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.

复杂的物理现象的可靠模拟可以使社会受益，并以无数方式满足人类的好奇心。示例范围从非常小的范围，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，例如由地震和海啸引起的自然灾害，以及我们对宇宙动力学和演变的理解。尽管计算能力正在增加，随着推动Exascale系统的推动，计算机硬件本身正在变得越来越多，并且难以有效使用，并且希望解决的模型所带来的挑战也在迅速增长。如果要实现新计算技术的巨大希望，则需要开发和部署更好的算法。该研究计划的重点是新的快速准确方法的发明，用于求解物理系统的全面模型，其中波传播起着核心作用，在上面概述的整个问题范围内应用。模拟波的主要障碍是大多数应用问题的多阶段性质。一方面，波的定义特征是它们相对于波长的长距离传播的能力，有效地导致了在无限域中构成的问题。另一方面，波与可能在波长尺度下或低于波长的介质相互作用。这种模型中的一个核心主题是非本地运营商的外观。该项目的主要目标是构建快速，准确和记忆效率的算法以评估它们。这包括（i）开发（i）有效和数学合理的一般波浪传播问题的域截断算法，目前仅适用于有限类别的系统，（ii）在数值构建波浪繁殖准确降低的材料中的材料和材料分离的材料分离的材料差异的情况下，在数值上构建波浪传播的准确降低级别的模型，该模型是有效的，该材料的分离率或有效地分离的材料（量表）量度分离的材料（量表）的分离量，或者是有效的量表。分数和其他操作员功能的算法。该项目的第二个目标是将强大和有效的离散方案扩展到二阶非线性波动方程，从行动原理中得出，尤其是在求解一般相对论和规格理论中求解方程的新方法。该奖项反映了NSF的法定任务，并通过评估了基金会的概述，并通过评估了基金会的范围和广泛的范围。