  Element Method. The Finite Volume Method guarantees local and global mass
conservation. A property not satisfied by the Finite Volume Method. On the down
side, the Finite Volume Method requires non trivial modifications to attain
high order approximations unlike the Finite Volume Method. It has been
contended that the Discontinuous Galerkin Method, locally conservative and high
order, is a natural progression for Coastal Ocean Modeling. Consequently, as a
primer we consider the vertical ocean-slice model with the inclusion of density
effects. To solve these non steady Partial Differential Equations, we develop a
pressure projection method for solution. We propose a Hybridized Discontinuous
Galerkin solution for the required Poisson Problem in each time step. The
purpose, is to reduce the computational cost of classical applications of the
Discontinuous Galerkin method. The Hybridized Discontinuous Galerkin method is
first presented as a general elliptic problem solver. It is shown that a high
order implementation yields fast and accurate approximations on coarse meshes.
