Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes - INSPIRE

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Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

Jul 31, 2017

35 pages

Published in:

Phys.Rev.D 97 (2018) 8, 084053

Published: May 1, 2018

e-Print:

1707.09910 [gr-qc]

DOI:

10.1103/PhysRevD.97.084053

View in:

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Abstract: (APS)

We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in nonconservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.

Note:

27 pages, 8 figures; v2 matches version published in PRD. Discussion improved and proof of strong hyperbolicity is now given for standard gauges

04.25.D-
04.25.dg

References(95)

Figures(31)

[1]

Evolution of binary black hole spacetimes

Frans Pretorius
(
- Caltech and
- Alberta U.
)

- Phys.Rev.Lett. 95 (2005) 121101
•
e-Print:
- gr-qc/0507014
•
DOI:
- 10.1103/PhysRevLett.95.121101

[2]

Accurate evolutions of orbiting black-hole binaries without excision

Manuela Campanelli
(
- Texas U., Brownsville and
- Texas U., Brownsville, CGWA
)
,
C.O. Lousto
(
- Texas U., Brownsville and
- Texas U., Brownsville, CGWA
)
,
P. Marronetti
(
- Florida Atlantic U.
)
,
Y. Zlochower
(
- Texas U., Brownsville and
- Texas U., Brownsville, CGWA
)

- Phys.Rev.Lett. 96 (2006) 111101
•
e-Print:
- gr-qc/0511048
•
DOI:
- 10.1103/PhysRevLett.96.111101

[3]

Gravitational wave extraction from an inspiraling configuration of merging black holes

- Phys.Rev.Lett. 96 (2006) 111102
•
e-Print:
- gr-qc/0511103
•
DOI:
- 10.1103/PhysRevLett.96.111102

[4]

Observation of Gravitational Waves from a Binary Black Hole Merger

LIGO Scientific and
Virgo

Collaborations

•

B.P. Abbott
(
- Caltech
)

et al.

- Phys.Rev.Lett. 116 (2016) 6, 061102
•
e-Print:
- 1602.03837
•
DOI:
- 10.1103/PhysRevLett.116.061102

[5]

Gravitation: An Introduction to Current Research

L. Witten
,
R. Arnowitt
,
S. Deser
,
C.W. Misner

[6]

Introduction to 3+1 Numerical Relativity

M. Alcubierre

[7]

Elements of Numerical Relativity and Relativistic Hydrodynamics: From Einstein’s Equations to Astrophysical Simulations

C. Bona
,
C. Palenzuela-Luque
,
C. Bona-Casas

[8]

Numerical Relativity: Solving Einstein’s Equations on the Computer

T.W. Baumgarte
,
S.L. Shapiro

[9]

Lecture Notes in Physics, Berlin Springer Verlag

E. Gourgoulhon

[10]

Relativistic Hydrodynamics

•
DOI:
- 10.1093/acprof:oso/9780198528906.001.0001

[11]

Numerical Relativity

M. Shibata

[12]

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

Olivier Sarbach
(
- IFM-UMSNH, Michoacan
)
,
Manuel Tiglio
(
- Maryland U.
)

- Living Rev.Rel. 15 (2012) 9
•
e-Print:
- 1203.6443
•
DOI:
- 10.12942/lrr-2012-9

[13]

Evolution of three-dimensional gravitational waves: Harmonic slicing case

Masaru Shibata
(
- Osaka U.
)
,
Takashi Nakamura
(
- Kyoto U., FIHS
)

- Phys.Rev.D 52 (1995) 5428-5444
•
DOI:
- 10.1103/PhysRevD.52.5428

[14]

On the numerical integration of Einstein's field equations

- Phys.Rev.D 59 (1998) 024007
•
e-Print:
- gr-qc/9810065
•
DOI:
- 10.1103/PhysRevD.59.024007

[15]

General Relativistic Collapse to Black Holes and Gravitational Waves from Black Holes

- Prog.Theor.Phys.Suppl. 90 (1987) 1-218
•
DOI:
- 10.1143/PTPS.90.1

[16]

