We focus on the numerical modelling of water waves by means of depth averaged models. We consider in particular PDE systems which consist in a nonlinear hyperbolic model plus a linear dispersive perturbation involving an elliptic operator. We propose two strategies to construct reduced order models for these problems, with the main focus being the control of the overhead related to the inversion of the elliptic operators, as well as the robustness with respect to variations of the flow parameters. In a first approach, only a linear reduction strategies is applied only to the elliptic component, while the computations of the nonlinear fluxes are still performed explicitly. This hybrid approach, referred to as pdROM, is compared to a hyper-reduction strategy based on the empirical interpolation method to reduce also the nonlinear fluxes. We evaluate the two approaches on a variety of benchmarks involving a generalized variant of the BBM-KdV model with a variable bottom, and a one-dimensional enhanced weakly dispersive shallow water system. The results show the potential of both approaches in terms of cost reduction, with a clear advantage for the pdROM in terms of robustness, and for the EIMROM in terms of cost reduction.

翻译：我们的重点是通过深度平均模型对水浪进行数字建模。我们特别考虑到由非线性双曲模型组成的PDE系统,以及涉及一个椭圆形操作员的线性分散扰动系统。我们建议采取两项战略,为这些问题建立减少定序模型,主要重点是控制与椭圆形操作员倒转有关的间接费用,以及在流动参数变化方面保持稳健。在第一个办法中,只有线性削减战略仅适用于椭圆形部分,而非线性通量的计算仍然明确进行。这种混合方法被称为双极管,与基于实验性内插法的大幅度削减战略相比较,以同时减少非线性通量。我们评估了两种基准,即BBMM-KdV模型具有可变底部的通用变量,以及单维度增强的微分差浅水系统。结果显示两种方法在成本削减方面的潜力,即非线性通流流流流,即称为PdROM方法,与基于实证的跨极化方法的超度战略相比较,以同时减少非线性通量性通量。我们评估了两种基准的基准,即BM-K-KdV模型在降低成本方面有明显的优势。