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重要的作用，并迅速取代制造和科学探索中大部分昂贵的原型和测试阶段，快速而强大的数值方法正在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求解一大类偏微分方程的快速、鲁棒的谱Galerkin方法，并将其应用于当前几个重要问题的研究，为一类偏微分方程的快速、精确的数值算法提供理论基础，在声散射、电磁散射、流体动力学和材料科学等领域具有重要的应用价值.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，并提供能力numericallyattack新兴的工程和科学应用所产生的挑战性问题。特别是，所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和复杂流体的流变和流体动力学性质提供有价值的信息。拟议的研究也将为本科生的研究项目，如软件开发，并行计算和数值模拟的机会。

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.