Numerical models of ocean circulation generally use finite difference and finite volume methods on structured rectangular grids, in present-day usage. In contrast, the more recent discontinuous Galerkin (DG) methods allow the possibility of high-order methods on unstructured grids while maintaining high locality. However, DG methods have some potential disadvantages in efficiency related to the number of unknowns and the maximum allowable time step. In a simple setting where DG and finite difference methods are equally applicable, an analysis of dispersion relations for inertia-gravity waves shows that the DG spatial discretization can be much more accurate than the standard finite difference methods that are widely used in ocean modeling. In some numerical computations, this advantage more than offsets the potential disadvantages mentioned above. The work described here is one piece of a larger project to evaluate the possibility of using DG methods in this area.