CDS&E: ECCS: Collaborative Research: PNPM Schemes Adapted for the First Time to Computational Electrodynamics for Solving 21st Century Problems

1966年，余甘恩发展时空计算算法，求解麦克斯韦方程组，用以研究电磁波的传播。他的方法发展成为现在所知的时域有限差分（FDTD）方法。在随后的几十年里，FDTD取得了进展，使其能够应用于电磁频谱的广泛问题，从低频（低于1 Hz）一直到可见光。目前，时域有限差分（FDTD）是对非常大、非常复杂的电磁波相互作用问题，特别是需要结合多物理场的问题进行建模不可缺少的工具。然而，FDTD正在显示出它的年龄。其基本二阶算法的精度和对光滑、非符合网格的材料界面建模的困难，已经成为严重的局限性。Co-PI Balsara最近发表了麦克斯韦方程组的所有高阶精确解的数学蓝图。这些方案将克服当前建模方法的局限性，同时保留其优点。由于这些高阶精确方案为所有意图和目的提供了麦克斯韦方程组的精确数值解，因此将有可能设计出比目前更隐身的航空航天和海军平台。同样，设计复杂的无线避碰和行人避碰交通系统也将成为可能，这些系统必须绝对是故障安全的，就像数百万辆自动驾驶汽车中使用的那样。PI Simpson将把高阶精确方案整合到她的“翻转”计算电动力学课程中，并将在YouTube上发布相应的视频讲座（任何人都可以免费访问）。同样，Balsara将在他的网站上发布新的章节、视频讲座和示例代码。一个简化版本的代码也将被开发出来，以帮助理工科本科生和高中生获得解决电磁问题的随时间变化的麦克斯韦方程组的实践经验。该项目的目标是开发计算电动力学的高阶算法，包括工程计算电动力学中必不可少的所有通用功能。Co-PI Balsara最近发表了麦克斯韦方程组高阶解的数学蓝图，称为多项式- n /多项式- m （PNPM）格式。与目前的麦克斯韦方程组数值求解技术相比，高阶PNPM格式具有几个关键优势，即：(1)可以提供麦克斯韦方程组的基本精确解；(2)每个波长只需要四个或五个网格单元；(3)全局保持散度约束（即全局满足高斯定律）；(4)可适应任意几何形状和非网格一致性材料界面；(5)保持不随精确度增加而减小的最大时间步长限制；(6)在超级计算机上是高度并行化的，因为处理器之间只需要共享一个数据平面。为了给研究界提供一个有效的模拟框架来解决广泛的应用问题，PNPM方法将被赋予处理完美匹配的层边界条件、色散介质和全场散射场平面波源条件的无缝策略。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In 1966, Kane Yee developed a space-time computational algorithm to solve Maxwell's equations, which are used to study electromagnetic wave propagation. His approach developed into what is now known as the finite-difference time-domain (FDTD) method. Through the ensuing decades, advances have been made to FDTD enabling it to be applied to a wide range of problems across the electromagnetic spectrum, literally from low frequencies (sub-1 Hz) all the way up to visible light. Currently, FDTD is an indispensable tool for modeling very large and very complex electromagnetic wave interaction problems, especially those problems requiring the incorporation of multiphysics. However, FDTD is showing its age. Its basic second-order algorithmic accuracy and difficulty in modeling smooth, non-grid-conforming material interfaces, have become serious limitations. Co-PI Balsara recently published a mathematical blueprint for entire classes of higher-order accurate solutions to Maxwell's equations. These schemes will overcome the limitations of current modeling approaches while also retaining their advantages. Since these high-order accurate schemes yield for all intents and purposes an exact numerical solution of Maxwell's equations, it will be possible to design more stealthy aerospace and naval platforms than at present. Similarly, it will be possible to design complex wireless collision-avoidance and pedestrian-avoidance transportation systems that must absolutely be fail-safe, such as those to be used in millions of self-driving cars. PI Simpson will incorporate the higher-order accurate schemes into her "flipped" course on computational electrodynamics and will post the corresponding video lectures on YouTube (freely accessible to anyone). Likewise, Co-PI Balsara will post new chapters, video lectures, and sample codes on his website. A simplified version of the codes will also be developed to help science and engineering undergraduates and high school students to get hands-on experience with the time-dependent Maxwell's equations for solving electromagnetic problems. The goal of this project is to develop higher-order algorithms for computational electrodynamics that include all the versatile features that are essential in engineering computational electrodynamics. Co-PI Balsara recently published a mathematical blueprint for higher-order solutions to Maxwell's equations, called polynomial-of-degree-N/polynomial-of-degree-M (PNPM) schemes. High-order PNPM schemes have several critical advantages relative to current numerical solution techniques for Maxwell's equations, Namely, high-order PNPM schemes: (1) can provide essentially exact solutions for Maxwell's equations; (2) require only four or five grid cells per wavelength; (3) preserve the divergence constraints globally (meaning Gauss' Laws are satisfied globally); (4) can be adapted to arbitrary geometries and non-grid-conforming material interfaces; (5) maintain a maximum time-step limit that does not diminish with increasing accuracy; (6) are highly parallelizable on supercomputers since only a single plane of data need to be shared between processors. To provide an effective simulation framework to the research community for solving a wide range of applications, the PNPM methods will be endowed with a seamless strategy for treating perfectly matched layer boundary conditions, dispersive media, and total-field scattered-field plane wave source conditions.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.