Black hole evolution with the BSSN system by pseudospectral methods

Abstract

We present a new pseudo-spectral code for the simulation of evolution systems that are second order in space. We test this code by evolving a non-linear scalar wave equation. These non-linear waves can be stably evolved using very simple constant or radiative boundary conditions, which we show to be well-posed in the scalar wave case. The main motivation for this work, however, is to evolve black holes for the first time with the BSSN system by means of a spectral method. We use our new code to simulate the evolution of a single black hole using all applicable methods that are usually employed when the BSSN system is used together with finite differencing methods. In particular, we use black hole excision and test standard radiative and also constant outer boundary conditions. Furthermore, we study different gauge choices such as 1 + log and constant densitized lapse. We find that these methods in principle do work also with our spectral method. However, our simulations fail after about 100M due to unstable exponentially growing modes. The reason for this failure may be that we evolve the black hole on a full grid without imposing any symmetries. Such full grid instabilities have also been observed when finite differencing methods are used to evolve excised black holes with the BSSN system.

Spectral Methods for Numerical Relativity

“…Their line element reads ds 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 , showing a coordinate singularity at the origin ( r = 0) and on the axis for which θ = 0, π . They are very useful in numerical relativity for the numerous sphere-like objects under study (stars, black hole horizons) and have mostly been implemented for shell-like domains [40, 109, 167, 219] and for spheres including the origin [44, 109]. …”

“…The situation becomes even more complicated with the setup of an artificial numerical boundary at a finite distance from the source, needing appropriate boundary conditions to control the physical wave content, and possibly to limit the growth of unstable modes. All these points have been extensively studied since 2000 by the Caltech/Cornell groups and their pseudospectral collocation code SpEC [125, 127, 187, 186, 138, 120, 126, 137, 49]; they were followed in 2004 by the Meudon group [37] and in 2006 by Tichy [219]. …”

“…Unfortunately, when numerically doing such free evolution, i.e. solving only for the dynamic equations, small violations of the constraints due to round-off errors appear to grow exponentially (for an illustration with spectral methods, see, e.g., [187, 219]). The opposite strategy is to discard some of the evolution equations, keeping the equations for the two physical dynamic degrees of freedom of the gravitational field, and to solve for the four constraint equations: this is a constrained evolution [37].…”

“…Since then, they have focused on the evolution of black-hole-binary systems, which has recently been simulated up to merger and ring down by Scheel et al [185]. Other groups (for instance Ansorg et al [10], Bartnik and Norton [21], Frauendiener [81] and Tichy [219]) have also used spectral methods to solve Einstein’s equations; Sections 5 and 6 are devoted to a more detailed review of these works.…”

Black-hole puncture initial data with realistic gravitational wave content

“…A golden age for numerical relativity is now emerging, in which multiple groups are using different computer codes to evolve BHBs for several orbits before plunge and merger [13,14,15,16,17,18,19,20,21]. Comparison of the numerical results obtained from these various codes has taken place [22,23,24], and comparison with PN inspiral waveforms has also been carried out with encouraging success [5,6,25,26].…”