Gravitational waves are expected to open up a new window for Astronomy in the near future. Numerical simulations are essential for this effort. By combining the results of simulations of systems containing compact stars or black holes with observational data we can hope to probe the dynamics of strong gravitational fields. The results will have significant impact on our understanding of the Universe. With the exponential growth of computer power, the numerical simulations of astrophysical interesting situations are becoming feasible. However, fully 3D numerical relativity simulations of gravitational-wave sources such as binary black holes or neutron stars are plagued by numerical instabilities whose origin is elusive. Well-posed ness of the continuum problem has been identified as a key ingredient for numerical stability. In this research project we shall formulate boundary conditions (both at the outer boundary of the numerical domain, and at interior boundaries where black holes are excised), which are both consistent with the constraints And mathematically well posed. Well-posed ness proofs for problems with boundaries are limited to smooth surfaces. However, a global spherical grid has coordinate singularities on which are hard to overcome, and is not well adapted for simulating binary merger containing excised regions. We shall explore a Cartesian main grid with overlapping grids adapted to the outer boundary and excision boundaries. Our aim is to construct stable and consistent numerical schemes, including boundary conditions, for binary black hole or neutron star simulations. As we proceed, we shall develop the mathematical theory of well-posed constraint-preserving boundary conditions, and in parallel develop the overlapping grids method by studying a variety of model problems ranging from the wave equation to full general relativity and from axisymmetry to 3D.
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