Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions

Abstract

We investigate numerical methods for wave equations in n + 2 spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on Sn, 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 , , 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.