Complex Singularities in Numerical Analysis and Nonlinear Dynamics

数值分析和非线性动力学中的复杂奇点

• 批准号: 1115277
• 负责人: Divakar Viswanath
• 金额: $ 26.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115277&HistoricalAwards=false
• 关键词: Complex; Singularities; Numerical; Analysis; Nonlinear

This project will investigate the precise analytic form of the complex singularities ofthe Lorenz system, the figure-eight solution of the three-body problem, and signals obtainedfrom the Navier-Stokes boundary layer. Both analytic and numerical techniqueswill be used. Research on the Lorenz system has the aim of understanding the analyticcontinuation of the Lorenz solutions to the entire complex plane. With regard to thefigure-eight solution, the project will show that the complex singularities of this systemhave an appealingly simple structure. The Navier-Stokes boundary layer has beenstudied primarily using techniques from spectral analysis. We will locate and elucidatesingularities of the analytic continuation of the signal in the complex plane, and andobtain a measurement as well as an understanding of the time scales imposed by theouter flow.This project endeavors to benefit the public in two respects. Firstly, we aim to obtainnew insights into the Navier-Stokes boundary layer which is of immense importance inengineering and meteorology. As an example of its importance, we mention that amajor part of the energy intake of automobiles is dissipated in the boundary layer.Secondly, we will write a new book that puts computer architecture at the heart ofscientific computing. This book will introduce a style of scientific computing that isdeeply informed by recent progress in computer architecture to a wider audience.

该项目将研究洛伦兹系统复杂奇点的精确解析形式、三体问题的八字形解以及从纳维-斯托克斯边界层获得的信号。将使用分析和数值技术。研究洛伦兹系统的目的是了解洛伦兹解对整个复平面的解析延拓。对于八字形解决方案，该项目将表明该系统的复杂奇点具有非常简单的结构。主要使用光谱分析技术来研究纳维-斯托克斯边界层。我们将定位并阐明复平面中信号的解析延拓的奇点，并获得对外部流所施加的时间尺度的测量和理解。该项目致力于在两个方面使公众受益。首先，我们的目标是获得对纳维-斯托克斯边界层的新见解，这在工程和气象学中非常重要。作为其重要性的一个例子，我们提到汽车吸收的大部分能量都在边界层中消散。其次，我们将写一本新书，将计算机体系结构置于科学计算的核心。本书将向更广泛的读者介绍一种科学计算风格，这种风格深受计算机体系结构最新进展的影响。