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

1966年，Kane Yee开发了一种时空计算算法来求解用于研究电磁波传播的麦克斯韦方程组。 他的方法发展成为现在所知的时域有限差分法（FDTD）。 在随后的几十年中，FDTD取得了进步，使其能够应用于电磁频谱中的各种问题，从低频（低于1 Hz）一直到可见光。 目前，时域有限差分法（FDTD）是模拟非常大和非常复杂的电磁波相互作用问题，特别是那些需要纳入多物理场的问题不可缺少的工具。 然而，FDTD显示出它的年龄。 其基本的二阶算法的精度和建模的困难，光滑，非网格一致的材料界面，已成为严重的限制。合作PI巴尔萨拉最近发表了一个数学蓝图，为整个类的高阶精确解麦克斯韦方程组。 这些方案将克服当前建模方法的局限性，同时也保留其优点。 由于这些高阶精确的格式可以得到麦克斯韦方程组的精确数值解，因此有可能设计出比目前更隐身的航空航天和海军平台。 同样，设计复杂的无线防撞和防撞交通系统也是可能的，这些系统必须绝对是故障安全的，比如那些将用于数百万辆自动驾驶汽车的系统。PI Simpson将把高阶精确方案纳入她关于计算电动力学的“翻转”课程，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，Co-PI Balsara将在他的网站上发布新的章节，视频讲座和示例代码。还将开发一个简化版本的代码，以帮助科学和工程本科生和高中生获得与时间有关的麦克斯韦方程组解决电磁问题的实践经验。 这个项目的目标是为计算电动力学开发高阶算法，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最近发表了一个数学蓝图，用于麦克斯韦方程的高阶解，称为N次多项式/M次多项式（PNPM）方案。高阶PNPM格式相对于现有的麦克斯韦方程数值解技术具有几个关键的优点，即：（1）可以提供麦克斯韦方程的基本精确解;（2）每个波长只需要四到五个网格单元;（3）全局保持发散约束（这意味着高斯定律是全局满足的）;（4）可以适用于任意几何形状和非网格一致的材料界面;（5）保持最大时间步长限制，该限制不随着精度的增加而减小;（6）在超级计算机上是高度可并行的，因为在处理器之间仅需要共享单个数据平面。 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，PNPM方法将被赋予一个无缝的策略来处理完全匹配的层边界条件，色散介质，和全场散射该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的评估被认为是值得支持的。影响审查标准。

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.