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We investigate numerical methods for wave equations in $n+2$ spacetimedimensions, written in spherical coordinates, decomposed in spherical harmonicson $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Suchan approach is useful when the full physical problem has spherical symmetry,for perturbation theory about a spherical background, or in the presence ofboundaries with spherical topology. The key numerical difficulty arises fromlower-order $1/r$ terms at the origin $r=0$. As a toy model for this, weconsider 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 theleading spherical harmonic index. We propose a class of summation by parts(SBP) finite differencing methods that conserve a discrete energy up toboundary terms, thus guaranteeing stability and convergence in the energy norm.We explicitly construct SBP schemes that are second and fourth-order accurateat interior points and the symmetry boundary $r=0$, and first and second-orderaccurate at the outer boundary $r=R$.

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