Testing the Cactus code on exact solutions of the Einstein field equations

The article presents a series of numerical simulations of exact solutions of the Einstein equations performed using the Cactus code, a complete 3-dimensional machinery for numerical relativity. We describe an application (``thorn'') for the Cactus code that can be used for evolving a variety of exact solutions, with and without matter, including solutions used in modern cosmology for modeling the early stages of the universe. Our main purpose has been to test the Cactus code on these well-known examples, focusing mainly on the stability and convergence of the code.Comment: 18 pages, 18 figures, Late

The status of numerical relativity

Numerical relativity has come a long way in the last three decades and is now reaching a state of maturity. We are gaining a deeper understanding of the fundamental theoretical issues related to the field, from the well posedness of the Cauchy problem, to better gauge conditions, improved boundary treatment, and more realistic initial data. There has also been important work both in numerical methods and software engineering. All these developments have come together to allow the construction of several advanced fully three-dimensional codes capable of dealing with both matter and black holes. In this manuscript I make a brief review the current status of the field.Comment: Report on plenary talk at the 17th International Conference on General Relativity and Gravitation (GR17), held at Dublin, Ireland, july 2004. Latex, 20 pages, 5 figure

Pathologies of hyperbolic gauges in general relativity and other field theories

We present a mathematical characterization of hyperbolic gauge pathologies in general relativity and electrodynamics. We show how non-linear gauge terms can produce a blow-up along characteristics and how this can be identified numerically by performing convergence analysis. Finally, we show some numerical examples and discuss the profound implications this may have for the field of numerical relativity.Comment: 5 pages, includes 2 figs. To appear in Phys.Rev.D Rapid Com

The appearance of coordinate shocks in hyperbolic formalisms of General Realtivity

I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Masso formalism with zero shift vector and show how shocks associated with two families of characteristic fields can develop. These shocks do not represent discontinuities in the geometry of spacetime, but rather regions where the coordinate system becomes pathological. For this reason I call them coordinate shocks. I show how one family of shocks can be eliminated by restricting the Bona-Masso slicing condition to a special case. The other family of shocks, however, can not be eliminated even in the case of harmonic slicing. I also show the results of numerical simulations in the special cases of a flat two-dimensional spacetime, a flat four-dimensional spacetime with a spherically symmetric slicing, and a spherically symmetric black hole spacetime. In all three cases coordinate shocks readily develop, confirming the predictions of the mathematical analysis. Although here I concentrate in the Bona-Masso formalism, the phenomena of coordinate shocks should arise in any other hyperbolic formalism. In particular, since the appearance of the shocks is determined by the choice of gauge, the results presented here imply that in any formalism the use of a harmonic slicing can generate shocks.Comment: REVTEX, 27 pages plus 11 Postscript figures. To be published in Phys. Rev.

Time-Symmetric ADI and Causal Reconnection: Stable Numerical Techniques for Hyperbolic Systems on Moving Grids

Moving grids are of interest in the numerical solution of hydrodynamical problems and in numerical relativity. We show that conventional integration methods for the simple wave equation in one and more than one dimension exhibit a number of instabilities on moving grids. We introduce two techniques, which we call causal reconnection and time-symmetric ADI, which together allow integration of the wave equation with absolute local stability in any number of dimensions on grids that may move very much faster than the wave speed and that can even accelerate. These methods allow very long time-steps, are fully second-order accurate, and offer the computational efficiency of operator-splitting.Comment: 45 pages, 19 figures. Published in 1994 but not previously available in the electronic archive

Regularization of spherical and axisymmetric evolution codes in numerical relativity

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, can not 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 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.Comment: 11 pages, 9 figures. Several changes. Main corrections are in eqs. (2.12) and (5.14). Accepted in Gen. Rel. Gra

Formulations of the 3+1 evolution equations in curvilinear coordinates

Following Brown, in this paper we give an overview of how to modify standard hyperbolic formulations of the 3+1 evolution equations of General Relativity in such a way that all auxiliary quantities are true tensors, thus allowing for these formulations to be used with curvilinear sets of coordinates such as spherical or cylindrical coordinates. After considering the general case for both the Nagy-Ortiz-Reula (NOR) and the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulations, we specialize to the case of spherical symmetry and also discuss the issue of regularity at the origin. Finally, we show some numerical examples of the modified BSSN formulation at work in spherical symmetry.Comment: 19 pages, 12 figure

Gauge conditions for long-term numerical black hole evolutions without excision

Numerical relativity has faced the problem that standard 3+1 simulations of black hole spacetimes without singularity excision and with singularity avoiding lapse and vanishing shift fail after an evolution time of around 30-40M due to the so-called slice stretching. We discuss lapse and shift conditions for the non-excision case that effectively cure slice stretching and allow run times of 1000M and more.Comment: 19 pages, 14 figures, REVTeX, Added a missing Acknowledgmen