Gauge Boundary conditions to mitigate center-of-mass drift in BBH simulations

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.

💡 Research Summary

The paper investigates a puzzling exponential drift of the coordinate centre‑of‑mass (CoM) observed in long‑duration binary‑black‑hole (BBH) simulations performed with the Spectral Einstein Code (SpEC). The authors first demonstrate that the drift is not a physical effect by extracting the global Poincaré charges (mass, linear momentum, angular momentum) at future null infinity using Cauchy‑Characteristic Extraction (CCE). While the coordinate CoM moves exponentially away from the origin, the corresponding mass‑dipole charge (the physical CoM) remains essentially constant, indicating that the observed motion is a pure gauge artifact.

The root cause is identified as imperfect absorption of gauge waves at the outer computational boundary. SpEC employs a Sommerfeld non‑reflective condition applied to the time derivative of the gauge source functions. This formulation is exact only for the monopole (ℓ = 0) mode when the source sits at the origin. Any displacement of the CoM generates higher‑order multipoles (ℓ ≥ 1) which the Sommerfeld condition does not fully absorb. Consequently, outgoing gauge waves are partially reflected back into the domain, and the reflected waves exert a force on the coordinate system that drives the CoM with exponential acceleration. The growth rate σ scales roughly as σ ∝ R⁻¹·⁴⁵, consistent with earlier observations that moving the boundary farther out reduces the drift.

To remedy this, the authors propose two complementary modifications to the boundary treatment. First, they introduce a shift parameter r₀ into the Sommerfeld operator, redefining it as L(r₀)ψ ≡ (∂ₜ + ∂_{|r−r₀|} + 1/|r−r₀|)ψ = 0. By choosing r₀ equal to the instantaneous CoM location, the monopole condition becomes exact for the displaced source, eliminating reflections of the ℓ = 0 mode. Second, they add an explicit CoM‑correction source term S_i = −κ v_i^{CoM} to the boundary condition, where κ is a small damping coefficient (empirically ≈10⁻³–10⁻⁴). This term directly damps the velocity induced by reflected gauge waves.

Numerical experiments with equal‑mass, non‑spinning BBH configurations and outer boundary radii of 100 M, 200 M, and 400 M confirm the efficacy of the new scheme. The modified boundary condition reduces the coordinate CoM displacement by several orders of magnitude (down to ≲10⁻⁴ M) while leaving the physical waveform, extracted gravitational‑wave strain, and Poincaré charges unchanged. The authors also discuss alternative strategies such as higher‑order absorbing boundary conditions, constraint‑preserving damping, NR‑post‑Minkowski matching, and Cauchy‑Characteristic Matching (CCM), positioning their approach as a simple yet powerful solution.

In conclusion, the exponential CoM drift in SpEC simulations originates from gauge‑wave reflections caused by an inadequate Sommerfeld boundary condition. By shifting the boundary operator to track the CoM and adding a modest damping source, the drift is effectively suppressed, enabling long‑term BBH simulations with smaller computational domains and reduced cost, without compromising the physical fidelity of the gravitational‑wave output.