3D simulations of Einstein's equations: Symmetric hyperbolicity, live gauges, and dynamic control of the constraints

Abstract

We present three-dimensional simulations of Einstein equations implementing a symmetric hyperbolic system of equations with dynamical lapse. The numerical implementation makes use of techniques that guarantee linear numerical stability for the associated initial-boundary value problem. The code is first tested with a gauge wave solution, where rather larger amplitudes and for significantly longer times are obtained with respect to other state of the art implementations. Additionally, by minimizing a suitably defined energy for the constraints in terms of free constraintfunctions in the formulation one can dynamically single out preferred values of these functions for the problem at hand. We apply the technique to fully three-dimensional simulations of a stationary black hole spacetime with excision of the singularity, considerably extending the lifetime of the simulations.

Relativistic MHD with adaptive mesh refinement

“…In particular, we have included fully dynamical general relativity in our code using the Einstein equations specified in [57]. In future work we hope to present evolutions of TOV stars as well as rotating, magnetized neutron stars.…”

Boundary conditions for the Einstein evolution system

“…(11)- (14), A kA B depends on the dynamical fields u α , and F A B depends on u α and ∂ k u α . We use upper case Latin indices such as A and B to label the constraint fields.…”

Hamiltonian relaxation

“…A method originally developed by Shibata and Nakamura [7] and later used by Baumgarte and Shapiro [8] (BSSN) is today the most commonly used for three-dimensional simulations. The literature offers an ever-growing list of new evolution formulations which can be divided in two groups: unconstrained [3,5,7,8,9,10,11,12,13,14,15,16,17,18,19] and constrained [20,21]. Some of these have been tested numerically under conditions that are either easy to implement numerically and / or have a high degree of symmetry.…”