Numerical Methods in General Relativity

鉴于国内外大规模建设引力波天文台的计划，迫切需要精确模拟黑洞碰撞等大质量宇宙事件的引力波发射。这需要对爱因斯坦场方程进行精确的数值模拟，这是一个极其复杂的偏微分方程系统，迄今为止，它一直抵制开发稳定的数值方法的尝试。在爱因斯坦方程数值方法的发展过程中，既需要对广义相对论和万有引力有深刻的物理理解，又需要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，与一组引力物理学家进行密切的跨学科合作和科学交流。在此期间，他将补充他在偏微分方程数值解方面的专业知识和经验，以及开发和实施爱因斯坦方程新数值方法所需的物理背景和理解。同时，他将向他的物理学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重力辐射模拟有关的两个方向。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网格求解器，以生成满足爱因斯坦方程隐含约束的物理相关初始数据。由此产生的初步数据将传播给从事重力辐射数值模拟的研究人员。其次，主要研究者和他的合作者将研究新的自适应非结构网格方法在进化问题上的应用，如不连续伽辽金方法，在爱因斯坦方程上的应用。这些方法不仅可以有效地解决复杂和高度可变的解结构，而且还可以处理与爱因斯坦方程有关的各种其他数值问题。这个项目将带来一个量子跃进在主要研究者的参与与数值相对论，将继续远远超出资助期。除了正在进行的合作之外，首席研究员还将开设研究生课程，并举办研讨会和讲座，以招募学生和博士后，并组建一个计算相对论小组。通过会议演讲、讲座和文章，他将致力于提高数值分析界对数值相对论问题和挑战的认识和兴趣。该IGMS项目由MPS多学科活动办公室（OMA）和数学科学司（DMS）联合支持。

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).