Fast Spectral Solvers for Partial Differential Equations in General Domains

小行星1714169 该项目涉及计算方法的开发和分析，能够预测物理系统的行为，包括流体流动，固体力学，声学和电磁学。 新的计算方法寻求有效地预测物理和工程系统的时间相关行为，并应用于医学，地震学，空气动力学和隐身等领域。 所有这些领域的一个共同点是这些系统固有的时间演化的复杂性。 心脏跳动时的血液流动、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：观察到的现象的时间演变。 本项目解决的正是这种时间依赖性的复杂性：傅立叶延拓（FC）和谱时间方法试图再现某些谱解算器的出色性能，这些谱解算器不幸地局限于简单的几何配置，用于在更一般的领域中进行仿真，同时大大减少了计算机存储器和处理时间--因此，有效地使得能够解决在高度重要的工程和科学兴趣的领域中的先前难以解决的问题。 一名研究生和一名本科生参加了该项目。 从数学的角度来看，这项工作旨在产生求解偏微分方程（PDE）在时域，可以提供，在快速计算时间，几乎无色散（光谱）PDE解决方案，一般复杂的三维结构。 因此，该项目扩展到一般的工程和科学配置的高品质，否则固有的光谱求解空间周期性结构。 FC和相关的频域方法已经在科学和工程中的许多重要问题上取得了巨大的成功。 这里研究的扩展旨在使时域求解器的普遍适用性的领域，从而扩展到时域的频谱求解器在频域领域的先前的成功。 由此产生的方法适用于材料科学，应用物理，电气工程，电子物理学和航空航天工程领域的重要问题。 一名研究生和一名本科生参加了该项目。

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.