High Order Numerical Methods for Gravitational Wave Computations

由于最近的多项突破，黑洞天体物理学的研究领域经历了一次重大转变——美国LIGO探测器发现了黑洞和中子星双星系统的引力波，获得了诺贝尔奖，事件视界望远镜首次拍摄到了黑洞的视界图像。引力波是爱因斯坦在一个世纪前预言的，在此之前从未被直接观测到。对这些来自紧凑双星系统的引力波的持续观测将用于获取宇宙中奇异天体（如黑洞和中子星）的额外信息。LIGO还产生了重要的衍生技术，并强烈吸引了公众对STEM学科的关注。这个提议的项目有助于开发先进的计算模型，这些模型将在LIGO和即将到来的LISA等太空任务的未来成功中发挥非常关键的作用。拟议项目的主要目标是开发新的计算技术，以满足LIGO和LISA数据分析工作所设定的高精度和高效率要求。该项目包括对学生（包括女性和少数族裔）的支持，因此直接有助于学生在重要的STEM领域获得指导、培训和保留。学生们培养的计算技能是广泛适用的，因此将使他们能够获得各种各样的职业选择，包括在国家需要的领域。私人机构以前的研究项目已经在一般媒体上讨论过，这项工作也有很大的潜力成功地向公众推广。提出的工作通过开发和调整用于引力波模拟的空间和时间演化方法来解决“宇宙之窗”的挑战。具体来说，Aim 1将开发一个一维不连续Galerkin方法来求解Teukolsky方程。该方法通过计算Dirac函数的导数作为域边界处的匹配条件，跟踪粒子并使其保持在域界面处。这种方法以极高的精度模拟了螺旋相位。然而，为了精确模拟暴跌和衰落阶段，我们需要一种能够处理狄拉克函数导数的激波捕获方案，并提供高效和准确的多维数值结果。为此，Aim 2将开发一个非常高阶的WENO解算器，该解算器将包括处理Dirac δ函数的三阶导数的能力，并在远离不连续的区域内实现高效。最后，必须针对目标1和目标2的空间方案定制高效、准确的时间演化方法。为此，目标3将针对目标1和目标2中的空间方案开发稳定有效的时间离散化。对于每个空间离散化，时间离散化方法，如龙格-库塔和多步龙格-库塔方法将被定制，使方法具有低存储、计算效率、小色散误差、小误差常数和适合空间离散化的稳定区域，以及（对于WENO）最优的SSP时间步长。提出的空间和时间离散化的发展将导致更有效的方法，可以准确和有效地处理长时间积分和狄拉克函数及其导数的存在。此外，能够在相当大的参数空间上产生波形的精确、有效的数值求解器的发展是引力波计算的一个重大进步，因此将对引力波科学领域产生重大影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research area of black hole astrophysics has experienced a major transformation as a result of multiple recent breakthroughs -- a Nobel Prize-winning discovery of gravitational waves from black hole and neutron star binary systems by the US LIGO detectors, and the first-ever image of the horizon of a black hole by the Event Horizon Telescope. Gravitational waves were predicted by Einstein himself a century ago and had never been directly observed before. Ongoing observations of these waves from compact binary systems will be used to obtain additional information about exotic astrophysical objects in the universe like black holes and neutron stars. LIGO has also generated significant spin-off technologies and strongly drawn public attention towards STEM disciplines. This proposed project aids in the development of advanced computational models that will play a very critical role in the future success of LIGO and upcoming space-borne missions like LISA. The main objective of the proposed project is to develop new computational techniques to meet the high-accuracy and high-efficiency requirements set by the LIGO and LISA data-analysis effort. This project includes support for students (including women and minorities) and therefore directly contributes to student mentorship, traineeship, and retention in an important STEM area. The computational skills that the students develop are broadly applicable, and therefore would allow them access to a variety of career options, including in areas of great national need. Previous research projects by the PIs have been discussed in the general media, and this work also has great potential at being successful for outreach to the general public.The proposed work addresses the "Windows on the Universe" challenge by developing and adapting spatial and time-evolution methods for use in gravitational wave simulations. Specifically, Aim 1 will develop a one dimensional discontinuous Galerkin method to solve the Teukolsky equations. This method tracks the particle and keeps it at the domain interfaces while computing the derivatives of the Dirac delta functions as matching conditions at the boundary of the domain. This approach simulates the in-spiral phase to extremely high accuracy. However, for an accurate simulation of the plunge and ring-down phase we require a shock capturing scheme that can handle derivatives of the Dirac delta function and provide highly efficient and accurate multi-dimensional numerical results. For this, Aim 2 will develop a very high order WENO solver that will include the ability to handle up to third derivatives of the Dirac delta function and be made highly efficient in the regions away from the discontinuity. Finally, efficient and accurate time evolution approaches must be tailored to the spatial schemes in Aims 1 and 2. For this, Aim 3 will develop stable and efficient time-discretizations tailored for the spatial schemes in Aims 1 and 2. For each spatial discretization, time-discretization approaches such as Runge-Kutta and multi-step Runge-Kutta methods will be tailored such that the methods are low storage, computationally efficient, have small dispersion errors, small error constants, and stability regions that are tailored to the spatial discretization, and (for WENO) optimal SSP time-steps. The proposed developments in both spatial and temporal discretizations will lead to more efficient methods that can accurately and efficiently handle long time-integration and the presence of Dirac delta functions and its derivatives. Furthermore, the development of an accurate, efficient numerical solver capable of generating waveforms over sizable portions of the parameter space is a major advance in the computation of gravitational waves, and will thus have a major impact on the field of gravitational wave science.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.