Three-dimensional initial data for the collision of two black holes

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Abstract
We describe three numerical approaches to the construction of three-dimensional initial data for the collision of two black holes. The first of our approaches involves finite differencing the 3 + 1 Hamiltonian constraint equation on a Čadež coordinate grid. The difference equations are then solved via the multigrid algorithm. The second approach also uses finite-difference techniques, but this time on a regular Cartesian coordinate grid. A Cartesian grid has the advantage of having no coordinate singularities. However, constant coordinate lines are not coincident with the throats of the black holes and, therefore, special treatment of the difference equations at these boundaries is required. The resulting equations are solved using a variant of line-successive overrelaxation. The third and final approach we use is a global, spectral-like method known as the multiquadric approximation scheme. In this case functions are approximated by a finite sum of weighted quadric basis functions which are continuously differentiable. We discuss particular advantages and disadvantages of each method and compare their performances on a set of test problems.