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

许多现代技术依赖于对物理现象的快速而准确的计算机模拟。例如，计算机模拟使高性能车辆、医疗设备、地球物理监测和预测以及无数其他社会健康和福利所依赖的应用的发展成为可能。一旦这样一个物理系统被数学方程系统建模，成功的模拟不仅依赖于强大的计算机硬件，还依赖于数学算法，这些算法可以利用计算机的高速数字运算来获得模型方程的精确解。虽然对于许多重要的物理系统存在有效的仿真算法，但仍有许多重要的应用尚未存在快速可靠的算法。此外，至关重要的是不仅要开发算法，还要对其性能进行稳健和严格的理解，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、改进和验证两类重要且具有挑战性的物理现象的仿真算法。第一个问题涉及波在复杂和无序介质中的传播，例如当电信号穿过半导体时在微小尺度上出现，当地震波穿过地球时在大尺度上出现，以及在许多其他相关制度中出现。第二个应用领域是在新兴的引力波天文学领域，由于最近探测到黑洞碰撞产生的引力波而成为可能，并且高度依赖于爱因斯坦广义相对论方程的数值模拟。该项目将推进用于模拟由偏微分方程建模的复杂物理现象的算法。工作的重点将放在波在复杂介质中的传播，例如，在含有杂质的半导体中电子的传导。这项工作将研究被称为特征函数局部化的显著效应，该效应在50多年前的诺贝尔奖获奖研究中首次被发现，并对许多涉及波传播的系统产生重大影响。到目前为止，准确预测和控制定位所需的理解是缺乏的，但最近的理论进步使其触手可及。这项研究的一个主要目标将是通过廉价的媒体处理快速准确地预测定位。这将为本地化控制开辟道路：为应用程序设计具有所需属性的媒体。第二个主要方向是对爱因斯坦广义相对论方程的模拟。我们的目标是为这些方程开发保留结构的有限元方法，从而有可能大大提高广义相对论模拟的准确性、效率和鲁棒性。这项工作将侧重于两种方法。首先是与Regge演算相关的方法，该方法是50多年前作为广义相对论的离散模拟而引入的，通过开发一组高阶Regge元素来克服Regge演算的低精度，并通过使用3+1分解来分离类时类空行为来攻击遇到的不稳定性。第二种方法将基于一个被称为爱因斯坦-比安奇方程的系统，它具有利用电磁系统数值模拟的进步来促进数值相对论的潜力。

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.