  We propose methods that augment existing numerical schemes for the simulation
of hyperbolic balance laws with Dirichlet boundary conditions to allow for the
simulation of a broad class of differential algebraic conditions. Our approach
is similar to that of Thompson (1987), where the boundary values were simulated
by combining characteristic equations with the time derivative of the algebraic
conditions, but differs in two important regards. Firstly, when the boundary is
a characteristic of one of the fields Thompson's method can fail to produce
reasonable values. We propose a method of combining the characteristic
equations with extrapolation which ensures convergence. Secondly, the
application of algebraic conditions can suffer from $O(1)$ drift-off error, and
we discuss projective time-stepping algorithms designed to converge for this
type of system. Test problems for the shallow water equations are presented to
demonstrate the result of simulating with and without the modifications
discussed, illustrating their necessity for certain problems.
