Constrained evolution in numerical relativity
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The strongest potential source of gravitational radiation for current and future detectors is the merger of binary black holes. Full numerical simulation of such mergers can provide realistic signal predictions and enhance the probability of detection. Numerical simulation of the Einstein equations, however, is fraught with difficulty. Stability even in static test cases of single black holes has proven elusive. Common to unstable simulations is the growth of constraint violations. This work examines the effect of controlling the growth of constraint violations by solving the constraints periodically during a simulation, an approach called constrained evolution. The effects of constrained evolution are contrasted with the results of unconstrained evolution, evolution where the constraints are not solved during the course of a simulation. Two different formulations of the Einstein equations are examined: the standard ADM formulation and the generalized Frittelli-Reula formulation. In most cases constrained evolution vastly improves the stability of a simulation at minimal computational cost when compared with unconstrained evolution. However, in the more demanding test cases examined, constrained evolution fails to produce simulations with long-term stability in spite of producing improvements in simulation lifetime when compared with unconstrained evolution. Constrained evolution is also examined in conjunction with a wide variety of promising numerical techniques, including mesh refinement and overlapping Cartesian and spherical computational grids. Constrained evolution in boosted black hole spacetimes is investigated using overlapping grids. Constrained evolution proves to be central to the host of innovations required in carrying out such intensive simulations.