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This paper is concerned with the Einstein equations in axisymmetric vacuumspacetimes. We consider numerical evolution schemes that solve the constraintequations as well as elliptic gauge conditions at each time step. We examinetwo such schemes that have been proposed in the literature and show that someof their elliptic equations are indefinite, thus potentially admittingnonunique solutions and causing numerical solvers based on classical relaxationmethods to fail. A new scheme is then presented that does not suffer from theseproblems. We use our numerical implementation to study the gravitationalcollapse of Brill waves. A highly prolate wave is shown to form a black holerather than a naked singularity.

Constrained evolution in axisymmetry and the gravitational collapse of prolate Brill waves