Efficient Algorithms for Computing Sommerfeld Integral Tails

Abstract

Abstract-Sommerfeld-integrals (SIs) are ubiquitous in the analysis of problems involving antennas and scatterers embedded in planar multilayered media. It is well known that the oscillating and slowly decaying nature of their integrands makes the numerical evaluation of the SI real-axis tail segment a very time consuming and computationally expensive task. Therefore, SI tails have to be specially treated. In this paper we compare two recently developed techniques for their efficient numerical evaluation. First, a partition-extrapolation method, in which the integration-then-summation procedure is combined with a new version of the weighted averages (WA) extrapolation technique, is summarized. The previous variants of WA technique are also discussed. Then, a review of double-exponential (DE) quadrature formulas for direct integration of the SI tails is presented. The efficient way of implementing the algorithms, their pros and cons, as well as comparisons of their performance are discussed in detail.

Weighted Averages and Double Exponential Algorithms

“…7 [10]. But this was always a delicate maneuver, due to the wild behavior of the Bessel functions in the complex plane.…”

“…As for the Sommerfeld tails, oscillating and eventually diverging but regular and smooth, they can be always solved with a WA algorithm [10]. However, Ooura's DE versions [26], [27] has paved the way for DE being also applied to the semi-infinite tails [29].…”

“…However, Ooura's DE versions [26], [27] has paved the way for DE being also applied to the semi-infinite tails [29]. Thus, two efficient options are now currently available for the numerical evaluation of complete&general Sommerfeld integrals: the DE+WA strategy or the DE+DE strategy [10]. Since they differ in the treatment of the tail, a canonical Sommerfeld integral with a standard tail can provide a perfect benchmark.…”

“…The decisive step here was to consider the implied integrals as defined in the Abel's sense and to apply to them the well known Cèsaro and Hölder means [8], [9]. In addition to Sommerfeld tails [10], generalized WA algorithms can cope now with divergent series [11] and with much more complex integrals, like those involving products of Bessel functions [12]. Thus, the transmutation of WA from a very specific tool for a particular electromagnetic problem to a generic numerical algorithm is being achieved.…”

“…A final development of paramount relevance for this paper is the transformation, obtained by T. Ooura, of the DE quadrature into a form that can be applied to Fourier transforms of slowly decaying functions [26], [27]. This modified DE version can be easily applied to the evaluation of Sommerfeld integral tails [10], [28], [29].…”