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

这一努力涉及到数学工具的开发，这些工具可用于对物理现象进行准确预测，并对电气工程等重大社会利益领域产生影响（optics，electronics，photonics）通信（天线）、大气科学、医学（断层摄影术、成像、超声波诊断和治疗、靶向给药）、军用和民用遥感（雷达、声纳、隐形）、可再生能源生产和能源政策（风力发电场的多普勒特征）等。为了清晰起见，我们可以将新方法简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，从而使以前不可行的模拟成为可能。虽然这种“平滑曲线表示”（或者，在数学术语中，高阶/谱方法）已经可用了很多年，所提出的工作的新奇在于，它使得能够以大大降低的计算成本利用这种高度精确的方法，并且用于高度复杂的工程问题--例如上面提到的那些--包括复杂的电子部件、整车等。本计画系关于发展与分析高效能、高精确度的数值演算法，以解偏微分方程式，并应用于材料科学与工程的广泛问题。所提出的算法强调的准确性，效率以及普遍适用性的基础上的频谱和高阶方法，相关的理论讨论，反过来，寻求提供必要的背景和性能保证。这一努力考虑了两个广泛的应用领域，即I.频域，时间谐波声学和电磁学，和II。时域偏微分方程是一般三维域中的偏微分方程，适用于声学、电磁学、弹性力学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.