Robust evolution system for numerical relativity

Arbona, A.; Bona, Carles; Massó, Joan; Stela, J.

doi:10.1103/physrevd.60.104014

Cited by 36 publications

(58 citation statements)

References 19 publications

(40 reference statements)

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“…(See also [16].) Similar variables have been introduced in [17,18,19,20,21]. Their evolution equation is obtained, as in the BSSN system, from commuting derivatives and then adding the momentum constraint.…”

Section: Introductionmentioning

confidence: 99%

Strongly hyperbolic second order Einstein’s evolution equations

2004

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BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformaltraceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudo-differential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints.

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Section: Introductionmentioning

confidence: 99%

Strongly hyperbolic second order Einstein’s evolution equations

2004

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“…In other 3+1 formulations, such as hyperbolic, conformal formulations and their combinations, the ADM equations are extended by introducing additional variables such as spatial derivatives of γ ij , traces of γ ij and K ij , conformal factors, by forming new combinations of these variables, or by modifying equations with the help of constraints [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. These modifications change the nature of the equations so that they can become more stable, and the integration can be prolonged [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Gauge stability of 3 + 1 formulations of general relativity

Khokhlov

Новиков²

2002

Class. Quantum Grav.

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We present a general approach to the analysis of gauge stability of 3+1 formulations of General Relativity (GR). Evolution of coordinate perturbations and the corresponding perturbations of lapse and shift can be described by a system of eight quasi-linear partial differential equations. Stability with respect to gauge perturbations depends on a choice of gauge and a background metric, but it does not depend on a particular form of a 3+1 system if its constrained solutions are equivalent to those of the Einstein equations. Stability of a number of known gauges is investigated in the limit of short-wavelength perturbations. All fixed gauges except a synchronous gauge are found to be ill-posed. A maximal slicing gauge and its parabolic extension are shown to be ill-posed as well. A necessary condition is derived for well-posedness of metric-dependent algebraic gauges. Well-posed metric-dependent gauges are found, however, to be generally unstable. Both instability and ill-posedness are associated with perturbations of physical accelerations of reference frames.

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“…(18), (19) and (20) since the derivatives of A l (r), B m (θ) and C n (φ) are known analytically. In order to evolve forward in time we can use any method of lines integrator.…”

Section: B Results For Non-linear Scalar Wavesmentioning

confidence: 99%

“…This is also how we have implemented this equation in our computer code, with the caveat that the Cartesian derivatives are computed using the expansions (18), (19) and (20). By this we mean that before each timestep derivatives like ∂ x ψ are computed using…”

Section: B Results For Non-linear Scalar Wavesmentioning

confidence: 99%

Black hole evolution with the BSSN system by pseudospectral methods

Tichy

2006

Phys. Rev. D

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We present a new pseudo-spectral code for the simulation of evolution systems that are second order in space. We test this code by evolving a non-linear scalar wave equation. These non-linear waves can be stably evolved using very simple constant or radiative boundary conditions, which we show to be well-posed in the scalar wave case. The main motivation for this work, however, is to evolve black holes for the first time with the BSSN system by means of a spectral method. We use our new code to simulate the evolution of a single black hole using all applicable methods that are usually employed when the BSSN system is used together with finite differencing methods. In particular, we use black hole excision and test standard radiative and also constant outer boundary conditions. Furthermore, we study different gauge choices such as 1 + log and constant densitized lapse. We find that these methods in principle do work also with our spectral method. However, our simulations fail after about 100M due to unstable exponentially growing modes. The reason for this failure may be that we evolve the black hole on a full grid without imposing any symmetries. Such full grid instabilities have also been observed when finite differencing methods are used to evolve excised black holes with the BSSN system.

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Robust evolution system for numerical relativity

Cited by 36 publications

References 19 publications

Strongly hyperbolic second order Einstein’s evolution equations

Strongly hyperbolic second order Einstein’s evolution equations

Gauge stability of 3 + 1 formulations of general relativity

Black hole evolution with the BSSN system by pseudospectral methods

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