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.
