Fast Spectral Solvers for Partial Differential Equations in General Domains

California Institute Of Technology, Pasadena CA

Investigators

Abstract

1714169 Bruno This project concerns development and analysis of computational methods enabling prediction of the behavior of physical systems including fluid flow, solid mechanics, acoustics, and electromagnetism. The new computational methods seek to efficiently predict the time-dependent behavior of physical and engineering systems, with application in areas such as medicine, seismology, aerodynamics, and stealth. A common thread in all of these areas relates to the complexity of the time evolution inherent in these systems. The flow of blood from a beating heart, the seismic waves arising from an earthquake, the fluid flow around an aircraft, and the behavior of radar technologies all share some of the same elements: the time evolution of the observed phenomena. It is this complexity in time-dependence that is addressed by this project: the Fourier Continuation (FC) and spectral time methodologies seek to reproduce the outstanding performance of certain spectral solvers, which are unfortunately restricted to simple geometric configurations, for simulations in a much more general realm, with greatly reduced computer memory and processing time -- thus, effectively enabling the solution of previously intractable problems in areas of highly significant engineering and scientific interest. A graduate student and an undergraduate student participate in the project. From a mathematical perspective, the effort seeks to produce solvers for partial differential equations (PDEs) in the time domain that can deliver, in fast computing times, nearly dispersionless (spectral) PDE solutions for general complex three-dimensional structures. The project thus extends to general engineering and scientific configurations the high quality otherwise inherent in spectral solvers for spatially periodic structures. The FC and related frequency-domain approaches have already been highly successful in a wide range of important problems in science and engineering. The extensions studied here seek to bring time-domain solvers to the realm of general applicability, thus extending to the time domain the previous successes in the area of spectral solvers in the frequency domain. The resulting methods are applied to important problems in the fields of materials science, applied physics, electrical engineering, geophysics, and aerospace engineering. A graduate student and an undergraduate student participate in the project.