The Hyperboloidal Numerical Evolution of a Good-Bad-Ugly Wave Equation

TL;DR

This paper develops a hyperboloidal numerical method for a wave system satisfying the weak-null condition, demonstrating regularization and convergence in spherical symmetry, with implications for Einstein equations at null infinity.

Contribution

It introduces a nonlinear variable change to regularize weak-null wave equations and implements a hyperboloidal evolution scheme with demonstrated convergence.

Findings

Successful regularization of the wave system

Clean convergence of the numerical scheme

Potential applicability to Einstein equations

Abstract

One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a naive choice of evolved variables. Here we consider a system of wave equations that satisfy the weak-null condition and serve as a model system with similar nonlinearities to those present in the Einstein field equations in generalized harmonic gauge. We show that these equations can be explicitly regularized by a nonlinear change of variables. Working in spherical symmetry, a numerical implementation of this model using compactified hyperboloidal slices is then presented. Clean convergence is found for the regularized system. Although more complicated, it is expected that general relativity can be treated similarly.