Accuracy Issues for Numerical Waveforms

TL;DR

This paper investigates the convergence properties of high-resolution numerical relativity simulations of black-hole binaries, revealing challenges in waveform accuracy and the effects of different formulations and techniques on error behavior.

Contribution

It provides a detailed analysis of convergence and error characteristics in high-order finite difference simulations of black-hole mergers, highlighting issues with waveform phase oscillations and comparing different formalisms.

Findings

Convergence of Hamiltonian constraint achieved at high resolutions.

Waveform phase oscillates with grid resolution, indicating stochastic errors.

Richardson extrapolation aligns cZ4 and BSSN waveform phases within 0.01 rad.

Abstract

We study the convergence properties of our implementation of the 'moving punctures' approach at very high resolutions for an equal-mass, non-spinning, black-hole binary. We find convergence of the Hamiltonian constraint on the horizons and the L2 norm of the Hamiltonian constraint in the bulk for sixth and eighth-order finite difference implementations. The momentum constraint is more sensitive, and its L2 norm shows clear convergence for a system with consistent sixth-order finite differencing, while the momentum and BSSN constraints on the horizons show convergence for both sixth and eighth-order systems. We analyze the gravitational waveform error from the late-inspiral, merger, and ringdown. We find that using several lower-order techniques for increasing the speed of numerical relativity simulations actually lead to apparently non-convergent errors. Even when using standard high-accuracy techniques, rather than seeing clean convergence, where the waveform phase is a monotonic function of grid resolution, we find that the phase tends to oscillate with resolution, possibly due to stochastic errors induced by grid refinement boundaries. Our results seem to indicate that one can obtain gravitational waveform phases to within 0.05 rad. (and possibly as small as 0.015 rad.), while the amplitude error can be reduced to 0.1%. We then compare with the waveforms obtained using the cZ4 formalism. We find that the cZ4 waveforms have larger truncation errors for a given resolution, but the Richardson extrapolation phase of the cZ4 and BSSN waveforms agree to within 0.01 rad., even during the ringdown.