  Numerical codes based on a direct implementation of the standard ADM
formulation of Einstein's equations have generally failed to provide long-term
stable and convergent evolutions of black hole spacetimes when excision is used
to remove the singularities. We show that, for the case of a single black hole
in spherical symmetry, it is possible to circumvent these problems by adding to
the evolution equations terms involving the constraints, thus adjusting the
standard ADM system. We investigate the effect that the choice of the lapse and
shift has on the stability properties of numerical simulations and thus on the
form of the added constraint term. To facilitate this task, we introduce the
concept of quasi well-posedness, a version of well-posedness suitable for
ADM-like systems involving second-order spatial derivatives.
