Fast Spectral Methods and their Applications

PI将开发和分析液晶流动和无界区域中的偏微分方程组的快速和准确的数值格式，并产生一个公开可用的高性能快速谱椭圆解算器软件包，这将是计算科学家和工程师的宝贵工具。PI还将实施建议的数值格式来研究当前感兴趣的几个重要问题，特别是液晶流动的耦合和缺陷运动，以及旋转流动中的动态控制和参数共振。所提出的方案的准确性和效率将使我们能够在合理的周转时间内模拟这些与时间相关的三维流动。计算机模拟在许多科学和工程分支中发挥着越来越大的作用，并正在迅速取代在任何产品推向市场之前所需的昂贵的原型和测试阶段。因此，快速可靠的数值方法/软件正在成为许多科学家和工程师，特别是流体力学和材料科学中的计算科学家不可或缺的工具。预计我们的数值模拟将有助于更好地理解复杂的物理和数学问题，并为先进材料和涡轮机械的设计提供有价值的信息。

The PI will develop and analyze fast and accurate numerical schemes for liquid crystal flows and for PDEs in unbounded domains, and produce a publicly available high performance software package of fast spectral elliptic solvers which will be a valuable tool for computational scientists and engineers. The PI will also implement the proposed numerical schemes to investigate several important problems of current interest, including in particular the coupling and defect motion of liquid crystal flows, and the dynamic control and parametric resonance in rotating flows. The accuracy and efficiency of the proposed schemes will allow us to simulate these three-dimensional time-dependent flows with a reasonable turn-over time.Computer simulations are playing an ever increasing role in many branches of science and engineering and are rapidly replacing much of the expensive prototyping and testing phase required before any product is brought to market. Consequently, fast and reliable numerical methods/software are becoming an indispensable tool for many scientists and engineers, especially for computational scientists in fluid dynamics and materials science. It is expected that our numerical simulations will contribute towards better understandings of the complex physical and mathematical problems, and provide valuable information for the design of advanced materials and of turbo-machinery.