Cauchy-perturbative matching and outer boundary conditions: Computational studies

Rezzolla, Luciano; Abrahams, Andrew M.; Matzner, Richard A.; Rupright, M. E.; Shapiro, Stuart L.

doi:10.1103/physrevd.59.064001

Cited by 51 publications

(90 citation statements)

References 26 publications

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“…Obviously, there is no reason to believe that the solution error should behave as ǫ(t − r)/r n for general choices of lapse and shift. Another approach to construct outer boundary conditions that has been used is to blend [21] the numerical solution to the analytic one beyond a certain radius, r. Finally, at the inner or excision boundary, the working assumption is that all of the fields have outgoing (into the hole) characteristics. We will show that for the gauge or coordinate conditions under consideration, this is indeed the situation.…”

Section: Theġ -K Equationsmentioning

confidence: 99%

Cure for unstable numerical evolutions of single black holes: Adjusting the standard ADM equations in the spherically symmetric case

et al. 2001

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Numerical codes based on a direct implementation of the standard ADM formulation of Einstein's equations have generally failed to provide long-term stable and convergent evolutions of black hole spacetimes when excision is used to remove the singularities. We show that, for the case of a single black hole in spherical symmetry, it is possible to circumvent these problems by adding to the evolution equations terms involving the constraints, thus adjusting the standard ADM system. We investigate the effect that the choice of the lapse and shift has on the stability properties of numerical simulations and thus on the form of the added constraint term. To facilitate this task, we introduce the concept of quasi well-posedness, a version of well-posedness suitable for ADM-like systems involving second-order spatial derivatives.

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Section: Theġ -K Equationsmentioning

confidence: 99%

Cure for unstable numerical evolutions of single black holes: Adjusting the standard ADM equations in the spherically symmetric case

et al. 2001

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“…In numerical relativity, boundary conditions have usually been rather crudely implemented. Some sort of outgoing radiation conditions are imposed on all components of the metric, or boundary conditions are based on an analytic exterior solution [12,21]. One attraction of hyperbolic methods has been the possibility of basing boundary conditions on the eigenmodes of the characteristic matrix.…”

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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“…In [46,47], boundary conditions for the full nonlinear Einstein equations on a finite domain are obtained by matching to exact solutions of the linearized field equations at the boundary. Alternatively, the interior code could be matched to an 'outer module' that solves the linearized field equations numerically [48][49][50][51]. Other approaches involve matching the interior nonlinear Cauchy code to an outer characteristic code (see [52] for a review) or using hyperboloidal spacetime slices that can be compactified towards null infinity (see [53] for a review).…”

Section: Discussionmentioning

confidence: 99%

Testing outer boundary treatments for the Einstein equations

Rinne¹,

Lindblom²,

Scheel³

2007

Class. Quantum Grav.

179

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Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a 'reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests are based on a first-order generalized harmonic formulation of the Einstein equations and are implemented using a pseudo-spectral collocation method. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar 0 , and control gauge reflections.

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Cauchy-perturbative matching and outer boundary conditions: Computational studies

Cited by 51 publications

References 26 publications

Cure for unstable numerical evolutions of single black holes: Adjusting the standard ADM equations in the spherically symmetric case

Cure for unstable numerical evolutions of single black holes: Adjusting the standard ADM equations in the spherically symmetric case

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

Testing outer boundary treatments for the Einstein equations

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