Numerical relativity with characteristic evolution, using six angular patches

TL;DR

This paper explores a six-patch angular coordinate system in characteristic numerical relativity, demonstrating significant error reductions over traditional two-patch methods through second- and fourth-order finite differencing.

Contribution

It introduces and compares a six-patch implementation with existing two-patch methods, showing substantial improvements in accuracy and error reduction.

Findings

Six-patch system reduces errors by a factor of 2 compared to two-patch at the same resolution.

Fourth-order finite differencing in six-patch further reduces errors by nearly 50.

Six-patch approach enhances the precision of characteristic numerical relativity simulations.

Abstract

The characteristic approach to numerical relativity is a useful tool in evolving gravitational systems. In the past this has been implemented using two patches of stereographic angular coordinates. In other applications, a six-patch angular coordinate system has proved effective. Here we investigate the use of a six-patch system in characteristic numerical relativity, by comparing an existing two-patch implementation (using second-order finite differencing throughout) with a new six-patch implementation (using either second- or fourth-order finite differencing for the angular derivatives). We compare these different codes by monitoring the Einstein constraint equations, numerically evaluated independently from the evolution. We find that, compared to the (second-order) two-patch code at equivalent resolutions, the errors of the second-order six-patch code are smaller by a factor of about 2, and the errors of the fourth-order six-patch code are smaller by a factor of nearly 50.