Hyperboloidal evolution of test fields in three spatial dimensions

TL;DR

This paper introduces a numerical method for solving hyperbolic PDEs on curved backgrounds using hyperboloidal compactification, effectively addressing boundary and radiation extraction issues at null infinity.

Contribution

It presents a novel implementation of hyperboloidal slicing and compactification in 3+1 formalism, enabling accurate boundary treatment and radiation extraction in numerical relativity.

Findings

Successful numerical tests with scalar waves on Minkowski and Schwarzschild backgrounds

Effective handling of null infinity and boundary conditions

Addressed implementation challenges for Einstein equations

Abstract

We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactification at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.