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，并迅速取代了制造业和科学探索中许多昂贵的原型和测试阶段，快速而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快速和鲁棒的谱伽辽金方法来求解一类大的偏微分方程，并应用它们来研究当前感兴趣的几个重要问题。提出的研究将导致一类偏微分方程的快速和准确的数值算法，应用于声学和电磁散射，流体动力学和材料科学。预计所提出的数值算法将允许在合理的周转时间内模拟三维时间相关问题，并提供数字攻击新兴工程和科学应用中出现的具有挑战性的问题的能力。特别是，提出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。拟议的研究也将为本科生的研究项目创造机会，如软件开发、并行计算和数值模拟。

As computer simulations are playing an ever increasing role in manybranches of science and engineering and are rapidly replacing much ofthe expensive prototyping and testing phases in manufacturing and inscience explorations, fast and robust numerical methods are becomingan indispensable tool for many scientists and engineers. The focus ofthis project is to design fast and robust spectral-Galerkin methodsfor solving a large class of partial differential equations, and applythem to investigate several important problems of current interest.The proposed research will result in fast and accurate numericalalgorithms for a class of partial differential equations withapplications in acoustic and electromagnetic scattering, fluiddynamics and materials science. It is expected that the proposed numerical algorithms will allowsimulations of three-dimensional time-dependent problems within areasonable turn-over time, and provide capabilities to numericallyattack challenging problems arising from emerging engineering andscientific applications. In particular, the proposed numericalsimulations will contribute towards better understandings of thecomplex physical and mathematical problems, and provide valuableinformation for the design of advanced materials and on therheological and hydrodynamic properties of complex fluids. Theproposed research will also generate opportunities for undergraduateresearch projects such as software development, parallel computationand numerical simulation.