Numerical Methods for Waves: Nonlocal, Nonlinear, and Multiscale Systems

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The reliable simulation of complex physical phenomena can benefit society and help satisfy human curiosity in countless ways. Examples range from the very small, such as the design of nanoscale devices and emerging applications of quantum systems, to the very large, such as natural disasters caused by earthquakes and tsunamis, as well as our understanding of the dynamics and evolution of the cosmos. Although computational capabilities are increasing, with a push towards exascale systems, the computer hardware itself is becoming more heterogeneous and difficult to use efficiently, and the challenges posed by the models one wishes to solve are also rapidly growing. The need to develop and deploy better algorithms is urgent if the tremendous promise of the new computing technologies is to be realized. This research program is focused on the invention of new fast and accurate methods for solving comprehensive models of physical systems where wave propagation plays a central role, with applications throughout the range of problems outlined above.A primary obstacle to simulating waves is the multiscale nature of most applied problems. On the one hand, the defining feature of waves is their ability to propagate long distances relative to their wavelength, effectively leading to problems posed on unbounded domains. On the other, waves interact with media that may vary at or below the wavelength scale. A central theme in such models is the appearance of nonlocal operators; one main goal of the project is the construction of fast, accurate, and memory-efficient algorithms to evaluate them. This includes the development of (i) effective and mathematically justified domain truncation algorithms for general wave propagation problems, which at present are only available for a limited class of systems, (ii) methods for numerically constructing accurate reduced-order models of wave propagation in the presence of subwavelength variations in material properties, capable of efficiently treating engineered materials without assumptions of scale separation or periodicity, and (iii) low-memory algorithms for fractional and other operator functions. The second goal of this project is the extension of robust and efficient discretization schemes to second-order nonlinear wave equations derived from action principles, yielding, in particular, new methods to solve equations arising in general relativity and gauge theories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.