Covariant formulations of BSSN and the standard gauge

J.David Brown
(
- North Carolina State U.
)

- Phys.Rev.D 79 (2009) 104029
•
e-Print:
- 0902.3652
•
DOI:
- 10.1103/PhysRevD.79.104029

[17]

General covariant evolution formalism for numerical relativity

C. Bona
(
- Balearic Islands U.
)
,
T. Ledvinka
(
- Charles U.
)
,
C. Palenzuela
(
- Balearic Islands U.
)
,
M. Zacek
(
- Charles U.
)

- Phys.Rev.D 67 (2003) 104005
•
e-Print:
- gr-qc/0302083
•
DOI:
- 10.1103/PhysRevD.67.104005

[18]

A Symmetry breaking mechanism for the Z4 general covariant evolution system

C. Bona
(
- Balearic Islands U.
)
,
T. Ledvinka
(
- Balearic Islands U.
)
,
C. Palenzuela
(
- Balearic Islands U.
)
,
M. Zacek
(
- Balearic Islands U.
)

- Phys.Rev.D 69 (2004) 064036
•
e-Print:
- gr-qc/0307067
•
DOI:
- 10.1103/PhysRevD.69.064036

[19]

Towards a gauge-polyvalent Numerical Relativity code

Daniela Alic
(
- Balearic Islands U.
)
,
Carles Bona
(
- Balearic Islands U.
)
,
Carles Bona-Casas
(
- Balearic Islands U.
)

- Phys.Rev.D 79 (2009) 044026
•
e-Print:
- 0811.1691
•
DOI:
- 10.1103/PhysRevD.79.044026

[20]

Constraint violation in free evolution schemes: Comparing BSSNOK with a conformal decomposition of Z4

Sebastiano Bernuzzi
(
- Jena U., TPI
)
,
David Hilditch
(
- Jena U., TPI
)

- Phys.Rev.D 81 (2010) 084003
•
e-Print:
- 0912.2920
•
DOI:
- 10.1103/PhysRevD.81.084003

[21]

Conformal and covariant formulation of the Z4 system with constraint-violation damping

Daniela Alic
(
- Potsdam, Max Planck Inst.
)
,
Carles Bona-Casas
(
- Balearic Islands U. and
- Amsterdam U.
)
,
Carles Bona
(
- Balearic Islands U.
)
,
Luciano Rezzolla
(
- Potsdam, Max Planck Inst. and
- Louisiana State U.
)
,
Carlos Palenzuela
(
- Canadian Inst. Theor. Astrophys. and
- Louisiana State U.
)

- Phys.Rev.D 85 (2012) 064040
•
e-Print:
- 1106.2254
•
DOI:
- 10.1103/PhysRevD.85.064040

[22]

Constraint damping of the conformal and covariant formulation of the Z4 system in simulations of binary neutron stars

- Phys.Rev.D 88 (2013) 6, 064049
•
e-Print:
- 1307.7391
•
DOI:
- 10.1103/PhysRevD.88.064049

[23]

Fully covariant and conformal formulation of the Z4 system in a reference-metric approach: comparison with the BSSN formulation in spherical symmetry

- Phys.Rev.D 89 (2014) 10, 104033
•
e-Print:
- 1403.3653
•
DOI:
- 10.1103/PhysRevD.89.104033

[24]

Final fate of compact boson star mergers

- Phys.Rev.D 95 (2017) 12, 124005
•
e-Print:
- 1705.01071
•
DOI:
- 10.1103/PhysRevD.95.124005

[25]

Constraint damping in the Z4 formulation and harmonic gauge

Carsten Gundlach
(
- Southampton U.
)
,
Jose M. Martin-Garcia
(
- Southampton U.
)
,
Gioel Calabrese
(
- Southampton U.
)
,
Ian Hinder
(
- Madrid, Inst. Estructura Materia
)

- Class.Quant.Grav. 22 (2005) 3767-3774
•
e-Print:
- gr-qc/0504114
•
DOI:
- 10.1088/0264-9381/22/17/025

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