Stable long-term evolution in numerical relativity

TL;DR

This paper identifies a late-time numerical instability in long-term black hole simulations caused by momentum constraint violations and proposes modified formulations that effectively stabilize the evolution.

Contribution

The authors introduce two modified CCZ4 schemes that prevent damping of momentum violations, significantly improving long-term stability in numerical relativity simulations.

Findings

The late-time instability is linked to accumulated momentum constraint violations.

Modified CCZ4 schemes effectively eliminate the instability.

Enhanced Hamiltonian constraint damping improves long-term simulation stability.

Abstract

We report on the potential occurrence of a numerical instability in the long-time simulation of black holes using the Baumgarte-Shapiro-Shibata-Nakamura formulation of numerical relativity, even in the simple set-up of a Schwarzschild black hole. Through extensive numerical experiments, we identify that this "late-time instability" arises from accumulated violations of the momentum constraint. To address this issue, we propose two modified versions of the so-called conformal covariant Z4 scheme, designed to propagate momentum constraint violations without damping. Our results demonstrate that these alternative formulations, which we refer to as CCZ4' and CCZ3, effectively resolve the late-time numerical instability not only in Schwarzschild spacetimes but also in black hole spacetimes with matter fields. Notably, by preventing damping of the momentum constraint violation, the Hamiltonian constraint damping can be significantly increased, which plays a crucial role in stabilizing long-term evolution in our proposed schemes.