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.
