Binary neutron star inspiral is the most certain source of gravitational waves detectable by Earth-based observatories like the US LIGO project, and simulations of such binaries should facilitate eventual detections. These simulations require initial conditions: solutions to the initial value problem of general relativity for the coupled gravity-matter system. The conformal thin sandwich method is an excellent approach for solving the initial value problem; however, although not an intrinsic assumption of the method, in practice the approach has assumed conformal flatness (as have other valuable approaches). Conformal flatness yields unphysical junk radiation. By numerically constructing helically symmetric solutions to the Einstein equations, the PI will extract initial data (or conformal thin sandwich trial data) which does not rely on conformal flatness, and therefore contains the correct initial gravitational wave content. The mixed PDEs arising from the helical reduction of the Einstein equations (or their approximation in the post-Minkowski formalism) will be solved with innovative techniques: sparse modal spectral-tau methods with new preconditioning strategies. In part, these strategies may rely on randomized algorithms for the interpolative decomposition. Spectral methods deliver superb accuracy for smooth problems(neutron star spacetimes are smooth almost everywhere), and sparsity affords a fast matrix-vector multiply when using a Krylov-subspace method to iteratively solve a linear system. Whereas the preconditioning of nodal (collocation) spectral methods is well studied, less is known about modal preconditioning. Our techniques have been successfully applied to models of the binary neutron star problem. Moreover, the problem's physical structure has already been explored with different, but limited, techniques.
This project is to combine two sets of techniques (each already developed) and further develop the first set (spectral-tau methods), in order to obtain new results for a leading problem in gravitational wave physics. The PI will develop these mathematical methods by applying them to the specific neutron star problem described above. This strategy of specificity is often used in the development of techniques, which then prove to be more general. Because the scientific problem is of great interest, much is known about it, and results therefore exist withwhich comparisons can be made. These comparisons facilitate the development of mathematical algorithms. Conversely, new mathematical methods deliver more and/or better solutions which enhances scientific understanding.
