Fixed mesh refinement in the characteristic formulation of General Relativity

TL;DR

This paper introduces a fixed mesh refinement technique within the characteristic formulation of General Relativity, enabling adaptive resolution and efficient simulations of gravitational phenomena, including critical behavior and energy conservation.

Contribution

The implementation of fixed mesh refinement in the characteristic formulation allows adaptive resolution in time, improving efficiency while accurately capturing complex gravitational dynamics.

Findings

Successfully replicates gravitational critical behavior

Achieves global energy conservation at black hole formation threshold

Offers improved runtime with results comparable to non-refined code

Abstract

We implement a spatially fixed mesh refinement under spherical symmetry for the characteristic formulation of General Relativity. The Courant-Friedrich-Levy (CFL) condition lets us deploy an adaptive resolution in (retarded-like) time, even for the nonlinear regime. As test cases, we replicate the main features of the gravitational critical behavior and the spacetime structure at null infinity using the Bondi mass and the News function. Additionally, we obtain the global energy conservation for an extreme situation, i.e. in the threshold of the black hole formation. In principle, the calibrated code can be used in conjunction with an ADM 3+1 code to confirm the critical behavior recently reported in the gravitational collapse of a massless scalar field in an asymptotic anti-de Sitter spacetime. For the scenarios studied, the fixed mesh refinement offers improved runtime and results comparable to code without mesh refinement.