A Fully Pseudospectral Scheme for Solving Singular Hyperbolic Equations

TL;DR

This paper demonstrates a fully pseudospectral numerical scheme for solving singular hyperbolic equations, achieving exponential convergence and avoiding artificial boundary conditions, with potential applications in General Relativity.

Contribution

It introduces a novel fully pseudospectral approach for hyperbolic equations on conformally compactified space-times, handling singularities at null infinity.

Findings

Achieved exponential spectral convergence in solutions.

Successfully handled singularities at null infinity.

Avoided artificial outer boundary conditions.

Abstract

With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving hyperbolic equations. The calculations are carried out within the framework of conformally compactified space-times. In our formulation, the equation becomes singular at null infinity and yields regular boundary conditions there. In this manner it becomes possible to avoid "artificial" conditions at some numerical outer boundary at a finite distance. We obtain highly accurate numerical solutions possessing exponential spectral convergence, a feature known from solving elliptic PDEs with spectral methods. Our investigations are meant as a first step towards the goal of treating time evolution problems in General Relativity with spectral methods in space and time.