Regularization of spherical and axisymmetric evolution codes in numerical relativity

TL;DR

This paper introduces a regularization method for spherical and axisymmetric evolution equations in numerical relativity, enabling more stable simulations of symmetric astrophysical phenomena by addressing coordinate singularities.

Contribution

It presents a simple regularization procedure for evolution equations in spherical and axisymmetric spaces, improving the stability of numerical relativity codes.

Findings

Regularized evolution equations are explicitly shown to be well-behaved.

Numerical examples demonstrate the effectiveness of the regularization.

The method prevents crashes caused by coordinate singularities.

Abstract

Several interesting astrophysical phenomena are symmetric with respect to the rotation axis, like the head-on collision of compact bodies, the collapse and/or accretion of fields with a large variety of geometries, or some forms of gravitational waves. Most current numerical relativity codes, however, can not take advantage of these symmetries due to the fact that singularities in the adapted coordinates, either at the origin or at the axis of symmetry, rapidly cause the simulation to crash. Because of this regularity problem it has become common practice to use full-blown Cartesian three-dimensional codes to simulate axi-symmetric systems. In this work we follow a recent idea idea of Rinne and Stewart and present a simple procedure to regularize the equations both in spherical and axi-symmetric spaces. We explicitly show the regularity of the evolution equations, describe the corresponding numerical code, and present several examples clearly showing the regularity of our evolutions.