Gauge Boundary conditions to mitigate CoM drift in BBH simulations

TL;DR

The study identifies the exponentially growing center-of-mass drift observed in long SpEC BBH simulations as a gauge artifact caused by reflections of gauge waves at the outer boundary under the Sommerfeld gauge condition. It derives a leading-order CoM equation of motion and demonstrates that embedding explicit CoM-correction source terms into the boundary condition can dramatically suppress the drift without impacting the physical waveform, as verified across equal-mass non-spinning and precessing cases. Two concrete implementations are shown: (1) setting $\bm{r_0} = \lambda \bm{r}_{\mathrm{s}}$, which yields tunable damping/oscillation behavior and, under suitable scaling $(\lambda-1) \sim R^{2}$, $R$-independent growth; and (2) using $\bm{r_0} = \bm{r}_{\mathrm{s}} + \eta R \dot{\bm{r}}_{\mathrm{s}}$ to realize exponential damping with controllable parameters. The results enable shorter outer boundaries and more efficient long-term BBH simulations, with waveform fidelity preserved, and open avenues for further improvements via Bayliss–Turkel, MPM, or CCM approaches in future work.

Abstract

Long-term numerical relativity (NR) simulations of binary black hole (BBH) systems in the Spectral Einstein Code (SpEC) code exhibit an unexpected exponential drift of the center-of-mass (CoM) away from the simulation's origin. In our work, we analyze this phenomenon and demonstrate that it is not a physical effect but rather a manifestation of a gauge artifact. The origin of this drift is the reflection of the gauge waves off the outer boundary of the computational domain. These reflections are introduced by inaccuracies in the gauge boundary condition, specifically, the application of the Sommerfeld condition to the time derivative of the gauge fields. Such an approach fails to completely suppress or correctly absorb the outgoing modes, thereby generating artificial feedback into the simulation. To mitigate this problem, we introduce a modified boundary condition that incorporates an explicit CoM correction source term designed to counteract the CoM motion. Our numerical experiments, performed with the SpEC code, reveal that this new boundary treatment reduces the CoM drift by several orders of magnitude compared to the standard implementation, and does not introduce any unwanted physical artifacts.

Figures (14)

• Figure 1: The CoM displacement from the origin for different outer boundary radii. The system is equal mass and non-spinning.
• Figure 2: The upper panel shows the evolution of CoM coordinates and the lower panel shows their corresponding CoM charge. The outer boundary radius is set as $100\,M$.
• Figure 3: Reflection coefficient $|\rho^\ell_{\ell' m'}|$ for different $\ell$. Given $m'=0, \ell'=\ell, k=1, t=0, \theta=\phi=0$.
• Figure 4: Reflection coefficient $|\rho|$ for a spherical wave whose source is localized at $\bm{\mathrm{r_s}}/R=0.15$, with $k=1, t=0, \theta=\phi=0$.
• Figure 5: A schematic diagram showing the domain setup and symbols.