A Solve-Then-Discretize Paradigm for Spectral Methods

谱方法的求解然后离散范式

基本信息

批准号：
1818757
负责人：
Alex Townsend
金额：
$ 30万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818757&HistoricalAwards=false
关键词：
Solve Discretize Paradigm Spectral Methods

项目摘要

Spectral methods are one of the big three technologies (along with finite differences and finite element methods) for the numerical solution of partial differential equations (PDEs) and are particularly powerful for fluid flow and airfoil simulations. This research project aims to develop a new infinite-dimensional framework for solving PDEs to derive competitive computational algorithms that preserve the continuum structure of differential operators, promising to overcome many of the hard-and-fast computational barriers with spectral discretizations. We aim to produce a collection of adaptive, robust, and industrial-strength iterative solvers for spectral methods to allow for the accurate resolution of fluid flows. We will also develop tools for computing the pseudospectra and continuous spectra of differential operators, facilitating improved understanding of inelastic scattering. The results will help to demonstrate that spectrally-accurate methods, when done carefully, are flexible, general, and powerful numerical tools in computational mathematics and engineering.The standard paradigm for solving a PDE is to first discretize the equation and then solve the resulting linear system. This approach has a number of drawbacks for spectral methods related to the design of preconditioners, the introduction of non-normality, and the perturbation of spectra. The infinite-dimensional framework under development in this project preserves the continuum structure of PDEs by avoiding the discretization of differential operators, and instead only discretizes smooth functions, such as the solution and the source terms of the PDE. Not working with finite sections of differential operators promises to enable us to develop robust Krylov-based iterative solvers, motivate preconditioners directly from the differential operator, compute the continuous part of the spectrum of operators, and develop a theoretical foundation for the adaptive resolution of solutions and eigenfunctions based on error analysis. We will apply these new tools to the numerical simulation of advection-dominated fluid flow as well as inelastic scattering.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

光谱方法是针对部分微分方程（PDE）的数值解（PDE）的三大技术（以及有限差异和有限元方法）之一，对于流体流量和机翼模拟而言特别强大。该研究项目旨在开发一个新的无限尺寸框架，以解决PDE，以得出竞争性计算算法，以保留差异操作员的连续结构，并有望克服许多具有光谱离散的硬及速度计算障碍。我们旨在生产一系列自适应，鲁棒和工业强度迭代的迭代求解器，用于光谱方法，以准确地分辨出流体流量。我们还将开发用于计算差异操作员的伪镜和连续光谱的工具，从而促进对非弹性散射的理解。结果将有助于证明，当仔细完成时，频谱准确的方法是柔性，一般且功能强大的数值工具，用于计算数学和工程。解决PDE的标准范例是首先离散方程，然后求解所得的线性系统。这种方法对于与预处理的设计，非正常性的引入以及光谱的扰动有关的光谱方法具有许多缺点。该项目正在开发的无限维框架通过避免差异操作员的离散化来保留PDE的连续结构，而仅将平滑功能离散，例如解决方案和PDE的源项。不与差异操作员的有限部分合作承诺使我们能够开发出强大的基于Krylov的迭代求解器，而是直接从差异操作员那里激励预先调查器，而是计算操作员频谱的连续部分，并为基于错误分析的解决方案和特征函数的适应性分辨率而建立理论基础。我们将把这些新工具应用于对流主导的流体流以及非弹性散射的数值模拟中。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，认为值得通过评估来获得支持。