Summation by parts methods for the spherical harmonic decomposition of the wave equation in arbitrary dimensions

TL;DR

This paper develops summation by parts finite difference methods for solving wave equations in spherical coordinates, ensuring stability and accuracy near the origin and boundaries, with applications in spherical symmetry problems.

Contribution

It introduces SBP finite differencing schemes tailored for wave equations in spherical coordinates, addressing the challenge of lower-order terms at the origin and guaranteeing energy conservation.

Findings

Constructed second and fourth-order accurate SBP schemes at the origin.

Achieved stability and convergence in the energy norm.

Provided explicit schemes for boundary accuracy.

Abstract

We investigate numerical methods for wave equations in $n+2$ spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Such an approach is useful when the full physical problem has spherical symmetry, for perturbation theory about a spherical background, or in the presence of boundaries with spherical topology. The key numerical difficulty arises from lower-order $1/r$ terms at the origin $r=0$. As a toy model for this, we consider the flat space linear wave equation in the form $\dot\pi=\psi'+p\psi/r$, $\dot\psi=\pi'$, where $p=2l+n$, and $l$ is the leading spherical harmonic index. We propose a class of summation by parts (SBP) finite differencing methods that conserve a discrete energy up to boundary terms, thus guaranteeing stability and convergence in the energy norm. We explicitly construct SBP schemes that are second and fourth-order accurate at interior points and the symmetry boundary $r=0$, and first and second-order accurate at the outer boundary $r=R$.