Testing outer boundary treatments for the Einstein equations

Abstract

Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a 'reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests are based on a first-order generalized harmonic formulation of the Einstein equations and are implemented using a pseudo-spectral collocation method. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar 0 , and control gauge reflections.

Catalog of 174 Binary Black Hole Simulations for Gravitational Wave Astronomy

“…Quasi-equilibrium [28,29] initial data are constructed [30] to solve the Einstein constraint equations [31] for binaries with low (∼10 −4 ) eccentricity [32][33][34] and are evolved using a generalized harmonic formulation [35][36][37][38] of Einstein's equations and damped harmonic gauge [39][40][41]. The adaptively-refined [42] grid extends from pure-outflow excision boundaries conforming to the shapes of the apparent horizons [24,41,43,44] to an artificial outer boundary where constraintpreserving boundary conditions [38,45,46] are imposed. After merger, the grid has only a single excision boundary [24,43].…”

Simulating binary neutron stars: Dynamics and gravitational waves

“…For the outer boundaries, we implement Sommerfeld boundary conditions and follow the prescription given in [48]. We have also used maximally dissipative boundary conditions, but found that they led to larger reflections at the boundaries which, in turn, corrupt the waveform extraction at late times.…”

Outer boundary conditions for Einstein's field equations in harmonic coordinates

“…This principle was used in [10] to assess the numerical performance of various boundary conditions. First, a reference solution is computed on a very large computational domain.…”

“…Finally, the solution on the smaller domain is compared with the reference solution, measuring the spurious reflections and constraint violations caused by the boundary conditions. Here we use the same test problem as in [10]. The initial data are taken to be a Schwarzschild black hole of mass M in Kerr-Schild coordinates with an outgoing odd-parity quadrupolar gravitational wave perturbation (satisfying the full nonlinear constraint equations).…”

“…Figure 2 shows the L ∞ norm of the difference U of the solution on the truncated domain with respect to the reference solution as a function of time. This quantity is obtained by taking a tensor norm of the differences in the metric and its first derivatives at each point [10]. We normalize U by the analogous difference of the perturbed initial data with respect to the unperturbed data.…”