Numerical Methods for Waves: Nonlocal, Nonlinear, and Multiscale Systems

波的数值方法：非局部、非线性和多尺度系统

基本信息

批准号：
2012296
负责人：
Thomas Hagstrom
金额：
$ 34.25万
依托单位：
Southern Methodist University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012296&HistoricalAwards=false
关键词：
Numerical Methods Waves Nonlocal Nonlinear

项目摘要

The reliable simulation of complex physical phenomena can benefit society and help satisfy human curiosity in countless ways. Examples range from the very small, such as the design of nanoscale devices and emerging applications of quantum systems, to the very large, such as natural disasters caused by earthquakes and tsunamis, as well as our understanding of the dynamics and evolution of the cosmos. Although computational capabilities are increasing, with a push towards exascale systems, the computer hardware itself is becoming more heterogeneous and difficult to use efficiently, and the challenges posed by the models one wishes to solve are also rapidly growing. The need to develop and deploy better algorithms is urgent if the tremendous promise of the new computing technologies is to be realized. This research program is focused on the invention of new fast and accurate methods for solving comprehensive models of physical systems where wave propagation plays a central role, with applications throughout the range of problems outlined above.A primary obstacle to simulating waves is the multiscale nature of most applied problems. On the one hand, the defining feature of waves is their ability to propagate long distances relative to their wavelength, effectively leading to problems posed on unbounded domains. On the other, waves interact with media that may vary at or below the wavelength scale. A central theme in such models is the appearance of nonlocal operators; one main goal of the project is the construction of fast, accurate, and memory-efficient algorithms to evaluate them. This includes the development of (i) effective and mathematically justified domain truncation algorithms for general wave propagation problems, which at present are only available for a limited class of systems, (ii) methods for numerically constructing accurate reduced-order models of wave propagation in the presence of subwavelength variations in material properties, capable of efficiently treating engineered materials without assumptions of scale separation or periodicity, and (iii) low-memory algorithms for fractional and other operator functions. The second goal of this project is the extension of robust and efficient discretization schemes to second-order nonlinear wave equations derived from action principles, yielding, in particular, new methods to solve equations arising in general relativity and gauge theories.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.

对复杂物理现象的可靠模拟可以造福社会，并以多种方式帮助满足人类的好奇心。例子范围从非常小的，例如纳米级设备的设计和量子系统的新兴应用，到非常大的，例如地震和海啸引起的自然灾害，以及我们对宇宙动力学和演化的理解。尽管计算能力不断增强，但随着百亿亿级系统的发展，计算机硬件本身变得更加异构且难以有效使用，并且人们希望解决的模型带来的挑战也在快速增长。如果要实现新计算技术的巨大前景，就迫切需要开发和部署更好的算法。该研究项目的重点是发明新的快速而准确的方法来解决波传播起着核心作用的物理系统的综合模型，并在上述问题范围内得到应用。模拟波的一个主要障碍是波的多尺度性质。最应用的问题。一方面，波的定义特征是它们能够相对于其波长传播长距离，这实际上导致了无界域上出现的问题。另一方面，波与可能在波长尺度或以下变化的介质相互作用。这些模型的一个中心主题是非本地运营商的出现。该项目的主要目标之一是构建快速、准确且节省内存的算法来评估它们。这包括开发（i）针对一般波传播问题的有效且数学上合理的域截断算法，该算法目前仅适用于有限类别的系统，（ii）在数学中构建准确的波传播降阶模型的方法材料特性中存在亚波长变化，能够在不假设尺度分离或周期性的情况下有效地处理工程材料，以及（iii）用于分数和其他算子函数的低内存算法。该项目的第二个目标是将稳健且高效的离散化方案扩展到从作用原理导出的二阶非线性波动方程，特别是产生求解广义相对论和规范理论中出现的方程的新方法。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。