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

TL;DR

This paper develops a stable 3D numerical framework for Einstein's equations using symmetric hyperbolic systems, live gauge conditions, and dynamic constraint control, enabling longer and more accurate black hole simulations.

Contribution

It introduces a numerically stable 3D simulation method with dynamic constraint management and live gauges for Einstein's equations, extending simulation lifetimes.

Findings

Enhanced stability and longer simulation times for black hole spacetimes.

Successful implementation of dynamic constraint control in 3D simulations.

Improved gauge wave test performance over existing methods.

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 constraint-functions 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.