Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives
Abstract
We introduce an O(M) algorithm for evaluating the azimuthal Fourier modes Gk,m, m = 0, 1, ..., M, of the three-dimensional Helmholtz Green's function with real wavenumber k, together with all their first- and second-order derivatives with respect to the cylindrical source and target coordinates. The cost is independent of both the wavenumber and the source-target separation, and high relative accuracy is retained even for modes whose magnitude is exponentially small. The method combines contour deformation at a few boundary modes with a boundary-value formulation of the five-term recurrence in the mode index. Derivative quantities are obtained from stable recurrences, adding only a small constant factor to the cost of Gk,m alone. Numerical experiments demonstrate high relative accuracy, linear scaling in M, and applications to modal boundary integral equation solvers for axisymmetric acoustic scattering, where the k-independent kernel evaluator makes dense per-mode linear algebra the dominant cost.
Versions
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BibTeX
@misc{zhang2026fast,
title = {{Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives}},
author = {Hanwen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.09952},
eprint = {2605.09952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.09952},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives
AU - Hanwen Zhang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2605.09952
UR - https://arxiv.org/abs/2605.09952
AB - We introduce an O(M) algorithm for evaluating the azimuthal Fourier modes G_k,m, m = 0, 1, ..., M, of the three-dimensional Helmholtz Green's function with real wavenumber k, together with all their first- and second-order derivatives with respect to the cylindrical source and target coordinates. The cost is independent of both the wavenumber and the source-target separation, and high relative accuracy is retained even for modes whose magnitude is exponentially small. The method combines contour deformation at a few boundary modes with a boundary-value formulation of the five-term recurrence in the mode index. Derivative quantities are obtained from stable recurrences, adding only a small constant factor to the cost of G_k,m alone. Numerical experiments demonstrate high relative accuracy, linear scaling in M, and applications to modal boundary integral equation solvers for axisymmetric acoustic scattering, where the k-independent kernel evaluator makes dense per-mode linear algebra the dominant cost.
ER -