Numerical Methods in General Relativity

广义相对论中的数值方法

• 批准号: 9972835
• 负责人: Douglas Arnold
• 金额: $ 10万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9972835&HistoricalAwards=false
• 关键词: Numerical; Methods; General; Relativity

In view of large-scale national and international programs to construct gravitational wave observatories, there is a pressing need for accurate simulation of the gravitational wave emission from massive cosmicevents such as inspiralling black hole collisions. This requires the accurate numerical simulation of the Einstein field equations, an extremely complex system of partial differential equations, which have thus far resisted attempts to develop stable numerical methods. Progress in the development of numerical methods for the Einstein equations will require both a deep physical understanding of general relativity and gravitation and the mastery of many areas of modern numerical analysis and scientific computation. The principalinvestigator will spend one year in residence at the Department of Physics of Penn State University, in a period of intense interdisciplinary collaboration and scientific interchange with a group of gravitational physicists. In this period he will complement his expertise and experience in the numerical solution of partial differential equations with the physical background and understanding necessary to develop and implement new numerical approaches to the Einstein equations. At the same time he will convey to his physicist collaborators recent advances in numerical analysis relevant to computational relativity.The interdisciplinary collaboration will focus on two directions relevant to the simulation of gravitational radiation. First, the principal investigator and his collaborators will apply recent advances in finite element technology, especially adaptive tetrahedral mesh refinement algorithms and multigrid solvers, to the generation of physically relevant initial data satisfying the constraints implied by the Einstein equations. The resulting initial data will be disseminated to the community of researchers working on numerical simulation of gravitational radiation. Second, the principal investigators and his collaborators will study the application of new adaptive unstructured mesh methods for evolution problems, such as discontinuous Galerkin methods, to the Einstein equations. These methods hold promise not only for the efficient resolution of the complex and highly variable solution structure, but also for dealing with a variety of other numerical problems that arise with the Einstein equations.This project will bring about a quantum leap in the principal investigator's involvement with numerical relativity which will continue well beyond the period of the grant. In addition to the ongoing collaboration, the principal investigator will offer a graduate course and give seminars and lectures, in order to recruit students and postdocs and to form a group in computational relativity. Through conference presentations, lectures, and articles he will work to develop awareness and interest in the numerical analysis community ofthe problems and challenges in numerical relativity. This IGMS project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).

鉴于国家和国际上大规模建设引力波观测站的计划，迫切需要精确模拟大质量宇宙事件（如螺旋黑洞碰撞）的引力波发射。这需要精确的数值模拟爱因斯坦场方程，一个极其复杂的系统的偏微分方程，迄今为止，抵制试图发展稳定的数值方法。 爱因斯坦方程数值方法的发展需要对广义相对论和引力有深刻的物理理解，并掌握现代数值分析和科学计算的许多领域。首席研究员将在宾夕法尼亚州立大学物理系居住一年，与一群引力物理学家进行激烈的跨学科合作和科学交流。 在此期间，他将补充他的专业知识和经验，在数值解偏微分方程的物理背景和必要的理解，以开发和实施新的数值方法的爱因斯坦方程。 与此同时，他将向他的物理学家合作者传达与计算相对论相关的数值分析的最新进展。跨学科合作将集中在与引力辐射模拟相关的两个方向。首先，首席研究员和他的合作者将应用有限元技术的最新进展，特别是自适应四面体网格细化算法和多重网格求解器，以生成满足爱因斯坦方程所隐含的约束的物理相关的初始数据。 由此产生的初始数据将分发给从事引力辐射数值模拟的研究人员。 其次，主要研究人员和他的合作者将研究新的自适应非结构化网格方法的应用，如不连续Galerkin方法，爱因斯坦方程的演化问题。 这些方法不仅可以有效地解决复杂多变的解结构问题，还可以用来处理爱因斯坦方程组中出现的其他各种数值问题。该项目将使首席研究员在数值相对论方面的研究出现一个巨大的飞跃，而这一飞跃将远远超出资助期。 除了正在进行的合作，主要研究者将提供一个研究生课程，并举行研讨会和讲座，以招募学生和博士后，并形成一个计算相对论小组。 通过会议演讲，讲座和文章，他将致力于发展认识和兴趣的数值分析社区的问题和挑战，在数值相对论。 该IGMS项目由MPS多学科活动办公室（OMA）和数学科学部（DMS）共同支持。