Numerical Solution of Partial Differential Equations and Applications

偏微分方程数值解及其应用

• 批准号: 0107233
• 负责人: Douglas Arnold
• 金额: $ 18万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2001-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107233&HistoricalAwards=false
• 关键词: Numerical; Solution; Partial; Differential; Equations

The investigator will devise, improve, and analyze methods for the numerical simulation of complex physical phenomena modeled by partial differential equations, emphasizing three main areas: mixed methods for elasticity equations, discontinuous Galerkin methods, and computational general relativity. For elasticity, the investigator will build on a recent breakthrough that enabled the construction of the first stable mixed finite element methods for the displacement-stress formulations with polynomial trial functions, and also work towards the development of simpler nonconforming mixed finite element methods and extensions to three dimensional elasticity problems. Concerning discontinuous Galerkin methods--finite element methods in which the approximating piecewise polynomial functions are discontinuous, with modifications incorporated into the variational formulation to achieve consistency--the investigator will work with his collaborators to build on recent work on the unification and classification of such methods to develop a unified approach to the analysis of and discrimination among a wide class of discontinuous Galerkin methods for elliptic equations. Application will also be made to the numerical simulation of elastic plates incorporating shear. The third and largest effort concerns the numerical solution of Einstein's field equations relating mass and the curvature of space-time. The emphasis here will be on understanding the fundamental properties of the Einstein equations most relevant to their numerical solution and the basic difficulties that have beset previous attempts at numerical simulations of them. The work will be guided by the goal of simulating the coalescence of inspiralling pairs of black holes and the resulting emission of gravitational radiation, which is a problem of fundamental importance to gravitational physics and also because such simulations will be essential to realization of a new generation of observatories based on gravitational wave detectors.Computer simulation is a key tool for the design and testing of complex engineering structures. In recent decades computer simulation has also joined experiment and theory as one of the main paradigms of scientific investigation. In both areas, many of the most complex systems are first modeled by systems of partial differential equations--in which the language of calculus is used to express the variations of the relevant physical quantities in space and time--and then these systems of differential equations must be approximated by numerical algorithms, which harness the power of modern computers to perform billions of arithmetical operations a seconds to extract the solutions to the equations to the required degree of accuracy. In recent decades the principles for the design and validation of such algorithms have been developed for many of the basic systems of differential equations encountered in science in technology, but many more complex systems have so far resisted effective computation, and that is the thrust of this research. A particular emphasis will be on numerical algorithms for accurate determination of the stresses internal to elastic structures, which is essential to building safe and economic engineering structures. A second emphasis will be on developing methods to simulate Einstein's equations of general relativity, especially for predicting the output of gravitational radiation--minute ripples that propagate on the curved surface of space-time--from massive cosmological events such as black hole collisions. Computer codes capable of making such predictions are needed to realize the effectiveness of a new type of observatory based on gravitational radiation currently being constructed, which will provide mankind with its first window on the dark matter that makes up 90% of the universe.

研究人员将设计、改进和分析由偏微分方程模拟的复杂物理现象的数值模拟方法，重点放在三个主要领域：弹性方程的混合方法、间断Galerkin方法和计算广义相对论。对于弹性力学，研究者将在最近的一项突破的基础上，为多项式试函数的位移-应力公式构造第一个稳定的混合有限元方法，并致力于发展更简单的非协调混合有限元方法，并将其扩展到三维弹性问题。关于不连续Galerkin方法--有限元方法，其中逼近的分段多项式函数是不连续的，并在变分公式中加入修改以达到一致性--研究者将与他的合作者合作，在最近关于这类方法的统一和分类的工作的基础上，开发一种统一的方法来分析和区分椭圆型方程的一大类不连续Galerkin方法。并将其应用于考虑剪切作用的弹性板的数值模拟。第三个也是最大的努力涉及爱因斯坦关于质量和时空曲率的场方程组的数值解。这里的重点将是理解与其数值解最相关的爱因斯坦方程的基本性质，以及以前试图对它们进行数值模拟时遇到的基本困难。这项工作将以模拟激发黑洞对的合并和由此产生的引力辐射的目标为指导，这是一个对引力物理至关重要的问题，也因为这种模拟将对实现基于引力波探测器的新一代天文台至关重要。计算机模拟是设计和测试复杂工程结构的关键工具。近几十年来，计算机模拟也作为科学研究的主要范式之一，与实验和理论相结合。在这两个领域，许多最复杂的系统首先由偏微分系统建模--其中使用微积分语言来表示相关物理量在空间和时间上的变化--然后这些微分方程组必须通过数值算法来近似，这些算法利用现代计算机的能力每秒执行数十亿次算术运算，以提取方程的解，以达到所需的精度。近几十年来，设计和验证这类算法的原则已经为科学上在技术上遇到的许多基本微分方程组制定了，但到目前为止，更复杂的系统一直在抵制有效的计算，这就是本研究的主旨。特别强调的是精确确定弹性结构内部应力的数值算法，这对于建造安全和经济的工程结构是必不可少的。第二个重点将是开发模拟爱因斯坦广义相对论方程的方法，特别是预测来自大规模宇宙事件(如黑洞碰撞)的引力辐射的输出。引力辐射是指在时空曲面上传播的微小涟漪。为了实现目前正在建设的基于引力辐射的新型天文台的有效性，需要能够进行此类预测的计算机代码，这将为人类提供了解暗物质的第一扇窗，暗物质构成了宇宙的90%。