Long term stable integration of a maximally sliced Schwarzschild black hole using a smooth lattice method

TL;DR

This paper demonstrates a stable numerical evolution of a maximally sliced Schwarzschild black hole over long times using a smooth lattice method that employs local coordinates and Bianchi identities without special techniques.

Contribution

The paper introduces a stable, unconstrained lattice-based numerical method for evolving Schwarzschild black holes, avoiding the need for special techniques.

Findings

No instability observed up to t=1000m

Stable evolution achieved with standard ADM equations

Method relies on lattice geometry, local coordinates, and Bianchi identities

Abstract

We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t=1000m. The principle features of our method are i) the use of a lattice to record the geometry, ii) the use of local Riemann normal coordinates to apply the 1+1 ADM equations to the lattice and iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form.