Analysing Ewald's method for the evaluation of Green's functions for periodic media

Abstract

Ewald's method is a technique for efficiently evaluating periodic Green's functions that is frequently used in the physics and engineering literature. The choice of the truncation and control parameters of the method is usually achieved by heuristic rules, and a rigorous numerical analysis of a fully discrete version of the method seems to be missing. In this paper, we analyse the truncation error of the method for the 2D and 3D periodic and biperiodic Green's functions of the Helmholtz equation, respectively, providing new, explicit and sharp bounds. These estimates are subsequently used to study the effect of choosing the method's control parameter and we give some recommendations for its choice. We present various numerical examples for the resulting method for evaluating the Green's functions. The results are also carried over to evaluating the partial derivatives.