Numerical Solution of Differential Equations in Mechanics

研究者将设计、改进和分析用偏微分方程模拟的复杂物理现象的数值模拟方法。将研究三个主要的应用领域，每个领域都需要特殊的方法：计算相对论，由不可压缩流体流动和电磁学引起的某些重要类别的微分方程，以及线弹性板。在计算相对论领域，这项工作的动机是利用目前正在建设中的LIGO探测器寻找引力辐射。LIGO是新一代引力波探测器的典型代表，它被设计用来探测引力波（迄今为止，由于其振幅小，引力波尚未被探测到）。如果能对探测到的波进行分析，以确定产生它的巨大而遥远的天文事件的性质（例如，一对黑洞的螺旋合并），那么LIGO将成为人类探索宇宙的第一个窗口，可以“看到”电磁频谱（光和无线电信号）之外的东西。提议的项目是设计有效的算法，使这种分析能够在高性能计算机上执行。这些算法是基于爱因斯坦广义相对论的基本方程。研究者将扩展和应用已经完成的约束方程的计算机代码，由爱因斯坦方程产生的纯空间偏微分方程，并将研究爱因斯坦进化方程的数值求解问题。这是一项高度跨学科的工作，涉及研究者、数学家和他的团队、物理学家和天文学家之间的广泛合作。第二个研究领域涉及多网格算法的发展和支持理论，以解决迄今为止一直抵制这种方法的某些类型的微分方程。多重网格算法，它将期望的解分解成许多不同尺度的分量，是解决多种微分方程的最有效算法之一。然而，对于一类由不可压缩连续体引起的重要问题和另一类由电磁学麦克斯韦方程组引起的问题，则需要新的多重网格方法。最后，研究者将合作撰写一本关于弹性板理论的专著，包括建模、分析和数值模拟。在过去的十年中，这些领域已经取得了广泛的进展，本书将作为该领域和弹性壳建模活跃领域的工作人员的参考，并作为该领域新研究人员的介绍。

9870399 ArnoldThe investigator will devise, improve, and analyze methods for the numerical simulation of complex physical phenomena modeled by partial differential equations. Three major areas of application will be studied, each requiring special methods: computational relativity, certain important classes of differential equations arising from incompressible fluid flow and from electromagnetism, and linearly elastic plates.In the area of computational relativity, the work is motivated by the search for gravitational radiation using the LIGO detector, presently under construction. LIGO is the prime example of a new generation of gravitational wave detectors, and is being designed to detect gravity waves (which have hitherto escaped detection due to their small amplitude). If the detected wave can be analyzed to determine the nature of the massive but distant astronomical event which gave birth to it (e.g., the spiraling coalescence of a pair of black holes), then LIGO will become mankind's first window on the universe that can "see" beyond the electromagnetic spectrum (light and radio signals). The proposed project is to devise efficient algorithms to enable such analyses to be performed on high performance computers. These algorithms are based on Einstein's equations underlying general relativity. The investigator will both extend and apply already completed computer codes for the constraint equations, purely spatial partial differential equations arising from the Einstein equations, and will also study the problem of numerically solving the Einstein evolution equations. This is highly interdisciplinary work, involving extensive collaboration between the investigator, a mathematician, and his group, and physicists and astronomers.The second area of study concerns the developments of multigrid algorithms and supporting theory for solving certain sorts of differential equations which have hitherto resisted such approaches. Multigrid algorithms, which exploit a decomposition of the desired solution into components on many different scales, are among the most efficient algorithms invented for solving many sorts of differential equations. However for an important class of problems arising from incompressible continua and another group of problems arising from Maxwell's equations of electromagnetism, new multigrid approaches are required.Finally the investigator will collaborate on a monograph on the theory of elastic plates, including modeling, analysis, and numerical simulation. There has been extensive progress in these areas over the past decade, and the book will serve as both a reference for workers in the field and in the active field of elastic shell modeling, and as an introduction for researchers new to the area.