Choosing splitting parameters and summation limits in the numerical evaluation of 1-D and 2-D periodic Green's functions using the Ewald method

Accurate and efficient computation of periodic free-space Green's functions using the Ewald method is considered for three cases: a 1-D array of line sources, a 1-D array of point sources, and a 2-D array of point sources. A limitation on the numerical accuracy when using the “optimum” E parameter (which gives optimum asymptotic convergence) at high frequency is discussed. A “best” E parameter is then derived to overcome these limitations. This choice allows for the fastest convergence while maintaining a specific level of accuracy (loss of significant figures) in the final result. Formulas for the number of terms needed for convergence are also derived for both the spectral and the spatial series that appear in the Ewald method, and these are found to be accurate in almost all cases.