Calculation of SPH and VSH Expansions

TL;DR

This paper introduces a spectrally accurate numerical method for computing spherical and vector spherical harmonic expansions of functions on a sphere, enabling precise simulations of acoustic and electromagnetic scattering problems.

Contribution

The paper develops a robust, high-accuracy spectral method for spherical harmonic expansions that efficiently handles oscillatory integrals, advancing computational techniques for 3D scattering simulations.

Findings

High accuracy achieved within reasonable computational time

Method effectively handles highly oscillatory integrals

Applicable to 3D acoustic and electromagnetic scattering simulations

Abstract

We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.