An energy-conserving and an energy-and-enstrophy conserving numerical schemes are derived by approximating the Hamiltonian formulation of the inviscid shallow water flows based on the vorticity-divergence variables. These schemes also conserve the first-order moments such as mass and vorticity, as usual. The Conservative properties of the schemes stem from the skew-symmetry and singularities of the Poisson brackets, which are carefully retained in the discrete approximations. The 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, which is a consequence of the use of the vorticity and divergence variables.

通过基于涡度-发散变量近似无粘性浅水流的哈密顿公式，导出了能量守恒和能量和熵守恒的数值方案。像往常一样，这些方案还保存了一阶矩，例如质量和涡量。方案的保守性质源于泊松括号的偏对称性和奇异性，它们在离散近似中被仔细保留。这些方案在有界或无界域上的非结构正交双网格上运行，并且它们还被证明具有与 Z 网格方案相同的最佳色散波关系，这是使用涡度和发散变量的结果.