Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A high-order perturbation of surfaces (HOPS) approach to Fokas integral equations: Three-dimensional layered-media scattering


Author: David P. Nicholls
Journal: Quart. Appl. Math. 74 (2016), 61-87
MSC (2010): Primary 65R20, 65N35
DOI: https://doi.org/10.1090/qam/1411
Published electronically: December 3, 2015
MathSciNet review: 3472520
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Abstract: In this paper we discuss a novel High-Order Perturbation of Surfaces (HOPS) method for the simulation of linear acoustic waves in a three-dimensional layered, periodic structure. The model we consider is that of linear time-harmonic wave propagation which generates three-dimensional, quasiperiodic, outgoing solutions of Helmholtz equations, coupled across irregular layer interfaces. We significantly enhance and stabilize the approach of the author and D. Ambrose, which is a formulation of the problem utilizing the integral equation formalism of Fokas and collaborators. The method is phrased in terms of interfacial variables resulting in a method which has an order of magnitude fewer unknowns than a volumetric approach to this problem. In contrast to classical integral equation formulations, the current contribution does not require specialized quadratures or periodized fundamental solutions. Additionally, as a result of the HOPS philosophy, our new approach is not only faster and better conditioned than the algorithm of the author and Ambrose, it also delivers the scattering returns of an entire family of solutions with a single simulation rather than requiring a new approximation for each profile of interest. Detailed numerical simulations are presented which demonstrate the efficiency, fidelity, and spectral accuracy which can be realized with this new methodology.


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Additional Information

David P. Nicholls
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
Email: davidn@uic.edu

DOI: https://doi.org/10.1090/qam/1411
Received by editor(s): April 25, 2014
Received by editor(s) in revised form: August 12, 2014
Published electronically: December 3, 2015
Article copyright: © Copyright 2015 Brown University


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