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

Ferhat T Celepcikay,David R Jackson,Donald R Wilton,Filippo Capolino

doi:10.1029/2007rs003820

Ferhat T Celepcikay, David R Jackson + Show 2 more

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https://doi.org/10.1029/2007rs003820

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Abstract

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.

Highlights

[1] 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
In the Ewald method, the FSPGF is expressed as the sum of a ‘‘modified spectral’’ and a ‘‘modified spatial’’ series
A value EL of the Ewald splitting parameter is obtained based on the number of significant figures L that may be lost

Summary

Introduction

[2] In applying numerical full wave methods like the Method of Moments (MoM) or Boundary Integral Equations (BIE) to periodic structures involving conducting or dielectric electromagnetic scatterers, fast and accurate means for evaluating the free-space periodic Green’s function (FSPGF) are often needed. [4] The method proposed and studied here limits the size of the largest terms in the series relative to that of the total Green’s function by modifying the value of the splitting parameter E to avoid undue loss of accuracy. A value EL of the Ewald splitting parameter is obtained based on the number of significant figures L that may be lost This ‘‘best’’ value, EL, yields the fastest convergence of the Ewald series while limiting the loss of significant figures to the user-defined value L. Both spatial and spectral representations of each Green’s function exist in the general form whereas those of the spectral representations are

À D array of point sources

À D array

À D array of point sources: ð6Þ

Results

Conclusions