Rapid solution of integral equations of scattering theory in two dimensions
Journal of Computational Physics
SIAM Journal on Scientific and Statistical Computing
Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs
SIAM Journal on Numerical Analysis
An adaptive spline wavelet ADI (SW-ADI) method for two-dimensional reaction-diffusion equations
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
A wideband fast multipole method for the Helmholtz equation in three dimensions
Journal of Computational Physics
Hi-index | 31.45 |
In this paper, we propose a parallel fast algorithm for computing the product of the discretized Helmholtz integral operator in layered media and a vector in O(N"qN"z^2N"xN"ylogN"xN"y) operations. Here N"xN"yN"z is the number of sources and N"q is the number of quadrature points used in the evaluation of the Sommerfeld integral in the definition of layered media Green's function (for problems in thin-layer media, N"z=O(1)). Such a product forms the key step of many iterative solvers (such as the Krylov subspace based GMRES and BiCGSTAB) for linear systems arising from the integral equation methods for the Helmholtz equations. The fast solver is based on two important techniques which reduce the cost of quadrature summations in the Sommerfeld contour integral for Green's functions in 3-D layered media. The first technique is the removal of surface pole effects along the real axis integration contour by identifying the pole locations with a discrete wavelet transform; In the second technique, we apply a window-based high frequency filter to shorten the contour length. As a result, the integral operator for the 3-D layered media can be efficiently written as a sum of 2-D Hankel cylindrical integral operators, and the latter can be calculated by either a tree-code or a 2-D wideband fast multipole method in a fast manner. Numerical results show the efficiency and parallelism of the proposed fast algorithm.