the waveform as opposed to the original approach via the Bondi news function. We develop the necessary analytic and computational formula to compute the $O(1/r)$ radiative part of ${\Psi }_4$ in terms of a conformally compactified treatment of null infinity. These methods are compared and calibrated in test problems based upon linearized waves.

Figure 1

Comparison of the $L_{\infty }$ error $\mathcal{E}(\Phi )$ vs $t$, for the highest resolution runs using the circular patch method and the cubed-sphere method.

Figure 2

The $L_{\infty }$ error $\mathcal{E}({ð}^2\Phi )$ vs $t$ for the highest resolution runs is compared using the circular patch method and the cubed-sphere method.

Figure 3

The error $\mathcal{E}(\overline{ð}{ð}^2\Phi )$ vs $t$ is compared for the highest resolution runs using the circular patch method and the cubed-sphere method.

Figure 4

2D snapshots of the error in $\overline{ð}{ð}^2\Phi$ at $t=120$ for the on the North hemisphere for the circular patch method (left plot), and for a cubed-sphere patch (right plot). For the sake of plot clarity these 2D snapshots use only every third data-point along each axis. The third angular derivatives are smooth for both methods.

Figure 5

The $L_{\infty }$ error $\mathcal{E}(J)$ plotted vs $x$ at $t=100$ for runs with the circular patch method (with and without dissipation) and with the square patch method.

Figure 6

Plots of the error $\mathcal{E}(J)$ at ${\mathcal{I}}^{+}$ vs $t$, for the circular patch with dissipation measured with the $L_2$ norm (left plot) and the $L_{\infty }$ norm (right plot). The error for the $M=100$ grid is rescaled and overlaid on the error for the $M=120$ grid to exhibit the second order convergence. The smaller $L_2$ error indicates that the maximum error arises near the equator.

Figure 7

Surface plot of the error in $J$ at $t=100$ for the circular patch run (with dissipation).

Figure 8

Surface plot of the error in $J$ at $t=100$ for the circular patch run without dissipation.

Figure 9

Surface plot of the error in $J$ at $t=100$ for the square patch run.

Figure 10

Plots of the $L_2$ errors vs $t$ for $N$ (left plot) and $N_{\Psi }$ (right plot) for the circular patch runs. The plots for the $M=100$ grid are rescaled to the $M=120$ grid. The plots are based upon output at integer values of $t$.

Figure 11

Surface plot of the error in $N$ at $t=100$ for the circular patch.

Figure 12

Surface plot of the error in $N$ at $t=100$ for the square patch.

Figure 13

Surface plot of the error in $N_{\Psi }$ at $t=100$ for the circular patch.

Figure 14

Surface plot of the error in $N_{\Psi }$ at $t=100$ for the square patch.