Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics

TL;DR

This paper introduces a spectral method using spin-weighted spherical harmonics for simulating tensorial fields on spherical manifolds, combining existing algorithms for stable and accurate numerical evolution.

Contribution

The paper develops a novel spectral method that integrates two existing algorithms to efficiently and accurately evolve tensor fields on the sphere.

Findings

Successfully simulated scalar and vector advection on the sphere.

Accurately modeled 2+1 Maxwell equations on a deformed sphere.

Demonstrated stability and efficiency of the combined numerical approach.

Abstract

Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through use of integer spin weighted spherical harmonics we present a method which allows for the implementation of arbitrary tensorial evolution equations. Our method combines two numerical techniques that were originally developed with different applications in mind. The first is Huffenberger and Wandelt's spectral decomposition algorithm to perform the mapping from physical to spectral space. The second is the application of Luscombe and Luban's method, to convert numerically divergent linear recursions into stable nonlinear recursions, to the calculation of reduced Wigner d-functions. We give a detailed discussion of the theory and numerical implementation of our algorithm. The properties of our method are investigated by solving the scalar and vectorial advection equation on the sphere, as well as the 2+1 Maxwell equations on a deformed sphere.