High Order Wave Equation Algorithms for the Frequency Domain

A defining feature of waves is their ability to carry information over large distances by propagating without changing their shape. It is this ability that allow waves to probe and image the human body, the interior of the earth and engineered structures like bridges and tunnels. Such images can then be turned into scientific and engineering knowledge that can be used to improve medical diagnostics and prevent failure of buildings and mechanical devices. 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 advanced materials that can enable better acoustic, elastic and electromagnetic components as well as faster and more accurate sensing technologies. Students will be trained as a part of this work. The research will further develop and apply advanced computational methods for solving systems of partial differential equations modeling wave propagation. The approximation methods will be designed to be robust and flexible while effectively utilizing emerging computational architectures. The research will dramatically improve the WaveHoltz method, a recently discovered idea that enables the use of time domain methods for wave equations to design frequency domain Helmholtz type solvers. WaveHoltz is remarkable in that its underlying linear operator corresponds to a symmetric positive definite matrix and allows a coercive problem to be solved rather than a highly indefinite Helmholtz problem. The research will analyze and develop wavefront preconditioners and deflation techniques for preconditioning WaveHoltz; design implicit and explicit error corrected methods for removing the temporal error in the WaveHoltz method; and consider multi-frequency versions of the WaveHoltz method. 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.