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

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

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.