Numerical Methods for Wave Equations in Time and Frequency Domain

An intrinsic feature of waves is their ability to propagate over large distances without changing their shape. This ability allows waves to carry information, be it through speech or electronic transmission of data. Waves can also be used to probe the interior of the earth, the human body or engineered structures like buildings or bridges.

This probing can be turned into images of the interior by the means of solving inverse problems, and in the extension, mitigate seismic hazards by accurate predictions of ground motion caused by earthquakes. In this project the principal investigator will develop computational simulation tools that increases our ability to exploit the properties of wave propagation for the common good.

The tools developed in the project can also be used to design modern materials with exotic properties that cannot be found in nature. Such metamaterials can enable better sensing technologies and faster acoustic and electromagnetic circuit components such as miniaturized speakers, 5G components and other millimeter wave technologies.

The research will use a new idea that enables the use of time domain methods for wave equations to design frequency domain Helmholtz type solvers. The approach is remarkable in that the underlying linear operator corresponds to a symmetric positive definite matrix allowing the solution of a coercive problem rather than an indefinite Helmholtz problem.

As the proposed Helmholtz solvers rely solely on evolving the wave equation they will be massively parallel, scalable and high order accurate. A goal of the research is to solve the Helmholtz equation in three dimensions at higher frequencies, and on a larger number of cores than is currently possible. The research will also seek to improve the time-step constraints of time domain discontinuous Galerkin methods by exploiting approximation spaces built on discrete periodic extensions from equidistant node data.

Such improvements will result in faster simulation times and more accurate predictions. Applications of the methods to modeling of micropolar materials and to simulation of seismic waves will be carried out in collaboration with researchers from academic institutions and national laboratories.

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.