Numerical Solution of Partial Differential Equations: Algorithms, Analysis, and Applications

偏微分方程的数值解：算法、分析与应用

• 批准号: 1719694
• 负责人: Douglas Arnold
• 金额: $ 30万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719694&HistoricalAwards=false
• 关键词: Numerical; Solution; Partial; Differential; Equations

Much of modern technology is dependent on fast and accurate computer simulation of physical phenomena. For example, computer simulations enable the development of high performance vehicles, medical devices, geophysical monitoring and prediction, and countless other applications on which the health and welfare of our society depend. Once such a physical system is modeled by a system of mathematical equations, successful simulation depends not only on powerful computer hardware but also on mathematical algorithms that can harness the computer's high speed number-crunching to obtain accurate solutions of model's equations. While effective simulation algorithms exist for many important physical systems, there remain many important applications for which fast and reliable algorithms do not yet exist. Moreover, it is crucial to develop not only the algorithms, but a robust and rigorous understanding of their performance, in order to be able to assess and certify their accuracy and delineate their limitations. This project will develop, improve, and validate simulation algorithms for two important and challenging classes of physical phenomena. The first concerns the propagation of waves through complex and disordered media, such as arises on tiny scales when electrical signals travel through semiconductors, on massive scales when seismic waves travel through the earth, and in many other relevant regimes. The second application area is in the burgeoning field of gravitational wave astronomy, made possible by the recent detection of gravitational waves from a black hole collision, and highly dependent on numerical simulation of Einstein's equations of general relativity. This project will advance the algorithms used to simulate complex physical phenomena modeled by partial differential equations. A key focus of effort will be on the propagation of waves through complex media, as, for example, the conduction of electrons in a semiconductor with impurities. This work will study the remarkable effect known as localization of eigenfunctions, first discovered over 50 years ago in Nobel prize winning research, and with major ramifications for many systems involving wave propagation. Until now, the understanding needed to accurately predict and control localization was lacking, but recent theoretical advances bring it within reach. A major goal of the research will be fast accurate prediction of localization from inexpensive processing of the media. This will then open the way for control of localization: the design of media with desired properties for applications. The second main direction concerns the simulation of Einstein's equations of general relativity. The goal will be to develop structure-preserving finite element methods for the equations, with the potential to greatly improve accuracy, efficiency, and robustness of simulations of general relativity. The work will focus on two types of approaches. First will be methods related to the Regge calculus introduced over 50 yeas ago as a discrete analogue of general relativity, overcoming the low accuracy of Regge calculus by developing a family of high order Regge elements and attacking the instabilities encountered, by using a 3+1 decomposition to separate time-like from space-like behavior. The second approach will be based on a system known as the Einstein-Bianchi equations, which hold the potential for leveraging advances in the numerical simulation of electromagnetic systems for the benefit of numerical relativity.

现代技术的大部分取决于物理现象的快速准确的计算机模拟。 例如，计算机模拟可以开发高性能车辆，医疗设备，地球物理监测和预测，以及我们社会健康和福利所依赖的无数其他应用。 一旦这样的物理系统由数学方程式系统建模，成功的仿真不仅取决于功能强大的计算机硬件，还取决于可以利用计算机高速数字处理的数学算法以获得模型方程的准确解决方案。 尽管许多重要的物理系统都存在有效的仿真算法，但仍然存在许多重要的应用，这些应用程序尚不存在，这些应用程序尚不存在。此外，不仅要开发算法，而且要对其性能进行牢固而严格的理解，以便能够评估和证明其准确性并描述其局限性。 该项目将开发，改进和验证两个重要且具有挑战性的物理现象类别的模拟算法。 第一个涉及通过复杂和无序的介质传播波浪的传播，例如当电信号穿过半导体，当地震波穿过地球以及许多其他相关方案时，电信号在大规模上传播时会出现。 第二个应用区域位于引力波天文学的迅速发展领域，这是由于最近发现的黑洞碰撞引力波，并且高度依赖于爱因斯坦一般相对性方程的数值模拟。该项目将推进用于模拟由部分微分方程建模的复杂物理现象的算法。努力的重点将是通过复杂介质传播波，例如，在具有杂质的半导体中传导电子。这项工作将研究被称为本本征的定位的显着效果，该作用最早是在50年前在诺贝尔奖获得研究中发现的，并且对许多涉及波浪传播的系统产生了重大影响。到目前为止，缺乏准确预测和控制本地化所需的理解，但是最近的理论进步使它触及了。这项研究的主要目标是快速准确地预测媒体廉价处理的本地化。然后，这将为控制本地化开辟道路：具有所需属性的媒体设计。 第二个主要方向涉及对爱因斯坦的一般相对论方程的模拟。目标是为方程开发具有结构的有限元方法，并有可能大大提高一般相对性模拟的准确性，效率和鲁棒性。这项工作将集中在两种类型的方法上。首先是与50年前引入的雷格微积分相关的方法，作为一般相对性的离散类似物，通过开发一个高阶regge元素的家族并攻击遇到的不稳定性的家族，通过使用3+1分解与太空行为分离，从而克服了Regge微积分的低准确性。第二种方法将基于一种称为爱因斯坦 - 比安奇方程的系统，该系统具有利用电磁系统数值模拟的进步的潜力，从而有利于数值相对性。

项目成果

Filoche et al. Reply:

费洛什等人。

• DOI: 10.1103/physrevlett.124.219702
• 发表时间: 2020
• 期刊: Physical Review Letters
• 影响因子: 8.6
• 作者: Filoche, M.;Arnold, D.;David, G.;Jerison, D.;Mayboroda, S.
• 通讯作者: Mayboroda, S.

Localization of eigenfunctions via an effective potential

• DOI: 10.1080/03605302.2019.1626420
• 发表时间: 2019-07-08
• 期刊: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
• 影响因子: 1.9
• 作者: Arnold, Douglas N.;David, Guy;Mayboroda, Svitlana
• 通讯作者: Mayboroda, Svitlana

COMPUTING SPECTRA WITHOUT SOLVING EIGENVALUE PROBLEMS

• DOI: 10.1137/17m1156721
• 发表时间: 2019-01-01
• 期刊: SIAM JOURNAL ON SCIENTIFIC COMPUTING
• 影响因子: 3.1
• 作者: Arnold, Douglas N.;David, Guy;Mayboroda, Svitlana
• 通讯作者: Mayboroda, Svitlana