  The DG algorithm is a powerful method for solving pdes, especially for
evolution equations in conservation form. Since the algorithm involves
integration over volume elements, it is not immediately obvious that it will
generalize easily to arbitrary time-dependent curved spacetimes. We show how to
formulate the algorithm in such spacetimes for applications in relativistic
astrophysics. We also show how to formulate the algorithm for equations in
non-conservative form, such as Einstein's field equations themselves. We find
two computationally distinct formulations in both cases, one of which has
seldom been used before for flat space in curvilinear coordinates but which may
be more efficient. We also give a new derivation of the ALE algorithm
(Arbitrary Lagrangian-Eulerian) using 4-vector methods that is much simpler
than the usual derivation and explains why the method preserves the
conservation form of the equations. The various formulations are explored with
some simple numerical experiments that also explore the effect of the metric
identities on the results.
