Fast Spectral-Galerkin Methods and Their Applicatons

PI将发展和分析快速和准确的谱方法来求解一大类偏微分方程组，并应用它们来研究当前感兴趣的几个重要问题，特别是多相复杂流动的界面动力学，分层剪切流动的稳定性和动力学，以及先进材料加工中的微观结构演变。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，并迅速取代了制造和科学探索中昂贵的原型和测试阶段，快速和健壮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，特别是对于流体力学和材料科学的计算科学家来说，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为先进材料的设计和分层剪切流的空间实验提供了有价值的信息。

The PI will develop and analyze fast and accurate spectral methodsfor solving a large class of partial differential equations and applythem to investigate several important problems of current interest,including in particular the inter-facial dynamics of multiphase complexflows, stability and dynamics of stratified shear flows andmicro-structural evolutions in advanced materials processing. Theproposed research will result in fast and accurate numericalalgorithms which will allow simulations of three-dimensionaltime-dependent problems within a reasonable turn-over time, andprovide capabilities to numerically attack challenging problemsarising from emerging engineering and scientific applications.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, especiallyfor computational scientists in fluid dynamics and materials science.It is expected that our numerical simulations will contribute towardsbetter understandings of the complex physical and mathematicalproblems, and provide valuable information for the designs of advancedmaterials and of a space experiment on stratified shear flows.