FRG: Collaborative Research: Developing Spectral Methods for Numerical Solutions of the Einstein Equations

在这项工作中，我们组建了一个新的跨学科团队，由计算数学家和天体物理学家组成，专注于以光谱多域方法为中心的新一代模拟工具的开发和分析。初步测试已经证实了这种技术的潜力，然而，仍然需要大量的算法开发，例如。，有效的非线性并行椭圆求解器，稳定技术，以及新的并行时间积分方法，以充分成熟。同时，还将关注爱因斯坦方程特有的自然问题，如初始数据问题、约束违反解的控制和合适的边界条件。通过目前部署的引力波探测器LIGO和不久的将来的LISA，数值相对论和求解爱因斯坦方程的计算方法的发展被注入了新的紧迫性。为了充分理解和分析用这些设备测量的信号，必须有新一代的计算工具可用于在很长时间内解决动态爱因斯坦方程。在这个跨学科的努力中，我们结合了计算数学和天体物理学的专业知识来实现这一目标，目的是模拟一对二元黑洞的完整动力学，据推测，正如广义相对论所预测的那样，它们是引力波的强大来源。因此，这项工作不仅将导致新的计算模拟技术的发展，而且还将导致测试人类已知的一些最基本的物理理论的有效性的能力。

In this effort, we form a new interdisciplinary team, comprised ofcomputational mathematicans and astrophysicists, to focus on the development amd analysis of a new generation of simulation tools centered around spectral multi-domain methods. Initial tests have confirmed the potential of such techniqueswhich, however, still require significant algorithmic developments,e.g., efficient nonlinear parallel elliptic solvers, stabilizationtechniques, and novel parallel temporal integration methods, to maturesufficiently. Simultaneously, attention will be paid to problems of anature more specific to the Einstein equations, e.g., the initial dataproblem, control of constraint violating solutions, and suitableboundary conditions.New urgency has been injected into numerical relativity and thedevelopment of computational methods for solving the Einsteinequations by the current deployment of gravitational wave detectors,LIGO and, in the near future, LISA. To fully understand and analyzethe signals measured with such facilities it is essential that a newgeneration of computational tools be available for solving the dynamical Einstein equations over very long times. In this interdiciplinaryeffort we combine expertize in computational mathematics andastrophysics to reach this goal, with the aim of modeling the fulldynamics of a binary pair of black holes, conjectured to be a strongsource of gravity waves as predicted by the general theory of relativity.Thus the effort will not only lead to the development of newcomputational simulation techniniques but also lead to the abilityto test the validity some of the most fundamental physical theoriesknown to mankind.