Carles Bona

General covariant evolution formalism for numerical relativity

A general-covariant extension of Einsteins field equations is considered with a view to numerical relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector

First order hyperbolic formalism for numerical relativity

{Z}_{\ensuremath{\mu}}.

Towards standard testbeds for numerical relativity

Einsteins solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition

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

{Z}_{\ensuremath{\mu}}=0.

A Symmetry breaking mechanism for the Z4 general covariant evolution system

The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.

Constraint-preserving boundary conditions in the Z4 numerical relativity formalism

The causal structure of Einstein’s evolution equations is considered. We show that in general they can be written as a first-order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution equations, which can lead to numerical inaccuracies, can be eliminated by using the Hamiltonian constraint. Furthermore, we show that the entire system is hyperbolic when the time coordinate is chosen in an invariant algebraic way, and for any fixed choice of the shift. This is achieved by using the momentum constraints in such a way that no additional space or time derivatives of the equations need to be computed. The slicings that allow hyperbolicity in this formulation belong to a large class, including harmonic, maximal, and many others that have been commonly used in numerical relativity. We provide details of some of the advanced numerical methods that this formulation of the equations allows, and we also discuss certain advantages that a hyperbolic formulation provides when treating boundary conditions. @S0556-2821~97!05616-6# PACS number~s!: 04.25.Dm

Robustness of the Blandford–Znajek mechanism

In recent years, many different numerical evolution schemes for Einsteins equations have been proposed to address stability and accuracy problems that have plagued the numerical relativity community for decades. Some of these approaches have been tested on different spacetimes, and conclusions have been drawn based on these tests. However, differences in results originate from many sources, including not only formulations of the equations, but also gauges, boundary conditions, numerical methods and so on. We propose to build up a suite of standardized testbeds for comparing approaches to the numerical evolution of Einsteins equations that are designed to both probe their strengths and weaknesses and to separate out different effects, and their causes, seen in the results. We discuss general design principles of suitable testbeds, and we present an initial round of simple tests with periodic boundary conditions. This is a pivotal first step towards building a suite of testbeds to serve the numerical relativists and researchers from related fields who wish to assess the capabilities of numerical relativity codes. We present some examples of how these tests can be quite effective in revealing various limitations of different approaches, and illustrating their differences. The tests are presently limited to vacuum spacetimes, can be run on modest computational resources and can be used with many different approaches used in the relativity community.

Robust evolution system for numerical relativity

We present a new formulation of the Einstein equations based on a conformal and traceless decomposition of the covariant form of the Z4 system. This formulation combines the advantages of a conformal decomposition, such as the one used in the BSSNOK formulation (i.e. well-tested hyperbolic gauges, no need for excision, robustness to imperfect boundary conditions) with the advantages of a constraintdamped formulation, such as the generalized harmonic one (i.e. exponential decay of constraint violations when these are produced). We validate the new set of equations through standard tests and by evolving binary black hole systems. Overall, the new conformal formulation leads to a better behavior of the constraint equations and a rapid suppression of the violations when they occur. The changes necessary to implement the new conformal formulation in standard BSSNOK codes are very small as are the additional computational costs.

Geometrically motivated hyperbolic coordinate conditions for numerical relativity : Analysis, issues and implementations

The general-covariant Z4 formalism is further analyzed. The gauge conditions are generalized with a view to Numerical Relativity applications and the conditions for obtaining strongly hyperbolic evolution systems are given both at the first and the second order levels. A symmetry-breaking mechanism is proposed that allows one, when applied in a partial way, to recover previously proposed strongly hyperbolic formalisms, like the BSSN and the Bona-Masso ones. When applied in its full form, the symmetry breaking mechanism allows one to recover the full five-parameter family of first order KST systems. Numerical codes based in the proposed formalisms are tested. A robust stability test is provided by evolving random noise data around Minkowski space-time. A strong field test is provided by the collapse of a periodic background of plane gravitational waves, as described by the Gowdy metric.

A 3+1 covariant suite of numerical relativity evolution systems

The constraint-preserving approach is discussed in parallel with other recent developments with the goal of providing consistent boundary conditions for numerical relativity simulations. The case of the first-order version of the Z4 system is considered, and constraint-preserving boundary conditions of the Sommerfeld type are provided. The stability of the proposed boundary conditions is related to the choice of the ordering parameter. This relationship is explored numerically and some values of the ordering parameter are shown to provide stable boundary conditions in the absence of corners and edges. Maximally dissipative boundary conditions are also implemented. In this case, a wider range of values of the ordering parameter is allowed, which is shown numerically to provide stable boundary conditions even in the presence of corners and edges.