Fast Spectral-Galerkin Methods and their Applications

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大的作用，并正在迅速取代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。这个项目的重点是设计求解一大类偏微分方程组的自动、快速和稳健的谱方法，并将它们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组提供快速、准确的数值算法，并在声学和电磁散射、复杂流体和材料科学中得到应用。特别是，提出的数值算法将使我们能够直接求解一些目前无法用现有数值算法处理的重要的高维方程。预计所提出的数值模拟将使我们能够以合理的CPU和周转时间来处理具有严格精度和/或内存要求的挑战性问题，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性和流体力学性质提供有价值的信息。这个项目的另一个重要目标是让研究生和本科生学习必要的计算和应用数学技能，以便他们能够在科学和工程领域取得成功。

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.