Fast Spectral-Galerkin Methods and their Applications

快速谱伽辽金方法及其应用

• 批准号: 0915066
• 负责人: Jie Shen
• 金额: $ 32.91万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0915066&HistoricalAwards=false
• 关键词: Fast; Spectral; Galerkin; Methods; their

As 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 phases in manufacturing and in science explorations, fast and robust numerical methods are becoming an indispensable tool for many scientists and engineers. The focus of this project is to design auurate, 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. The proposed research will result in fast and accurate numerical algorithms for a class of partial differential equations with applications in acoustic and electromagnetic scattering, complex fluids and materials science. In particular, the proposed numerical algorithms will make it possible to directly solve some important high-dimensional equations which are currently not treatable with existing numerical algorithms.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 and undergraduate students in learning necessary skills of computational and applied mathematics so that they can pursue a successful career in sciences and engineering.

随着计算机模拟在科学和工程的许多分支中发挥着越来越重要的作用，并迅速取代了制造和科学探索中许多昂贵的原型和测试阶段，快速和强大的数值方法正成为许多科学家和工程师不可或缺的工具。 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。该研究将为一类偏微分方程提供快速准确的数值算法，并将在声和电磁散射、复杂流体和材料科学中应用。 特别是，所提出的数值算法将有可能直接解决一些重要的高维方程，目前无法处理与现有的数值算法。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/或内存要求的挑战性问题，在CPU和周转时间的合理成本，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以及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和本科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。