Summation by parts methods for the spherical harmonic decomposition of the wave equation in arbitrary dimensions
This work provides stable numerical methods for wave equations in spherical coordinates, relevant for problems with spherical symmetry or boundaries, but the results are incremental as they extend existing SBP techniques to a specific coordinate system.
The paper develops summation by parts (SBP) finite difference methods for the spherical harmonic decomposition of the wave equation in arbitrary dimensions, achieving stability and convergence in the energy norm. The proposed schemes are second and fourth-order accurate at interior points and the symmetry boundary, and first and second-order accurate at the outer boundary.
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 $\dotπ=ψ'+pψ/r$, $\dotψ=π'$, 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 $r=R$.