Computational models of ocean circulation have applications ranging from global climate modeling to limited-area modeling of near-shore regions. This process requires the numerical solution of a system of partial differential equations describing fluid flow, as adapted to this situation. Solving such a system numerically requires discretizations both in space and in time. In the case of the horizontal spatial discretization, there are known deficiencies in some traditional approaches that are widely used in operational ocean circulation models. One such deficiency is that nonphysical grid-scale numerical noise can arise in certain situations. Another is a potentially inaccurate propagation of gravity waves and/or Rossby waves, depending on the discretization chosen and the relation between the grid resolution and a length scale known as the Rossby radius. A relatively new discretization method for partial differential equations, the discontinuous Galerkin method, has a potential of overcoming the above deficiencies.