PDE solvers: Frequency-domain, time-domain and hybrids---with applications to materials science and engineering

这项工作涉及开发可用于对物理现象作出准确预测的数学工具，这些工具对下列重大社会领域有影响：电气工程(光学、电子学、光子学)、通信(天线)、大气科学、医学(断层扫描、成像、诊断和治疗超声、靶向药物输送)、军事和民用遥感(雷达、声纳、隐形)、可再生能源生产和能源政策(风电场的多普勒特征)等。为了清晰起见，新方法过于简化，可以被可视化为使用某种计算版本的法国曲线工具，而不是直尺，这样可以更准确地表示物理现实，并大大降低计算成本--以至于以前不可行的模拟成为可能。虽然这种“光滑曲线表示法”(或者，在数学术语中，高阶/谱方法)已经存在多年，但拟议工作的新奇之处在于，它能够在极大地降低计算成本的情况下利用这种高精度方法，并用于高度复杂的工程问题--如上面提到的那些--包括复杂的电子元件、整车等。从技术上讲，该项目致力于开发和分析求解偏微分方程(PDE)的高性能、高精度的数值算法，并应用于材料科学和工程中的广泛问题。所提出的算法强调在频谱和高阶方法的基础上的准确性、效率以及普遍适用性；相关的理论讨论反过来又试图提供必要的背景和性能保证。这项工作考虑了两个广泛的应用领域，即i.频域、时间谐声学和电磁学，以及ii.一般三维域中的时间域偏微分方程组，适用于声学、电磁学、弹性以及可压缩和不可压缩流体动力学。

This effort concerns development of mathematical tools which can be used to make accurate predictions about physical phenomena, with impact on areas of significant societal interest such as electrical engineering (optics, electronics, photonics) communications (antennas), atmospheric science, medicine (tomography, imaging, diagnostic and therapeutic ultrasound, targeted drug delivery), military and civilian remote sensing (radar, sonar, stealth), renewable energy production and energy policy (Doppler signature of wind farms) etc. The methodologies to be pursued represent a change in paradigm in the mathematical approach. Oversimplifying for the sake of clarity, the new methods can be visualized as using some sort of a computational version of the french curve tool rather than a straight ruler in such a way that much more accurate representations of physical reality as well as greatly reduced computing costs result -- to the point that previously unfeasible simulations become possible. While such "smooth-curve representations" (or, in mathematical nomenclature, high-order/spectral methods) have been available for many years, the novelty of the proposed work is that it enables utilization of such highly accurate methodologies at vastly reduced computing costs and for highly complex engineering problems--such as those mentioned above--including complex electronic components, full vehicles, etc.Technically, this project concerns development and analysis of high-performance, highly accurate numerical algorithms for solution of Partial Differential Equations (PDE), with application to a wide range of problems in materials science and engineering. The proposed algorithms emphasize accuracy, efficiency as well as generally applicability on the basis of spectral and high-order methodologies; the associated theoretical discussions, in turn, seek to provide necessary background and performance guarantees. This effort considers two broad application areas, namely, I. Frequency-domain, time-harmonic acoustics and electromagnetism, and II. Time-domain PDE in general three-dimensional domains, with applicability to acoustics, electromagnetism and elasticity as well as compressible and incompressible fluid-dynamics.