Mathematical Sciences: Approximation of the Global Attractors of Evolution Equations

数学科学：进化方程全局吸引子的近似

• 批准号: 9404340
• 负责人: Michael Jolly
• 金额: $ 14.06万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404340&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Approximation; Global; Attractors

Jolly The investigator and his colleague test approximate inertial manifolds (AIMs) on nonlinear dissipative evolution equations, including the Kuramoto-Sivashinsky, Navier-Stokes, and Lorenz equations. The emphasis is on capturing elements of the global attractor, and not on solving the initial value problem over short time intervals. The phase space dimension reduction offered by AIMs is exploited to visualize global bifurcations involving two-dimensional stable manifolds, which would be of higher dimension in traditional Galerkin discretizations. Of particular interest is an algebraic AIM, which offers arbitrary accuracy, at a fixed dimension. Increased accuracy in this case is achieved by increasing the degree of a polynomial that implicitly defines the AIM. The investigators explore various time discretizations that take advantage of the particular features of the associated differential-algebraic system. This research has an impact on how the long term behavior of physical systems is determined by computers. Part of the work is devoted to comparing the efficiency of alternative algorithms to standard ones used in high performance computing. Another is to visualize certain critical phenomena, which could not be seen with traditional methods, regardless of the computational effort. While the various methods will be tested on particular mathematical models of combustion, fluid flow, and the weather, they are applicable to a wide range of problems that fit into a general framework.

研究者和他的同事在非线性耗散演化方程上测试近似惯性流形（AIMs），包括Kuramoto-Sivashinsky、Navier-Stokes和Lorenz方程。重点是捕获全局吸引子的元素，而不是在短时间间隔内解决初始值问题。利用AIMs提供的相空间降维特性，可以可视化二维稳定流形的全局分岔，而传统的伽辽金离散方法需要更高的维数。特别有趣的是代数AIM，它在固定维度上提供任意精度。在这种情况下，通过增加隐式定义AIM的多项式的程度来提高准确性。研究人员探索了利用相关微分代数系统的特殊特征的各种时间离散化。这项研究对计算机如何决定物理系统的长期行为产生了影响。部分工作致力于将替代算法与高性能计算中使用的标准算法的效率进行比较。另一个是可视化某些关键现象，这是传统方法无法看到的，无论计算努力如何。虽然各种方法将在燃烧、流体流动和天气的特定数学模型上进行测试，但它们适用于适合一般框架的广泛问题。