  This is the first paper in a series aimed to implement boundary conditions
consistent with the constraints' propagation in 3D numerical relativity. Here
we consider spherically symmetric black hole spacetimes in vacuum or with a
minimally coupled scalar field, within the Einstein-Christoffel symmetric
hyperbolic formulation of Einstein's equations. By exploiting the
characteristic propagation of the main variables and constraints, we are able
to single out the only free modes at the outer boundary for these problems. In
the vacuum case a single free modes exists which corresponds to a gauge
freedom, while in the matter case an extra mode exists which is associated with
the scalar field. We make use of the fact that the EC formulation has no
superluminal characteristic speeds to excise the singularity. We present a
second-order, finite difference discretization to treat these scenarios, where
we implement these constraint-preserving boundary conditions, and are able to
evolve the system for essentially unlimited times. As a test of the robustness
of our approach, we allow large pulses of gauge and scalar field enter the
domain through the outer boundary. We reproduce expected results, such as
trivial (in the physical sense) evolution in the vacuum case (even in gauge-
dynamical simulations), and the tail decay for the scalar field.
