A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

TL;DR

This paper introduces a spectral numerical method for solving the wave equation for divergence-free vectors and symmetric tensors inside a sphere, utilizing spherical harmonics to efficiently evolve divergence-free degrees of freedom.

Contribution

The paper presents a novel spectral decomposition approach that simplifies the evolution of divergence-free vector and tensor fields in spherical coordinates, with explicit recovery of full fields at each step.

Findings

Effective spectral method for divergence-free wave equations

Accurate numerical tests with Chebyshev-tau method

Clear correspondence with poloidal-toroidal decomposition

Abstract

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal-toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.