Modifications for numerical stability of black hole evolution

We experiment with several new modifications to the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein’s field equation, and demonstrate how these modifications affect the stability of numerical black hole evolution. With these modifications, we obtain accurate and stable simulations of both single excised Kerr-Schild black holes and punctured binary black holes.

Figure 1

The root mean square (rms) of the change in the trace of extrinsic curvature between consecutive time steps as functions of time for the cases listed in Table . The usual BSSN formulation is used in Case STD. The modifications suggested in 13 are used in Case YBS. The modifications suggested in this work are activated one by one from Case E1 to Case E6. Case E5 has the same modifications with Case E4 except with a different value of $\sigma$ used in Eq. (46). Case N7 has the same modifications with Case E6 except with no symmetry for the computational domain. There is no instability shown in Cases E5, E6, and N7 throughout these runs.

Figure 2

The monitored quantities as functions of time for Case E6. The upper-left panel compares different integrals for the ADM mass. The lower-left panel compares different integrals for the angular momentum. The upper-right panel shows the L2 norms of the Hamiltonian constraint $\mathcal{H}$ and the momentum constraint ${\mathcal{M}}^x$. The lower-right panel shows a log plot of the rms of the changes in the lapse and the trace of extrinsic curvature between consecutive time steps.

Figure 3

The rms of the change in the trace of extrinsic curvature between consecutive time steps as functions of time with different resolutions.

Figure 4

Comparison of $\ell =2,m=2$ mode of ${\Psi }_4$ for the usual BSSN formulation and the one with our new modification. The outermost boundary locates at $517.12$. And the detector locates at $R=50$. The (red) dashed line corresponds to the result with our new modification and the solid line corresponds to the result with the usual BSSN formulation. The results from the two formulations are quite consistent. But for the late ringdown phase, shown in the enlarged subplots, our new modification gives a much smoother result.