  Techniques are developed for projecting the solutions of symmetric hyperbolic
evolution systems onto the constraint submanifold (the constraint-satisfying
subset of the dynamical field space). These optimal projections map a field
configuration to the ``nearest'' configuration in the constraint submanifold,
where distances between configurations are measured with the natural metric on
the space of dynamical fields. The construction and use of these projections is
illustrated for a new representation of the scalar field equation that exhibits
both bulk and boundary generated constraint violations. Numerical simulations
on a black-hole background show that bulk constraint violations cannot be
controlled by constraint-preserving boundary conditions alone, but are
effectively controlled by constraint projection. Simulations also show that
constraint violations entering through boundaries cannot be controlled by
constraint projection alone, but are controlled by constraint-preserving
boundary conditions. Numerical solutions to the pathological scalar field
system are shown to converge to solutions of a standard representation of the
scalar field equation when constraint projection and constraint-preserving
boundary conditions are used together.
