Mathematical Sciences: Approximation of the Global Attractors of Evolution Equations

Jolly 研究人员和他的同事测试非线性耗散演化方程的近似惯性流形（AIM），包括Kuramoto-Sivashinsky，Navier-Stokes和Lorenz方程。 重点是捕捉全局吸引子的元素，而不是在短时间间隔内解决初值问题。 AIMs提供的相空间降维被用来可视化涉及二维稳定流形的全局分叉，这在传统的Galerkin离散中是高维的。 特别感兴趣的是一个代数AIM，它提供了任意的精度，在一个固定的尺寸。 在这种情况下，通过增加隐式定义AIM的多项式的次数来提高精度。 研究人员探索各种时间离散化，利用相关的微分代数系统的特定功能。 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。 部分工作致力于比较替代算法的效率，用于高性能计算的标准。 另一个是将某些关键现象可视化，无论计算量如何，这些现象都无法用传统方法看到。 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。

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.