Fast spectral methods and their applications

快速光谱方法及其应用

• 批准号: 1620262
• 负责人: Jie Shen
• 金额: $ 18万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620262&HistoricalAwards=false
• 关键词: Fast; spectral; methods; their; applications

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. Solutions for many problems of interest exhibit local singular behaviors which contaminate the accuracy of usual spectral/spectral-element methods. Many complex systems exhibiting anomalous diffusion can be better modeled with fractional partial differential equations, which are numerically challenging due to their nonlocal nature. Spectral/spectral-element methods usually lead to dense or block dense and ill-conditioned matrices that are difficult and expensive to solve. The focus of this project is to design, analyze, and implement fast and robust spectral methods for a class of numerically challenging problems. The 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, and will contribute to a better understanding of some fundamental issues in materials science and fluid dynamics through fast and accurate numerical simulations. Another important goal of this project is to engage graduate students in learning necessary skills of computational and applied mathematics so that they can pursue a successful career in sciences and engineering.The PI will address these issues with the following tasks: (i) develop effective Muntz Galerkin method to deal with problems with singular solutions; (ii) develop efficient and accurate spectral methods for solving a class of fractional differential equations by constructing special basis functions with generalized Jacobi and Laguerre functions, and derive corresponding error estimates; (iii) develop direct structured solvers with optimal computational complexity for dense or block dense linear systems arising from spectral/spectral-element discretization; and (iv) develop efficient spectral algorithms for solving the phase-field model of electro-magnetic couplings in ferroelectric and multiferroic nanostructures. The research will result in fast and stable direct spectral/spectral-element solvers for a class of partial differential equations as well as fast Jacobi/spherical harmonic transforms. This project will also result in a set of computational modules to efficiently and accurately solve the coupled nonlinear system for the phase-field model of electro-magnetic couplings in ferroelectric and multiferroic nanostructures.

光谱方法是用于应用数学和科学计算的一类技术，用于求解某些微分方程。有关许多感兴趣问题的解决方案表现出局部奇异行为，这些行为污染了通常的光谱/光谱元素方法的准确性。许多表现出异常扩散的复杂系统可以通过分数偏微分方程进行更好的建模，由于其非局部性质，它们在数值上具有挑战性。光谱/光谱元素方法通常会导致难以解决的量很难且昂贵。该项目的重点是设计，分析和实施一类数字具有挑战性的问题的快速且强大的光谱方法。数值模拟将使我们能够处理具有严格准确性和/或内存要求的具有挑战性的问题，并在CPU和周转时间合理成本，并通过快速，准确的数值模拟来更好地理解材料科学和流体动力学中的某些基本问题。该项目的另一个重要目标是吸引研究生学习计算和应用数学的必要技能，以便他们可以从事科学和工程领域的成功职业。PI将通过以下任务解决这些问题：（i）开发有效的Muntz Galerkin方法来处理具有单数解决方案的问题； （ii）开发有效而准确的光谱方法来通过使用广义的jacobi和laguerre函数构建特殊基础函数来求解一类分数微分方程，并得出相应的误差估计； （iii）开发具有最佳计算复杂性的直接结构化求解器，以用于光谱/光谱元素离散化引起的致密或块致密的线性系统； （iv）开发有效的光谱算法，用于求解铁电和多性纳米结构中电磁耦合的相位模型。这项研究将导致一类偏微分方程以及快速的雅各比/球形谐波变换的快速和稳定的直接光谱/光谱元素求解器。该项目还将导致一组计算模块有效，准确地求解在铁电和多种纳米结构中电磁耦合的相位模型的耦合非线性系统。