Constraint preserving boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura formulation in spherical symmetry

TL;DR

This paper develops and tests boundary conditions for the BSSN formulation in spherical symmetry that preserve constraints and improve the stability and accuracy of long-term Einstein evolution simulations.

Contribution

It introduces two new approaches for constraint-preserving boundary conditions in the BSSN formulation under spherical symmetry, enhancing simulation stability.

Findings

Boundary conditions prevent spurious reflections.

Simulations remain accurate and stable over long durations.

Solutions satisfy constraints up to the boundary.

Abstract

We introduce a set of constraint preserving boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein evolution equations in spherical symmetry, based on its hyperbolic structure. While the outgoing eigenfields are left to propagate freely off the numerical grid, boundary conditions are set to enforce that the incoming eigenfields don't introduce spurious reflections and, more importantly, that there are no fields introduced at the boundary that violate the constraint equations. In order to do this we adopt two different approaches to set boundary conditions for the extrinsic curvature, by expressing either the radial or the time derivative of its associated outgoing eigenfield in terms of the constraints. We find that these boundary conditions are very robust in practice, allowing us to perform long lasting evolutions that remain accurate and stable, and that converge to a solution that satisfies the constraints all the way to the boundary.