  This is the second in a series of papers describing a 3+1 computational
scheme for the numerical simulation of dynamic black hole spacetimes. We
discuss the numerical time-evolution of a given black-hole-containing initial
data slice in spherical symmetry. We avoid singularities via the "black-hole
exclusion" or "horizon boundary condition" technique, where the slices meet the
black hole's singularity, but on each slice a spatial neighbourhood of the
singularity is excluded from the domain of the numerical computations.
  After first discussing some of the key design choices which arise with the
black hole exclusion technique, we then give a detailed description of our
numerical evolution scheme for spherically symmetric scalar field evolution,
assuming that a black hole is already present on the initial slice. We use a
free evolution, with Eddington-Finkelstein-like coordinates and the inner
boundary placed at a fixed coordinate radius well inside the horizon.
  Our numerical scheme is based on the method of lines (MOL), where spacetime
PDEs are first finite differenced in space only, yielding a system of coupled
ODEs for the time evolution of the field variables along the spatial-grid-point
world lines. These ODEs are then time-integrated by standard methods. We use
4th order finite differencing in both space and time, with 5 and/or 6 point
spatial molecules (off-centered near the grid boundaries), and a Runge-Kutta
time integrator. The spatial grid is smoothly nonuniform, but not adaptive.
  We present numerical black hole + scalar field evolutions showing that this
scheme is stable, can evolve "forever" (we have gone to t > 4000m), and is very
accurate. At a resolution Delta_r/r = 3% near the horizon, typical errors in
g_ij(K_ij) at t=100m are <= 1e-5(3e-7), and the energy constraint is < 3e-5.
