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.

翻译：使用双曲浅水矩方程进行地球物理流模拟时，需要高效离散化潜在的大规模偏微分方程组（即矩系统）。这要求定制化的模型降阶技术，既能实现高效精确的模拟，又能保证质量守恒等物理特性。本文首次提出适用于双曲浅水矩方程的模型降阶方法，并实现了质量守恒。该方法通过对模型进行宏观-微观双尺度分解（分为宏观守恒部分与微观非守恒部分），随后仅在微观（非守恒）部分采用POD-Galerkin或动态低秩近似进行模型降阶。数值实验展示了新模型降阶方法的性能，包括高精度、快速计算时间以及保证的守恒性与一致性特征。