Multidomain Spectral Methods and Radiation Boundary Conditions with Applications in Numerical Relativity

该奖项支持应用数学和计算物理之间的研究。数值相对论的一个直接目标是模拟轨道双星，例如两个黑洞，以便计算分析引力波探测器(如激光干涉引力波天文台)输出所需的辐射波形。演化爱因斯坦方程的计算挑战也推动了数值和应用分析方面的现代研究。这项研究将集中在数值相对论(爱因斯坦方程的计算机解)和辐射边界条件的谱方法上，主要但不限于爱因斯坦方程。光谱方法提供了极高的效率、精度和内插自由度。后者意义重大，因为坐标灵活性在爱因斯坦的理论中是至高无上的。辐射边界条件允许在具有人工边界的有限计算区域上模拟(引力或非引力)波，其规范是计算数学中的一个基本问题，在科学上有着广泛的应用。通过发展具有明确应用的新的谱方法，研究的目标是解决一个从现有方法和计算机资源的角度来看难以解决的基波问题，即不等质量双星的有效时间积分。光谱方法的研究计划还包括探索不连续伽辽金方法对主要的轨道双星移动穿孔技术的适用性。辐射边界条件的研究注重理论理解和有效的数值实现，考虑了标量波、麦克斯韦方程和爱因斯坦方程。这些研究的潜在应用包括计算声学和电磁学中的大纵横比现象，数值相对论中的极端质量比双星和旋转黑洞，以及完全非线性爱因斯坦方程的依赖于历史的辐射边界条件。

This award supports research at the interface between applied mathematics and computational physics. An immediate goal of numerical relativity is the simulation of orbiting binaries, such as two black holes, in order to compute the radiation waveforms necessary to analyze output from gravitational wave detectors like the Laser Interferometric Gravitational Wave Observatory. The computational challenge of evolving the Einstein equations also drives modern research in numerical and applied analysis. This research will focus on spectral methods for numerical relativity (computer solution of the Einstein equations) and radiation boundary conditions, chiefly, but not exclusively, for the Einstein equations. Spectral methods offer superb efficiency, accuracy, and interpolation freedom. The latter is significant, since coordinate flexibility is paramount in Einstein's theory. Radiation boundary conditions allow for wave simulation (gravitational or otherwise) on finite computational domains with artificial boundaries, and their specification is a fundamental problem in computational mathematics with broad application in the sciences. Through the development of novel spectral methods with explicit applications, the goal of the research is to solve a fundamental wave problem which is intractable from the standpoint of current methods and computer resources, namely the efficient time integration of unequal mass binaries. The research program on spectral methods also includes exploration of the applicability of discontinuous Galerkin methods to the predominant moving puncture technique for orbiting binaries. Emphasizing both theoretical understanding and efficient numerical implementation, the research on radiation boundary conditions considers the scalar wave, Maxwell, and Einstein equations. Potential applications of the proposed investigations include high-aspect-ratio phenomena in computational acoustics and electromagnetics, extreme-mass-ratio binaries and rotating black holes in numerical relativity, and history-dependent radiation boundary conditions for the full nonlinear Einstein equations.