In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data.

在本文中，我们介绍了一些在一般非结构网格上的新的高阶离散公式，这些公式特别设计用于基于属于完全非线性和弱分散浅水方程组的偏微分方程来研究非旋转自由表面流。通过使用最近的一系列优化的渐近等效方程，我们受益于线性色散算子的简化分析结构，可以方便地将模型重新构造为经典非线性浅水方程，并补充了多个代数源项，这些项在全球范围内都说明了非静水问题通过引入辅助耦合变量来实现效果。主要流量变量的高阶离散近似是通过RK-DG方法获得的，而辅助变量的轨迹则通过具有高阶HDG公式的二阶线性椭圆子问题的解析来近似估计网格骨架。与以前的方法相比，混合未知数和局部后处理的组合使用显着有助于减少全局耦合的未知数。然后，将所提出的公式扩展到三个参数增强的Green-Naghdi方程的更复杂的族。所得的数值模型已通过多个基准进行了验证，这些基准涉及非线性波的转换和在不同地形上的传播，显示出良好的收敛性，并且与几组实验数据非常吻合。