Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing

TL;DR

This paper introduces a minimal extension to the moving-puncture BAM code by replacing bulk spatial derivatives with sixth-order finite difference operators, while retaining fourth-order dissipation and RK4 time stepping. The authors implement careful AMR strategies, including buffer zones and sixth-order interpolation, to maintain stability across refinement boundaries. In long equal-mass binary evolutions, they demonstrate near-sixth-order convergence of the gravitational-wave phase over about nine orbits, achieving a phase error growth described by δφ ≈ 0.0117 exp(0.003 t/M) and a merger-time phase accuracy around 4 M, at roughly 11k CPU hours. The results indicate substantial improvements in phase accuracy for long BBH inspirals with a moderate computational cost increase, and point to future applications to unequal masses and spins as well as further stability analyses.

Abstract

We describe a modification of a fourth-order accurate ``moving puncture'' evolution code, where by replacing spatial fourth-order accurate differencing operators in the bulk of the grid by a specific choice of sixth-order accurate stencils we gain significant improvements in accuracy. We illustrate the performance of the modified algorithm with an equal-mass simulation covering nine orbits.

Figures (6)

• Figure 1: Left panel: Coordinate tracks of the puncture location of one black hole for simulations $\{64,72,80\}$. Only in the last few orbits are differences between the three runs discernable. Right panel: the waveform plotted as the real part of $r h_{22}$, as defined in Ajith:2007qp.
• Figure 2: Left panel: coordinate distance of the black holes for the fourth order version of the $48$-configuration and sixth-order simulations $\{48,64,80\}$ in the order of increasing merger time. Right panel: the gravitational wave phase for the same runs. The $72$ simulation would not be distinguishable from the $80$ simulation on the scale shown here.
• Figure 3: Convergence test for the gravitational wave phase. Plotted are the difference between the $72$ and $80$ runs, and the difference between the $64$ and $72$ runs rescaled for sixth-order convergence. Also shown is the convergence factor divided by $6$, which shows a "glitch" around the time that the phase increases very sharply, and the error estimate after performing Richardson extrapolation. The left panel shows a linear scaling, the right panel shows the same plot with a logarithmic scaling to emphasize the slow but clean exponential growth $\delta \varphi = 0.0117 \exp{ 0.003 t/M}$ of the phase error at intermediate times.
• Figure 4: The $l=2,m=2$ mode of the wave signal is split into the absolute value of $\Psi_{4,22}$ (left panel) and the wave frequency $\omega$ (right panel). Both panels show the simulations $\{64,72,80\}$, aligned in time to coincide at the peak of $\vert \Psi_4 \vert$. The curves are clipped at early times, where they are very noisy due to the smallness of the signal and finite differencing error in computing the wave frequency from the phase.
• Figure 5: Convergence plot for the wave amplitude $\vert {\psi_4}_{22}\vert$ in the $l=2,m=2$ mode. Both panels show the difference between the $72$ and $80$ runs and differences between the $64$ and $72$ runs rescaled for sixth-order convergence. In the left panel data at different resolutions are compared at the same coordinate time, which leads to a seeming loss of convergence near the radiation peak, which is due to the relatively large phase error. In the right panel the data are compared at the same value of the gravitational wave phase, which restores clean sixth order convergence.