Constraint damping in the Z4 formulation and harmonic gauge

By C. Gundlach, Gioel Calabrese, Ian Hinder, J. Martin-Garcia. 2005

223 citations recorded by the source corpus.

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 (1999 J. Math. Phys. 40 909). 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3a26bda98299b8ea12dcfa1a93e3300c3d792e5e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. Jorge E Santos
2. Benson Way
3. B. Kelly
4. J. V. Meter
5. J. Centrella
6. Ó. J. Dias
7. John G. Baker
8. Mark A. Scheel

In Nuclear and High Energy Physics

1. Jorge E Santos
2. Benson Way
3. Ó. J. Dias
4. J. Centrella
5. J. V. Meter
6. B. Kelly
7. John G. Baker
8. Robert Owen

In Statistical and Nonlinear Physics

1. Benson Way
2. Ó. J. Dias
3. Jorge E Santos
4. P. Hintz
5. Pau Figueras
6. Eugene A. Lim
7. Mark A. Scheel
8. András Vasy

Citations in the corpus

• Numerical methods for finding stationary gravitational solutions (2015, background): “These include the characteristic formalism [2, 3], the ADM, BSSN, Z4 formalisms [4–10], and the generalized harmonic gauge formulation [11–16].”
• The global non-linear stability of the Kerr-de Sitter family of black holes (2016, background): “The key insight, which has been exploited before in the numerics literature [66], [117], is that one can modify the gauged Einstein equation by additional terms, preserving its hyperbolic nature, to arrange for constraint damping, which says that solutions of the correspondingly modified constraint ”
• GRChombo: Numerical relativity with adaptive mesh refinement (2015, methodology): “More recently, other refined formulations of the Einstein equation based on the Z4 system [30, 36] have been proposed, most notably the Z4c formulation [37] and the CCZ4 formulation(5) [29].”
• A new generalized harmonic evolution system (2005, methodology): “This new firstorder GH system therefore has the same constraint suppression properties as the second-order system of Gundlach et al [15] and Pretorius [14].”
• Black-hole binaries, gravitational waves, and numerical relativity (2010, background): “The subsequent introduction of constraint-damping terms, which tend to drive the constraints towards zero, further ensured stability (Gundlach et al., 2005).”