Hyperbolic formulations and numerical relativity: II. asymptotically constrained systems of Einstein equations

Yoneda, Gen; Shinkai, H.

doi:10.1088/0264-9381/18/3/307

Cited by 26 publications

(59 citation statements)

References 35 publications

(57 reference statements)

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“…We note that these guidelines are confirmed numerically for wave propagation in the Maxwell system and in the Ashtekar version of the Einstein system [12], and also for error propagation in Minkowskii space-time using adjusted ADM systems [16]. Supporting theorems for above (A) was recently discussed [31].…”

Section: Constraint Propagation Analysis In Flat Space-time a Psupporting

confidence: 64%

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Advantages of a modified ADM formulation: Constraint propagation analysis of the Baumgarte-Shapiro-Shibata-Nakamura system

Yoneda

Shinkai

2002

Phys. Rev. D

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(gr-qc/0204002) to appear in Phys. Rev. D.)Several numerical relativity groups are using a modified ADM formulation for their simulations, which was developed by Nakamura et al (and widely cited as Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is shown to be more stable than the standard ADM formulation in many cases, and there have been many attempts to explain why this re-formulation has such an advantage. We try to explain the background mechanism of the BSSN equations by using eigenvalue analysis of constraint propagation equations. This analysis has been applied and has succeeded in explaining other systems in our series of works. We derive the full set of the constraint propagation equations, and study it in the flat background space-time. We carefully examine how the replacements and adjustments in the equations change the propagation structure of the constraints, i.e. whether violation of constraints (if it exists) will decay or propagate away. We conclude that the better stability of the BSSN system is obtained by their adjustments in the equations, and that the combination of the adjustments is in a good balance, i.e. a lack of their adjustments might fail to obtain the present stability. We further propose other adjustments to the equations, which may offer more stable features than the current BSSN equations.

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Section: Constraint Propagation Analysis In Flat Space-time a Psupporting

confidence: 64%

“…Through the series of studies [3,6,12,16], we propose a systematic treatment for constructing a robust evolution system against perturbative error. We call it an asymp-totically constrained (or asymptotically stable) system if the error decays itself.…”

Section: Introductionmentioning

confidence: 99%

Advantages of a modified ADM formulation: Constraint propagation analysis of the Baumgarte-Shapiro-Shibata-Nakamura system

Yoneda

Shinkai

2002

Phys. Rev. D

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“…Another possibility is to take as the variables the components of an orthonormal triad and the projection of the extrinsic curvature onto the triad (or the Ashtekar variables [15][16][17]). The triad formalism, because the extrinsic curvature tensor projected on the triad is symmetric, has fewer variables than the mixed coordinate component formalism.…”

Section: B Relevance Of Results and Future Directionsmentioning

confidence: 99%

“…Adding these energy constraint terms to the AY formulation is a special case of the more general Kidder, Scheel, and Teukolsky [19] schemes. Shinkai and Yoneda [15][16][17] analyzed the stability and accuracy properties of first order hyperbolic systems using Ashtekar's connection variables in plane-symmetric spacetimes, and found that the addition of multiples of the constraints to the dynamical equations improved accuracy and stability. These results have been extended to ADM systems of equations [18].…”

Section: Introductionmentioning

confidence: 99%

Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1D colliding gravitational plane waves

Bardeen

Buchman

2002

Phys. Rev. D

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We investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a first-order 3+1 approach to the Einstein equations, with basic variables some combination of first derivatives of the spatial metric and components of the extrinsic curvature tensor. Hyperbolic schemes, specifically variations on schemes proposed by Bona and Massó and Anderson and York, are compared with variations of the Arnowitt-Deser-Misner formulation. Modifications of the three basic schemes include raising one index in the metric derivative and extrinsic curvature variables and adding a multiple of the energy constraint to the extrinsic curvature evolution equations. Redundant variables in the Bona-Massó formulation may be reset frequently or allowed to evolve freely. Gauge conditions which simplify the dynamical structure of the system are imposed during each time step, but the lapse and shift are reset periodically to control the evolution of the spacetime slicing and the longitudinal part of the metric. We show that physically correct boundary conditions, satisfying the energy and momentum constraint equations, generically require the presence of some ingoing eigenmodes of the characteristic matrix. Numerical methods are developed for the hyperbolic systems based on decomposing flux differences into linear combinations of eigenvectors of the characteristic matrix. These methods are shown to be second-order accurate, and in practice second-order convergent, for smooth solutions, even when the eigenvectors and eigenvalues of the characteristic matrix are spatially varying.

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“…Another possibility is the construction of asymptotically constrained systems that seek to control the violation of constraints by having the constraint surface as an attractor. This includes the λ−system [21,78] and a family of adjusted systems [35,87,88,86]. All these have shown several advantages when compared to the standard ADM formulation [76,67].…”

Section: Introductionmentioning

confidence: 99%

On the hyperbolicity and stability of $$3+1$$ 3 + 1 formulations of metric f(R) gravity

Mongwane

2016

Gen Relativ Gravit

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3+1 formulations of the Einstein field equations have become an invaluable tool in Numerical relativity, having been used successfully in modeling spacetimes of black hole collisions, stellar collapse and other complex systems. It is plausible that similar considerations could prove fruitful for modified gravity theories. In this article, we pursue from a numerical relativistic viewpoint the 3 + 1 formulation of metric f (R) gravity as it arises from the fourth order equations of motion, without invoking the dynamical equivalence with Brans-Dicke theories. We present the resulting system of evolution and constraint equations for a generic function f (R), subject to the usual viability conditions. We confirm that the time propagation of the f (R) Hamiltonian and Momentum constraints take the same Mathematical form as in general relativity, irrespective of the f (R) model. We further recast the 3+1 system in a form akin to the BSSNOK formulation of numerical relativity. Without assuming any specific model, we show that the ADM version of f (R) is weakly hyperbolic and is plagued by similar zero speed modes as in the general relativity case. On the other hand the BSSNOK version is strongly hyperbolic and hence a promising formulation for numerical simulations in metric f (R) theories.

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Hyperbolic formulations and numerical relativity: II. asymptotically constrained systems of Einstein equations

Cited by 26 publications

References 35 publications

Advantages of a modified ADM formulation: Constraint propagation analysis of the Baumgarte-Shapiro-Shibata-Nakamura system

Advantages of a modified ADM formulation: Constraint propagation analysis of the Baumgarte-Shapiro-Shibata-Nakamura system

Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1D colliding gravitational plane waves

On the hyperbolicity and stability of $$3+1$$ 3 + 1 formulations of metric f(R) gravity

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