二维浅水线性化矩方程：双曲性与良平衡格式

中文摘要 AI 辅助

我们研究了二维水平方向上浅水线性化矩方程的旋转不变性、双曲性以及良平衡离散化。尽管直接扩展具有实特征速度，但其方向矩阵可能缺乏完整的特征基。我们构造了一个旋转不变的修正，并证明了其在所有正水深状态下的全局双曲性。对于具有Navier-slip摩擦的非平坦地形上的准二维流动，我们发展了基于中点配点构造的离散移动平衡态精确保持的一阶和二阶路径守恒有限体积格式。数值测试确认了平衡态保持至舍入误差、光滑稳态分支的二阶逼近，以及二阶格式对测试扰动的近二阶精度。与三维两相OpenFOAM模拟的比较展示了不变的浅水极限，并表明保留矩改善了弯曲旋转速度剖面的重构，而径向运动仍存在差异。

英文摘要

We study rotational invariance, hyperbolicity, and well-balanced discretization of shallow water linearized moment equations in two horizontal dimensions. Although the direct extension has real characteristic speeds, its directional matrices can lack a complete eigenbasis. We construct a rotationally invariant modification and prove its global hyperbolicity for all states with positive water depth. For quasi-two-dimensional flows over non-flat topography with Navier-slip friction, we develop first- and second-order path-conservative finite-volume schemes that exactly preserve a discrete moving equilibrium constructed by midpoint collocation. Numerical tests confirm equilibrium preservation to roundoff, second-order approximation of smooth stationary branches, and near-second-order accuracy of the second-order scheme for the tested perturbations. Comparisons with three-dimensional two-phase OpenFOAM simulations illustrate the invariant shallow-water limit and show that retaining moments improves the reconstruction of curved rotational velocity profiles, while differences remain in radial motion.

发表机构

• Michigan State University（密歇根州立大学）
• University of Delaware（特拉华大学）

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