Einstein equations in the null quasi-spherical gauge III: numerical algorithms

We describe numerical techniques used in our construction of a 4th order in time evolution for the full Einstein equations, and assess the accuracy of some representative solutions. The scheme employs several novel geometric and numerical techniques, including a geometrically invariant coordinate gauge, which leads to a characteristic-transport formulation of the underlying hyperbolic system, combined with a “method of lines” evolution; a convolution spline for radial interpolation, regridding, differentiation and noise suppression; representations using spin-weighted spherical harmonics; and a spectral preconditioner for solving a class of 1st order elliptic systems on S. Initial data for the evolution is unconstrained, subject only to a mild size condition. For sample initial data of “intermediate” strength (19% of the total mass in gravitational energy), the code is accurate to 1 part in 10, until null time z = 55M when the coordinate condition breaks down. This project has been supported by Australian Research Council grants A69330046 and A69802586. 1 [email protected] 2 [email protected]