Strongly hyperbolic second order Einstein’s evolution equations

Nagy, Gabriel; Ortiz, Omar E.; Reula, Oscar

doi:10.1103/physrevd.70.044012

Cited by 99 publications

(214 citation statements)

References 23 publications

(68 reference statements)

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“…Secondly, there is a use of the momentum constraint in the evolution equations for the three new firstorder variablesΓ i . According to [10], the latter play a critical role in the well-posed properties of the BSSN equations in a pseudospectral sense, whereas the former is irrelevant to well-posedness in such a sense. This accounts for the difference in the well-posedness of the first-order reduction of BSSN as compared to ADM (but of course, the comparison is not completely fair because BSSN is already a partial reduction).…”

Section: Discussionmentioning

confidence: 99%

Potential for ill-posedness in several second-order formulations of the Einstein equations

Frittelli

2004

Phys. Rev. D

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Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false.

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Section: Discussionmentioning

confidence: 99%

Potential for ill-posedness in several second-order formulations of the Einstein equations

Frittelli

2004

Phys. Rev. D

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“…This procedure enforces both the parity conditions and the conditions arising from local flatness at the origin and the axis of symmetry. We paid particular attention to the fact that our regularization procedure is independent of the system of evolution equations chosen, explicitly showing this in the case of the ADM formulation, as well as a strongly hyperbolic formulation similar to that of Nagy, Ortiz and Reula [15] (slightly modified in order to have all the dynamical variables well defined in curvilinear coordinates).…”

Section: Discussionmentioning

confidence: 99%

“…Many different well posed formulations of the 3+1 evolution equations have been proposed in the literature. For simplicity, here we will follow the work of Nagy and collaborators [15], but will adapt it to the case of axial symmetry.…”

Section: B Hyperbolic Evolution Systemmentioning

confidence: 99%

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Regularization of spherical and axisymmetric evolution codes in numerical relativity

2007

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Several interesting astrophysical phenomena are symmetric with respect to the rotation axis, like the head-on collision of compact bodies, the collapse and/or accretion of fields with a large variety of geometries, or some forms of gravitational waves. Most current numerical relativity codes, however, cannot take advantage of these symmetries due to the fact that singularities in the adapted coordinates, either at the origin or at the axis of symmetry, rapidly cause the simulation to crash. Because of this regularity problem it has become common practice to use full-blown Cartesian three-dimensional codes to simulate axi-symmetric systems. In this work we follow a recent idea of Rinne and Stewart and present a simple procedure to regularize the equations both in spherical and axi-symmetric spaces. We explicitly show the regularity of the evolution equations, describe the corresponding numerical code, and present several examples clearly showing the regularity of our evolutions.

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“…The BSSN formulation is one of many modifications to the ADM system that can yield a strongly (or even symmetric) hyperbolic problem when coupled to some gauge [60].…”

Section: The Lash Formulationmentioning

confidence: 99%

Stability of the puncture method with a generalized Baumgarte-Shapiro-Shibata-Nakamura formulation

2011

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The puncture method for dealing with black holes in the numerical simulation of vacuum spacetimes is remarkably successful when combined with the BSSN formulation of the Einstein equations. We examine a generalized class of formulations modelled along the lines of the Laguna-Shoemaker system and including BSSN as a special case. The formulation is a two parameter generalization of the choice of variables used in standard BSSN evolutions. Numerical stability of the standard finite difference methods is proven for the formulation in the linear regime around flat space, a special case of which is the numerical stability of BSSN. Numerical evolutions are presented and compared with a standard BSSN implementation. Surprisingly, a significant portion of the parameter space yields (long-term) stable simulations, including the standard BSSN formulation as a special case. Furthermore, non-standard parameter choices typically result in smoother behaviour of the evolution variables close to the puncture.

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Strongly hyperbolic second order Einstein’s evolution equations

Cited by 99 publications

References 23 publications

Potential for ill-posedness in several second-order formulations of the Einstein equations

Potential for ill-posedness in several second-order formulations of the Einstein equations

Regularization of spherical and axisymmetric evolution codes in numerical relativity

Stability of the puncture method with a generalized Baumgarte-Shapiro-Shibata-Nakamura formulation

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