Constraint damping in the Z4 formulation and harmonic gauge

Abstract

We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a λ-system as suggested by Brodbeck et al. We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.

Comparing gravitational waveform extrapolation to Cauchy-characteristic extraction in binary black hole simulations

“…[14-16, 66, 67] and references therein. This code evolves a first-order representation [68] of the generalized harmonic system [69][70][71] with constraint damping [68,71,72]. Outgoing-wave boundary conditions [17,68,73] designed to preserve the constraints [74][75][76][77][78][79][80] are imposed at the outer boundary.…”

Generalized harmonic formulation in spherical symmetry

“…[24]). In our implementation we follow Pretorius [9,11] by adding constraint damping terms in a fashion inspired by studies of the so-called γ -systems [25,26]. The modified equations take the form…”

“…hypersurfaces, which can be written as 10) and κ is an adjustable parameter that controls the damping timescale. Specifically, as discussed in [26], small constraint perturbations about a fixed background decay exponentially with a characteristic timescale of order κ. We note that the constraint damping term contains only first derivatives of the metric and hence does not affect the principal (hyperbolic) part of the equations.…”

Understanding possible electromagnetic counterparts to loud gravitational wave events: Binary black hole effects on electromagnetic fields

“…Our code implements both systems of equations where the constraints, as mentioned, are kept under control via different but related damping mechanisms: constraint damping for the Einstein equations [18] and the extended divergence cleaning for the Maxwell equations as explained in the previous section. We adopt boundary conditions defined via a combination of Sommerfeld and constraint preserving boundary conditions [21] for both systems.…”