The evolution of hyperboloidal data with the dual foliation formalism: Mathematical analysis and wave equation tests

TL;DR

This paper introduces a novel dual foliation formalism combined with hyperboloidal coordinates for evolving hyperboloidal initial data, enabling accurate treatment of null-infinity and effective wave equation simulations.

Contribution

It develops a new approach integrating dual foliation and hyperboloidal slices, allowing stable evolution at null-infinity and dynamic slice adjustment via eikonal equation solutions.

Findings

Outgoing waves are absorbed at null-infinity with minimal reflection.

Errors decrease rapidly with increased resolution in numerical tests.

The method successfully evolves nonlinear wave equations violating the null-condition.

Abstract

A long-standing problem in numerical relativity is the satisfactory treatment of future null-infinity. We propose an approach for the evolution of hyperboloidal initial data in which the outer boundary of the computational domain is placed at infinity. The main idea is to apply the `dual foliation' formalism in combination with hyperboloidal coordinates and the generalized harmonic gauge formulation. The strength of the present approach is that, following the ideas of Zenginoglu, a hyperboloidal layer can be naturally attached to a central region using standard coordinates of numerical relativity applications. Employing a generalization of the standard hyperboloidal slices, developed by Calabrese et. al., we find that all formally singular terms take a trivial limit as we head to null-infinity. A byproduct is a numerical approach for hyperboloidal evolution of nonlinear wave equations violating the null-condition. The height-function method, used often for fixed background spacetimes, is generalized in such a way that the slices can be dynamically `waggled' to maintain the desired outgoing coordinate lightspeed precisely. This is achieved by dynamically solving the eikonal equation. As a first numerical test of the new approach we solve the 3D flat space scalar wave equation. The simulations, performed with the pseudospectral bamps code, show that outgoing waves are cleanly absorbed at null-infinity and that errors converge away rapidly as resolution is increased.