An explicit harmonic code for black-hole evolution using excision

TL;DR

This work demonstrates a stable, explicit-in-time, finite-difference evolution of black holes using a generalized harmonic formulation with excision. By combining constraint damping, carefully chosen gauge conditions, and SBP/mesh-refinement techniques, the authors validate their approach through Schwarzschild evolutions, head-on mergers, and a QC-0 inspiral, observing that individual MOTS can touch and intersect after common horizon formation. The results confirm the method's viability and highlight MOTS dynamics consistent with recent mathematical insights, while also pointing to avenues for improvement such as constraint-preserving outer boundaries and optimized dissipation and boundary blending. Overall, the study advances the practical capability of harmonic-Excision codes to simulate fully nonlinear black-hole spacetimes and motivates cross-code comparisons for robust gravitational-wave predictions.

Abstract

We describe an explicit in time, finite-difference code designed to simulate black holes by using the excision method. The code is based upon the harmonic formulation of the Einstein equations and incorporates several features regarding the well-posedness and numerical stability of the initial-boundary problem for the quasilinear wave equation. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code. Such tests range from the evolution of isolated black holes to the head-on collision of two black holes and then to a binary black hole inspiral and merger. Besides assessing the accuracy of the code, the inspiral and merger test has revealed that individual apparent horizons can touch and even intersect. This novel feature in the dynamics of the marginally trapped surfaces is unexpected but consistent with theorems on the properties of apparent horizons.

Figures (6)

• Figure 1: Error $r_{_{\rm AH}}/M - 2$ in the areal radius of the apparent horizon as a function of time for evolutions of Schwarzschild initial data at 4 different resolutions. The main figure shows $r_{_{\rm AH}}/M - 2$, while the inset shows $(0.1M/h)^4 \times (r_{_{\rm AH}}/M - 2)$, demonstrating that the error shows fourth-order convergence to zero as the resolution is increased.
• Figure 2: Scaled harmonic constraint $(0.1M/h)^4 \times C^0$ at $t=200M$ along the $x$ axis, for evolutions of Schwarzschild initial data at 4 different resolutions.
• Figure 3: Masked 2-norm of the harmonic constraint $\| C^0\|_2$ as a function of the finest grid resolution $h$, at selected times, for evolutions of Schwarzschild initial data at 4 different resolutions. Note that the $t=200M$, $t=500M$, and $t=1000M$ curves are almost coincident (because the norms are almost time-independent over this range of times).
• Figure 4: Zerilli waveform for the head-on problem, extracted at $R=60M$. The outer boundary in this test was at $L=144 M$.
• Figure 5: are $(x,y,t)$. The overlap of the MOTS is clearly visible. Note that these surfaces are actually smooth everywhere; their apparent non-smoothness in some parts of this figure is an artifact of the perspective projection.