An efficient solver for problems of scattering by bodies of revolution
YoungAe Han
TL;DR
The paper advances a high-order, axisymmetric solver for scattering by penetrable bodies of revolution by reformulating the 3-D Lippmann-Schwinger problem into axisymmetric Legendre modes and accelerating angular and radial integrations with Fast Legendre Transforms. It achieves near $O(N\log^2 N)$ complexity per GMRES iteration and demonstrates spectral convergence for globally smooth refractive indices, with convergence limited by the smoothness of the solution otherwise. Numerical experiments on spheres and annuli validate accuracy, robustness to geometry, and scalability, highlighting the method's capacity to deliver high-precision results for axisymmetric scattering problems. Theoretical analysis connects convergence to the decay of Legendre coefficients and tail errors, linking regularity to performance and guiding parameter choices for practical implementations.
Abstract
This proposal relates to the design, analysis and application of a novel numerical scheme for the solution of axisymmetric scattering problems. To this end, a procedure is introduced to iteratively evaluate the solution of the Lippmann-Schwinger integral equation in $O(N\log^2 N)$ operations, where $N$ is the number of the discretization points. The method achieves its efficiency through the use of the addition theorem and Fast Legendre Transforms (FLT). For globally smooth sound velocities/refractive indexes the method is spectrally accurate. More generally the order of convergence is tied to and in fact, limited by, the smoothness of the solution.