Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

使用双曲浅水矩方程进行地球物理流模拟需要对潜在的大型偏微分方程系统（即所谓的矩系统）进行有效离散。这需要定制的模型降阶技术，以实现高效、准确的模拟，同时保证质量守恒等物理特性。在本文中，我们开发了双曲浅水矩方程的第一个模型简化并实现了质量守恒。这是通过将模型宏微观分解为宏观（保守）部分和微观（非保守）部分来完成的，随后仅在微观（非保守）部分上使用 POD-Galerkin 或动态低秩近似进行模型简化。保守）部分。数值实验展示了新模型简化方法的性能，包括高精度和快速计算时间以及保证的守恒性和一致性特性。