Accurate black hole evolutions by fourth-order numerical relativity

TL;DR

This work addresses the need for accurate gravitational-wave templates from binary black-hole mergers by developing a $4^{th}$-order numerical relativity framework, LazEv, built around the BSSN formulation and MoL integration. The authors validate their approach with Apples With Apples tests and demonstrate substantial improvements in waveform accuracy for head-on BBH collisions when using fourth-order spatial differencing outside horizons and second-order differencing inside, aided by a Local Order Reduction strategy. Key contributions include a modular, high-order-capable framework, demonstration of $4^{th}$-order convergence on gauge and Gowdy tests, and enhanced waveform fidelity at typical resolutions, with quantified radiated energy and quasinormal-mode fits. The findings imply significant gains in accuracy and efficiency for BBH simulations, paving the way for orbiting binaries, constraint-preserving boundaries, and adaptive mesh refinement in future work.

Abstract

We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher order finite differencing for the BSSN formulation of Einstein's equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution.

Figures (27)

• Figure 1: The $L_2$ norm of $\delta \gamma_{xx} = \gamma_{xx} - \gamma_{xx}^{analytic}$, rescaled by $\rho^4/16$, for the one-dimensional 'Gauge Wave' test with $A=0.01$. Note the near perfect overlap for 630 crossing times and that the larger $\rho$ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the error is smaller than $10^{-4}$ (i.e. $1\%$ of $A$).
• Figure 2: The $L_2$ norm of ${\cal H}$, rescaled by $\rho^4/16$, for the one-dimensional 'Gauge Wave' test with $A=0.01$. Note the near perfect overlap for 800 crossing times and that the larger $\rho$ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the Hamiltonian constraint is smaller than $10^{-4}$ (i.e. $1\%$ of $A$)
• Figure 3: The $L_2$ norm of ${\cal H}$, rescaled by $\rho^4/16$, for the one-dimensional 'Gauge Wave' test with $A=0.1$. Note the near perfect overlap for 80 crossing times and that the larger $\rho$ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the Hamiltonian constraint is smaller than $10^{-3}$ (i.e. $1\%$ of $A$).
• Figure 4: The $L_2$ norm of ${\cal H}$, rescaled by $\rho^4/16$, for the two-dimensional 'Gauge Wave' test with $A=0.1$. Note the near perfect overlap for 55 crossing times and that the larger $\rho$ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the Hamiltonian constraint is smaller than $10^{-3}$ (i.e. $1\%$ of $A$).
• Figure 5: The $L_2$ norm of ${\cal H}$, rescaled by $\rho^4/16$, for the one-dimensional 'Gowdy Wave' test. Note the good agreement between the curves for 1000 crossing times and that the evolution is backwards in time