Construction of three-dimensional black-hole initial data via multiquadrics

Numerical solutions of the 3+1 Hamiltonian constraint equation for single-black-hole initial data are presented. When expressed in Cartesian coordinates the solutions for the conformal factor are fully three-dimensional (3D), and therefore the approach described here can straightforwardly produce initial data for two or more black holes with arbitrary positions, spins, and linear momenta. The numerical method we use is known as the multiquadric approximation scheme, in which a continuous function, f, is written as f(x,y,z)=${\mathit{a}}_1$+ ${\mathit{tsum}}_{\mathit{j}=2\mathrm{}}^{\mathit{N}}$ ${\mathrm{a}}_{\mathit{j}}$g${\mathrm{̃}}_{\mathit{j}}$(x,y,z), where the ${\mathit{a}}_{\mathit{j}}$’s are constants, the g${\mathrm{̃}}_{\mathit{j}}$’s are radial basis functions (multiquadrics), and N is the total number of grid points used. Despite numerical problems with ill-conditioning of the matrix, with care the method is capable of producing highly accurate results and it has a number of distinct advantages over finite-difference methods for problems with complicated boundaries, as is the case here. Typically we can reproduce the York-Bowen analytic solution for a boosted black hole to less than 0.5% maximum error with N=1057 in 3D. Where no analytic solutions exist comparisons of our 3D results with previous axisymmetric 2D numerical calculations show very good agreement.