Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives
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| Format: | Preprint |
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2026
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| _version_ | 1866916000042057728 |
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| author | Zhang, Hanwen |
| author_facet | Zhang, Hanwen |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09952 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives Zhang, Hanwen Numerical Analysis Computational Physics 65D20, 65R20, 65D32 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. |
| title | Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives |
| topic | Numerical Analysis Computational Physics 65D20, 65R20, 65D32 |
| url | https://arxiv.org/abs/2605.09952 |