Hybrid Frequency-Time Analysis and Numerical Methods for Time-Dependent Wave Propagation

Citation

Anderson, Thomas Geoffrey (2021) Hybrid Frequency-Time Analysis and Numerical Methods for Time-Dependent Wave Propagation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/hmv1-r869. https://resolver.caltech.edu/CaltechTHESIS:09042020-172204130

Abstract

This thesis focuses on the solution of causal, time-dependent wave propagation and scattering problems, in two- and three-dimensional spatial domains. This important and long-lasting 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 "time-stepping" in one form or another, which involves local-in-time approximation of the evolution of the solution of the partial differential equation (PDE) based on the immediate time history and temporal finite-difference 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, time-domain boundary conditions, and absorbing boundary conditions which often need to be addressed.

To sidestep many of these problems, this thesis develops a novel highly-efficient approach for time-dependent wave scattering problems employing the global-in-time techniques of Fourier transformation and leading to a frequency/time hybrid method for the time-dependent wave equation. Thus, relying on Fourier Transformation in time and utilizing a fixed (time-independent) number of frequency-domain solutions, the method evaluates the desired time-domain 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, time-domain integral equations, and convolution-quadrature.

The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate 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 Laplace-based method, and only briefly point out its essential features and associated challenges.

The proposed frequency/time Fourier-transform methods for obstacle scattering problems are easily generalizable to any linear partial differential equation in the time domain for which frequency-domain solutions can readily be obtained, including e.g. the time-domain Maxwell equations, the linear elasticity equations, inhomogeneous and/or frequency-dependent 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 time-asymptotics, compute two-dimensional solutions at O(1) cost despite the very slow time-decay that takes place in the two-dimensional 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 well-known 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 frequency-domain estimates (developed previously in the context of numerical analysis of frequency-domain problems) on integral operators in the high-frequency 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 scattered-field 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 radically-new framework for the solution of time-dependent wave propagation and scattering problems.

Thesis Files