  An energy-conserving and an energy-and-enstrophy conserving numerical schemes
are derived, by approximating the Hamiltonian formulation, based on the Poisson
brackets and the vorticity-divergence variables, of the inviscid shallow water
flows. The conservation of the energy and/or enstrophy stems from skew-symmetry
of the Poisson brackets, which is retained in the discrete approximations.
These schemes operate on unstructured orthogonal dual meshes, over bounded or
unbounded domains, and they are also shown to possess the same optimal
dispersive wave relations as those of the Z-grid scheme.
