Dealing with the center and boundary problems in 1D Numerical Relativity

TL;DR

This paper addresses stability issues in 1D numerical relativity simulations caused by coordinate singularities, proposing a new method leveraging redundant quantities to improve stability and boundary condition implementation.

Contribution

It introduces a novel approach using redundant variables from hyperbolic Einstein formulations to enhance stability and boundary handling in 1D numerical relativity codes.

Findings

Significant stability improvements in simulations of boson stars.

Enhanced boundary condition implementation demonstrated.

Reduction of instabilities caused by coordinate singularities.

Abstract

Instabilities in finite difference codes due to the singularity of spherical coordinates at the center are studied. In typical Numerical Relativity applications, standard regularization techniques by themselves do not ensure long term stability. A proposal to remedy that problem is presented, which takes advantage of redundant quantities introduced in recent hyperbolic formulations of Einstein's evolution equations. The results are discussed through the example case of a boson star, where a significant improvement in the implementation of boundary conditions is also presented.