Citation
Anderson, Thomas Geoffrey (2021) Hybrid FrequencyTime Analysis and Numerical Methods for TimeDependent Wave Propagation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/hmv1r869. https://resolver.caltech.edu/CaltechTHESIS:09042020172204130
Abstract
This thesis focuses on the solution of causal, timedependent wave propagation and scattering problems, in two and threedimensional spatial domains. This important and longlasting problem has attracted a great deal of interest reflecting not only its use as a model problem but also the prevalence of wave phenomena in diverse areas of modern science, technology and engineering. Essentially all prior methods rely on "timestepping" in one form or another, which involves localintime approximation of the evolution of the solution of the partial differential equation (PDE) based on the immediate time history and temporal finitedifference approximation. In addition to the need to manage the accumulation of (dispersion) error and the burdensome increase in computational cost over time, there are additionally difficult issues of stability, timedomain boundary conditions, and absorbing boundary conditions which often need to be addressed.
To sidestep many of these problems, this thesis develops a novel highlyefficient approach for timedependent wave scattering problems employing the globalintime techniques of Fourier transformation and leading to a frequency/time hybrid method for the timedependent wave equation. Thus, relying on Fourier Transformation in time and utilizing a fixed (timeindependent) number of frequencydomain solutions, the method evaluates the desired timedomain evolution with errors that both, decay faster than any negative power of the temporal sampling rate, and that, for a given sampling rate, are additionally uniform in time for all time. The fast error decay guarantees that high accuracies can be attained on the basis of relatively coarse temporal and frequency discretizations. The uniformity of the error for all time with fixed sampling rate, a property known as dispersionlessness, plays a crucial role, together with other properties of the Fourier transform, in enabling the evaluation of solutions for long times at O(1) cost. In particular, this thesis demonstrates the significant advantages enjoyed by the proposed methods over alternative approaches based on volumetric discretizations, timedomain integral equations, and convolutionquadrature.
The approach relies on two main elements, namely, 1) A smooth timewindowing methodology that enables accurate bandlimited representations for arbitrarilylong time signals, and 2) A novel Fourier transform approach which, in a timeparallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrallyaccurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider in detail the Laplacebased method, and only briefly point out its essential features and associated challenges.
The proposed frequency/time Fouriertransform methods for obstacle scattering problems are easily generalizable to any linear partial differential equation in the time domain for which frequencydomain solutions can readily be obtained, including e.g. the timedomain Maxwell equations, the linear elasticity equations, inhomogeneous and/or frequencydependent dispersive media, etc. Further, the proposed approach can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping—that is, solution sampling at any given time T at O(1)bounded sampling cost, for arbitrarily large values of T, and without requirement of evaluation of the solution at intermediate times. In particular, effective algorithms are introduced that, relying on use of timeasymptotics, compute twodimensional solutions at O(1) cost despite the very slow timedecay that takes place in the twodimensional case.
A significant portion of this thesis is devoted to a theoretical study of the validity of a certain stopping criterion used by the algorithm, which guarantees that certain field contributions can safely be neglected after certain stopping times. Roughly speaking, the theoretical results guarantee that, after the incident field is turned off, the magnitude of the future scattering density (and thus the magnitudes of the fields) can be estimated by the magnitude of the integral density over a time period comparable to the time required by a wave to travel a distance equal to the diameter of the scatterer. The criterion, which is crucial in ensuring the O(1) computational cost of the algorithm, is closely related to the wellknown scattering theory developed in the 1960s and '70s by Lax, Morawetz, Phillips, Strauss and others. Our approach to the decay problem is based on use of frequencydomain estimates (developed previously in the context of numerical analysis of frequencydomain problems) on integral operators in the highfrequency regime for obstacles of various trapping classes. In particular, our theory yields, for the first time, decay estimates for a class of connected trapping obstacles: all previous estimates of scatteredfield decay for connected obstacles are restricted to nontrapping structures.
In all, the proposed approach leverages the power of the Fourier transformation together with a range of newly developed spectrally convergent numerical methods in both the frequency and time domain and a variety of novel theoretical results in the general area of scattering theory to produce a radicallynew framework for the solution of timedependent wave propagation and scattering problems.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Partial differential equations, fourier transform, scattering theory, numerical analysis, integral equations  
Degree Grantor:  California Institute of Technology  
Division:  Engineering and Applied Science  
Major Option:  Applied And Computational Mathematics  
Awards:  Ben P.C. Chou Doctoral Prize in IST, 2021.  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  17 August 2020  
Funders: 
 
Record Number:  CaltechTHESIS:09042020172204130  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:09042020172204130  
DOI:  10.7907/hmv1r869  
Related URLs: 
 
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  13864  
Collection:  CaltechTHESIS  
Deposited By:  Thomas Anderson  
Deposited On:  11 Sep 2020 15:57  
Last Modified:  03 Nov 2021 18:47 
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