  We expand upon our previous analysis of numerical moving-puncture simulations
of the Schwarzschild spacetime. We present a derivation of the family of
analytic stationary 1+log foliations of the Schwarzschild solution, and outline
a transformation to isotropic-like coordinates. We discuss in detail the
numerical evolution of standard Schwarzschild puncture data, and the new
time-independent 1+log data. Finally, we demonstrate that the moving-puncture
method can locate the appropriate stationary geometry in a robust manner when a
numerical code alternates between two forms of 1+log slicing during a
simulation.
