Numerical Methods in General Relativity

鉴于国内和国际上建立引力波观测站的大规模计划，迫切需要对黑洞碰撞等大质量宇宙事件的引力波发射进行准确模拟。这需要对爱因斯坦场方程进行准确的数值模拟，这是一个极其复杂的偏微分方程组，到目前为止，这些方程一直抵制开发稳定的数值方法的尝试。在发展爱因斯坦方程的数值方法方面，既需要对广义相对论和引力有深刻的物理理解，也需要掌握现代数值分析和科学计算的许多领域。这位首席研究员将在宾夕法尼亚州立大学物理系驻留一年，期间与一群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求解方面的专业知识和经验，以及开发和实施新的爱因斯坦方程数值方法所需的物理背景和理解。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上。首先，主要研究人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解算器，来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传播给从事引力辐射数值模拟工作的研究人员社区。第二，主要研究人员和他的合作者将研究新的自适应非结构网格方法在演化问题中的应用，例如间断伽辽金方法，用于爱因斯坦方程。这些方法不仅有望有效地解决复杂和高度可变的解结构，而且还可以处理爱因斯坦方程中出现的各种其他数值问题。这个项目将使主要研究人员在数值相对论方面的参与出现巨大飞跃，这种飞跃将持续到拨款期限之后很久。除了正在进行的合作外，首席研究人员还将开设研究生课程，并举办研讨会和讲座，以招收学生和博士后，并组成计算相对论小组。通过会议报告、演讲和文章，他将致力于提高数值分析社区对数值相对论中的问题和挑战的认识和兴趣。该IGMS项目由重大计划多学科活动办公室(OMA)和数学科学处(DMS)共同支持。

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).