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，三体问题的数字8解，以及从Navier-Stokes边界层获得的信号。将使用解析和数值技术。研究洛伦兹系统的目的是理解整个复平面的洛伦兹解的解析延拓。关于数字8的解决方案，该项目将展示这个系统的复杂奇点有一个吸引人的简单结构。纳维-斯托克斯边界层主要是用光谱分析技术来研究的。我们将定位和阐明信号在复平面上的解析延拓的奇异性，并获得外部流施加的时间尺度的测量和理解。本项目致力于从两个方面造福公众。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具有巨大的重要性。作为其重要性的一个例子，我们提到汽车的大部分能量消耗在边界层中。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。这本书将向更广泛的读者介绍一种深受计算机体系结构最新进展影响的科学计算风格。