Efficient computation of the 2D periodic Green’s function using the Ewald method

An efficient computation of the periodic Helmholtz Green’s function for a 2D array of point sources using the Ewald method is presented. Limitations on the numerical accuracy when using the “optimum” E parameter (which gives optimum asymptotic convergence) at high frequency are discussed. A “best” E parameter is then derived to overcome these limitations, which allows for the fastest convergence while maintaining a specific level of accuracy (loss of significant figures) in the final result. The actual loss of significant figures has been verified through numerical simulations. Formulas for the number of terms needed for convergence have also been derived for both the spectral and the spatial series that appear in the Ewald method and are found to be accurate in almost all cases.