Hyperboloidal layers for hyperbolic equations on unbounded domains

TL;DR

This paper introduces hyperboloidal layers, a novel numerical method that employs coordinate transformations and compactification to solve hyperbolic equations on unbounded domains without artificial boundaries, demonstrated through Maxwell and wave equations.

Contribution

The paper presents a new hyperboloidal layer method combining coordinate transformation and compactification for efficient unbounded domain simulations.

Findings

Effective for Maxwell equations in 1D

Spectral techniques work for 3D wave equations

Avoids artificial boundary issues

Abstract

We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.