Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives
This work provides a fast and accurate method for computing Green's function modes, which is a key bottleneck in modal boundary integral equation solvers for axisymmetric scattering problems.
The paper presents an O(M) algorithm for evaluating azimuthal Fourier modes of the 3D Helmholtz Green's function and their derivatives, achieving high accuracy independent of wavenumber and source-target separation. Numerical experiments confirm linear scaling and applicability to axisymmetric acoustic scattering.
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.