A multi-block infrastructure for three-dimensional time-dependent numerical relativity

TL;DR

This paper presents a flexible multi-patch infrastructure for 3D numerical relativity simulations, demonstrating stability, accuracy, and efficiency in evolving Einstein's equations and gravitational waves.

Contribution

It introduces a multi-patch framework with penalty boundary conditions, enabling stable, accurate 3D simulations of Einstein's equations and gravitational waves.

Findings

Multi-patch systems compete with fixed mesh refinement.

Stable long-term evolutions of gauge waves achieved.

Accurate gravitational waveforms extracted from 3D simulations.

Abstract

We describe a generic infrastructure for time evolution simulations in numerical relativity using multiple grid patches. After a motivation of this approach, we discuss the relative advantages of global and patch-local tensor bases. We describe both our multi-patch infrastructure and our time evolution scheme, and comment on adaptive time integrators and parallelisation. We also describe various patch system topologies that provide spherical outer and/or multiple inner boundaries. We employ penalty inter-patch boundary conditions, and we demonstrate the stability and accuracy of our three-dimensional implementation. We solve both a scalar wave equation on a stationary rotating black hole background and the full Einstein equations. For the scalar wave equation, we compare the effects of global and patch-local tensor bases, different finite differencing operators, and the effect of artificial dissipation onto stability and accuracy. We show that multi-patch systems can directly compete with the so-called fixed mesh refinement approach; however, one can also combine both. For the Einstein equations, we show that using multiple grid patches with penalty boundary conditions leads to a robustly stable system. We also show long-term stable and accurate evolutions of a one-dimensional non-linear gauge wave. Finally, we evolve weak gravitational waves in three dimensions and extract accurate waveforms, taking advantage of the spherical shape of our grid lines.