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变革--美国LIGO探测器发现了来自黑洞和中子星双星系统的引力波，以及事件视界望远镜首次拍摄到黑洞的地平线图像，这是诺贝尔奖得主。引力波是一个世纪前由爱因斯坦自己预测的，以前从未被直接观测到。对这些来自致密双星系统的波的持续观测将被用来获得关于宇宙中奇异天体的额外信息，如黑洞和中子星。LIGO还产生了大量的副产品技术，并引起了公众对STEM学科的强烈关注。这一拟议的项目有助于开发先进的计算模型，这些模型将对LIGO和即将到来的LISA等空间任务的未来成功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数据分析工作设定的高精度和高效率要求。该项目包括对学生(包括妇女和少数群体)的支持，因此直接有助于在一个重要的STEM领域提供学生指导、培训和留住学生。学生发展的计算技能广泛适用，因此将使他们能够获得各种职业选择，包括在国家需求巨大的领域。PIS以前的研究项目已经在一般媒体上进行了讨论，这项工作在成功推广到普通公众方面也具有巨大的潜力。拟议的工作通过开发和调整用于引力波模拟的空间和时间演变方法来应对“宇宙之窗”的挑战。具体地说，目标1将发展一维间断Galerkin方法来求解Teukolsky方程。这种方法跟踪粒子并将其保持在区域界面上，同时计算狄拉克增量函数的导数作为区域边界处的匹配条件。该方法模拟螺旋内相位，具有极高的精度。然而，为了准确地模拟俯冲和衰荡阶段，我们需要一个激波捕捉方案，它可以处理狄拉克增量函数的导数，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求解器，该解算器将包括处理狄拉克增量函数的高达三阶导数的能力，并在远离间断的区域变得非常高效。最后，必须为目标1和目标2中的空间方案量身定制高效和准确的时间演化方法。为此，目标3将为目标1和目标2中的空间方案开发稳定和高效的时间离散。对于每个空间离散，时间离散方法，如Runge-Kutta方法和多步Runge-Kutta方法，将被量身定制，以使这些方法具有低存储量、计算效率、具有小的色散误差、小的误差常数和为空间离散量身定制的稳定区域，并且(对于WENO)SSP时间步长最优。在空间和时间离散方面的拟议发展将导致更有效的方法，可以准确和高效地处理长时间积分和Dirac Delta函数及其导数的存在。此外，开发一种准确、高效的数值求解器，能够在参数空间的相当大一部分上生成波形，是引力波计算方面的一大进步，因此将对引力波科学领域产生重大影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.