Constraint-preserving boundary conditions in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>first-order approach

Bona, Carles; Bona-Casas, Carles

doi:10.1103/physrevd.82.064008

Cited by 13 publications

(15 citation statements)

References 25 publications

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“…In [25] a natural candidate, namely a conformal decomposition of the Z4 formulation [26][27][28][29][30][31][32], Z4c was identified, and indeed found to give favorable results over those of BSSNOK in the context of spherical symmetry. Like that of GHG, the constraint subsystem of the Z4 formulation has a trivial wavelike principal part.…”

Section: Introductionmentioning

confidence: 99%

Numerical stability of the Z4c formulation of general relativity

Cao

Hilditch²

2012

Phys. Rev. D

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We study numerical stability of different approaches to the discretization of a conformal decomposition of the Z4 formulation of general relativity. We demonstrate that in the linear, constant coefficient regime a novel discretization for tensors is formally numerically stable with a method of lines time-integrator. We then perform a full set of apples with apples tests on the non-linear system, and thus present numerical evidence that both the new and standard discretizations are, in some sense, numerically stable in the non-linear regime. The results of the Z4c numerical tests are compared with those of BSSNOK evolutions. We typically do not employ the Z4c constraint damping scheme and find that in the robust stability and gauge wave tests the Z4c evolutions result in lower constraint violation at the same resolution as the BSSNOK evolutions. In the gauge wave tests we find that the Z4c evolutions maintain the desired convergence factor over many more light-crossing times than the BSSNOK tests. The difference in the remaining tests is marginal.Comment: submitted to PR

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Section: Introductionmentioning

confidence: 99%

Numerical stability of the Z4c formulation of general relativity

Cao

Hilditch²

2012

Phys. Rev. D

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“…For BSSNOK radiation controlling constraint preserving boundary conditions have been proposed [23] but to our knowledge have not been successfully used in numerical applications. With these considerations in mind, a conformal decomposition of the Z4 formulation [24][25][26][27][28][29][30] was proposed in Ref. [1].…”

Section: Introductionmentioning

confidence: 99%

Compact binary evolutions with the Z4c formulation

Hilditch¹,

Bernuzzi²,

Thierfelder³

et al. 2013

Phys. Rev. D

195

206

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Numerical relativity simulations of compact binaries with the Z4c and BSSNOK formulations are compared. The Z4c formulation is advantageous in every case considered. In simulations of non-vacuum spacetimes the constraint violations due to truncation errors are between one and three orders of magnitude lower in the Z4c evolutions. Improvements are also found in the accuracy of the computed gravitational radiation. For equal-mass irrotational binary neutron star evolutions we find that the absolute errors in phase and amplitude of the waveforms can be up to a factor of four smaller. The quality of the Z4c numerical data is also demonstrated by a remarkably accurate computation of the ADM mass from surface integrals. For equal-mass non-spinning binary puncture black hole evolutions we find that the absolute errors in phase and amplitude of the waveforms can be up to a factor of two smaller. In the same evolutions we find that away from the punctures the Hamiltonian constraint violation is reduced by between one and two orders of magnitude. Furthermore, the utility of gravitational radiation controlling, constraint preserving boundary conditions for the Z4c formulation is demonstrated. The evolution of spacetimes containing a single compact object confirm earlier results in spherical symmetry. The boundary conditions avoid spurious and non-convergent effects present in high resolution runs with either formulation with a more naive boundary treatment. We conclude that Z4c is preferable to BSSNOK for the numerical solution of the 3+1 Einstein equations with the puncture gauge

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“…For numerical studies, see [249, 104, 40, 404, 405, 98, 287, 244, 378, 253, 61, 362, 35, 33, 368, 57, 56], especially [366] and [369] for a comparison between different boundary conditions used in numerical relativity and [365] for a numerical implementation of higher absorbing boundary conditions. For review articles on the IBVP in general relativity, see [372, 355, 435].…”

Section: Boundary Conditions For Einstein’s Equationsmentioning

confidence: 99%

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Sarbach

Tiglio

2012

Living Rev. Relativ.

151

168

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Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

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Constraint-preserving boundary conditions in the3+1first-order approach

Cited by 13 publications

References 25 publications

Numerical stability of the Z4c formulation of general relativity

Numerical stability of the Z4c formulation of general relativity

Compact binary evolutions with the Z4c formulation

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

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