Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media

TL;DR

This paper introduces two novel fast multipole algorithms for 3D Helmholtz equations in layered media, utilizing Taylor expansions and complex image approximations to achieve efficient and accurate solutions with linear complexity.

Contribution

The paper develops two new fast multipole algorithms based on Taylor expansion techniques for layered media Helmholtz equations, improving computational efficiency and accuracy.

Findings

Algorithms achieve O(N) complexity.

Numerical tests validate accuracy.

Methods are effective for layered media.

Abstract

In this paper, we develop fast multipole methods for 3D Helmholtz kernel in layered media. Two algorithms based on different forms of Taylor expansion of layered media Green's function are developed. A key component of the first algorithm is an efficient algorithm based on discrete complex image approximation and recurrence formula for the calculation of the layered media Green's function and its derivatives, which are given in terms of Sommerfeld integrals. The second algorithm uses symmetric derivatives in the Taylor expansion to reduce the size of precomputed tables for the derivatives of layered media Green's function. Numerical tests in layered media have validated the accuracy and O(N) complexity of the proposed algorithms.