A spectral solver for evolution problems with spatial S3-topology

TL;DR

This paper presents a spectral collocation method tailored for solving initial value problems of PDEs on 3-sphere domains, with applications to cosmological Einstein equations and symmetric tensorial evolutions.

Contribution

It introduces a novel spectral solver for PDEs on 3-spheres, addressing implementation issues and extending to symmetric tensorial equations in cosmology.

Findings

Successfully applied to cosmological Einstein equations

Demonstrated effectiveness for symmetric tensorial PDEs

Analyzed numerical stability and accuracy

Abstract

We introduce a single patch collocation method in order to compute solutions of initial value problems of partial differential equations whose spatial domains are 3-spheres. Besides the main ideas, we discuss issues related to our implementation and analyze numerical test applications. Our main interest lies in cosmological solutions of Einstein's field equations. Motivated by this, we also elaborate on problems of our approach for general tensorial evolution equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy symmetry here.