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

Understand

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.