Taylor expansion based fast multipole method for 3-D Helmholtz equations in layered media

In this paper, we develop a Taylor expansion (TE) based fast multipole method (FMM) for low frequency 3D Helmholtz Green's function in layered media. Two forms of Taylor expansions, with either non-symmetric or symmetric derivatives of layered media Green's functions, are used for the implementations of the proposed TE-FMM. In the implementation with non-symmetric derivatives, an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives. Meanwhile, the implementation based on symmetric derivatives is more robust and pre-computed tables for the high order derivatives in translation operators are used. Numerical tests in layered media have validated the accuracy and O(N) complexity of the proposed algorithms. (C) 2019 Elsevier Inc. All rights reserved.