Stable and efficient evaluation of periodized Green's functions for the Helmholtz equation at high frequencies

We present a new algorithm for the evaluation of the periodized Green's function for Helmholtz equation in two and three dimensions. A variety of classical algorithms (based on spatial and spectral representations, Ewald transformation, etc.) have been implemented in the past to evaluate such acoustic fields. As we show however, these methods become unstable and/or impractically expensive as the frequency of use of the sources increases. Here we introduce a new numerical scheme that overcomes some of these limitations allowing for simulations at unprecedented frequencies. The method is based on a new integral representation derived from the classic spatial form, and on suitable further manipulations of the relevant integrands to render the integrals amenable to efficient and accurate approximations through standard quadrature formulas. We include a variety of numerical results that demonstrate that our algorithm compares favorably with every classical method both for points close to the line where the poles are located and at high-frequencies while remaining competitive with them in every other instance.