Fast Spectral Methods and their Applications

本项目的重点是设计精确、快速和鲁棒的谱方法来求解一大类偏微分方程，并将其应用于研究当前感兴趣的几个重要问题。更准确地说，它是建议发展（i）自适应稀疏谱方法解决一类高维问题，（ii）谱精确的计划，在空间-时间的一类抛物型偏微分方程，（iii）强大的和准确的谱方法的电磁散射，特别是分层介质和更高的波数。以及（iv）应用所提出的快速准确的光谱方法来研究流体动力学和材料科学中的新兴问题，例如先进材料加工中的液滴形成&、细丝断裂和铁电&多铁性纳米结构。预计所提出的数值模拟将使我们能够处理具有严格的精度和/或存储器要求的挑战性问题，具有合理的CPU成本和周转时间，有助于更好地理解复杂的物理和数学问题，并为先进材料的设计和复杂流体的流变学和流体动力学特性提供有价值的信息。该项目的另一个重要目标是让研究生学习计算和应用数学的必要技能，以便他们能够在科学和工程领域取得成功。

The focus of this project is to design accurate, fast and robust spectral methods for solving a large class of partial differential equations, and apply them to investigate several important problems of current interest. More precisely, it is proposed to develop (i) adaptive sparse spectral methods for solving a class of higher-dimensional problems; (ii) spectrally-accurate schemes in space-time for a class of parabolic type PDEs; (iii) robust and accurate spectral methods for electromagnetic scattering, particularly for layered medium and with higher wave numbers; and (iv) to apply the proposed fast and accurate spectral methods to investigate emerging problems in fluid dynamics and materials science such as drop formation & filament breakup and ferroelectric & multiferroic nanostructures in advanced materials processing.It is expected that the proposed numerical simulations will enable us to handle challenging problems having stringent accuracy and/or memory requirements with a reasonable cost in CPU and turn-around time, contribute towards better understandings of the complex physical and mathematical problems, and provide valuable information for the design of advanced materials and on the rheological and hydrodynamic properties of complex fluids. Another important goal of this project is to engage graduate students in learning necessary skills of computational and applied mathematics so that they can pursue a successful career in sciences and engineering.