Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations

We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a li...

This is the first of a series of papers describing a numerical implementation of the conformally rescaled Einstein equation, an implementation designed to calculate asymptotically flat spacetimes, especially spacetimes containing black holes. In the present paper we derive the new first order time evolution equations to be used in the scheme. These time evolution equations can either be written in symmetric hyperbolic or in flux-conservative form. Since the conformally resc...

This paper focuses on the imposition of boundary conditions for numerical relativity simulations of black holes. This issue is used to motivate the discussion of a new hyperbolic formulation of 3+1 general relativity. The paper will appear in the Proceedings of the Les Houches School on Astrophysical Sources of Gravitational Radiation, 1995, edited by J.-A. Marck and J.-P. Lasota to be published by Springer-Verlag.

Combining deeper insight of Einstein's equations with sophisticated numerical techniques promises the ability to construct accurate numerical implementations of these equations. We illustrate this in two examples, the numerical evolution of ``bubble'' and single black hole spacetimes. The former is chosen to demonstrate how accurate numerical solutions can answer open questions and even reveal unexpected phenomena. The latter illustrates some of the difficulties encountered i...

The parabolic-hyperbolic form of the constraints is integrated numerically. The applied numerical stencil is $4^{th}$ order accurate (in the spatial directions) while 'time'-integration is made by using the method of lines with a $4^{th}$ order accurate Runge-Kutta scheme. The proper implementation of the applied numerical method is verified by convergence tests and monitoring the relative and absolute errors determined by comparing numerical and analytically known solutions ...

We present the first stable dynamical numerical evolutions of the Einstein equations in terms of a conformally rescaled metric on hyperboloidal hypersurfaces extending to future null infinity. Axisymmetry is imposed in order to reduce the computational cost. The formulation is based on an earlier axisymmetric evolution scheme, adapted to time slices of constant mean curvature. Ideas from a previous study by Moncrief and the author are applied in order to regularize the formal...

Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of ...

We present a novel implicit numerical implementation of the parabolic-hyperbolic formulation of the constraints of general relativity. The proposed method is unconditionally stable, has the advantage of not requiring the imposition of any boundary conditions in the strong field regime, and offers a holistic (all inclusive) approach to the construction of single and binary black hole initial data. The new implicit solver is extensively tested against known exact black hole sol...

This is the second in a series of papers describing a 3+1 computational scheme for the numerical simulation of dynamic black hole spacetimes. We discuss the numerical time-evolution of a given black-hole-containing initial data slice in spherical symmetry. We avoid singularities via the "black-hole exclusion" or "horizon boundary condition" technique, where the slices meet the black hole's singularity, but on each slice a spatial neighbourhood of the singularity is excluded f...

We present a new class of 3D black hole initial data sets for numerical relativity. These data sets go beyond the axisymmetric, ``gravity wave plus rotating black hole'' single black hole data sets by creating a dynamic, distorted hole with adjustable distortion parameters in 3D. These data sets extend our existing test beds for 3D numerical relativity, representing the late stages of binary black hole collisions resulting from on-axis collision or 3D spiralling coalescence, ...