Constrained evolution in axisymmetry

This paper is concerned with the Einstein equations in axisymmetric vacuum spacetimes. We consider numerical evolution schemes that solve the constraint equations as well as elliptic gauge conditions at each time step. We examine two such schemes that have been proposed in the literature and show that some of their elliptic equations are indefinite, thus potentially admitting nonunique solutions and causing numerical solvers based on classical relaxation methods to fail. A new scheme is then presented that does not suffer from these problems. We use our numerical implementation to study the gravitational collapse of Brill waves. A highly prolate wave is shown to form a black hole rather than a naked singularity. PACS numbers: 04.25.D−, 04.20.Cv, 04.20.Ex, 04.25.dc (Some figures in this article are in colour only in the electronic version)