Fast Spectral Solvers for Partial Differential Equations in General Domains

1714169Bruno这个项目涉及开发和分析能够预测物理系统行为的计算方法，包括流体流动、固体力学、声学和电磁学。新的计算方法旨在有效地预测物理和工程系统的时间依赖行为，应用于医学、地震学、空气动力学和隐身等领域。所有这些领域的共同线索都与这些系统中固有的时间演化的复杂性有关。从跳动的心脏流出的血液，地震产生的地震波，飞机周围的流体流动，以及雷达技术的行为都有一些相同的元素：所观察到的现象的时间演变。这个项目要解决的就是这种时间依赖性的复杂性：傅里叶延拓（FC）和频谱时间方法试图重现某些频谱解算器的杰出性能，这些解算器不幸局限于简单的几何构型，用于更一般领域的模拟，大大减少了计算机内存和处理时间——因此，有效地解决了以前在高度重要的工程和科学兴趣领域棘手的问题。一名研究生和一名本科生参与了这个项目。从数学的角度来看，这项工作旨在产生时域偏微分方程（PDE）的求解器，可以在快速计算时间内为一般复杂三维结构提供几乎无色散（谱）的PDE解决方案。因此，该项目扩展到一般工程和科学配置的高质量，否则固有的光谱求解空间周期结构。FC和相关的频域方法已经在科学和工程领域的许多重要问题上取得了巨大的成功。本文研究的扩展是为了将时域解算器扩展到普遍适用的领域，从而将之前在频域频谱解算器领域的成功扩展到时域。由此产生的方法被应用于材料科学、应用物理、电气工程、地球物理和航空航天工程等领域的重要问题。一名研究生和一名本科生参与了这个项目。

1714169Bruno 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.