Equilibrium stability analysis of hyperbolic shallow water moment equations

In this paper, we analyze the stability of equilibrium manifolds of hyperbolic shallow water moment equations. Shallow water moment equations describe shallow flows for complex velocity profiles which vary in vertical direction and the models can be seen as extensions of the standard shallow water equations. Equilibrium stability is an important property of balance laws that determines the linear stability of solutions in the vicinity of equilibrium manifolds, and it is seen as a necessary condition for stable numerical solutions. After an analysis of the hyperbolic structure of the models, we identify three different stability manifolds based on three different limits of the right-hand side friction term, which physically correspond to water-at-rest, constant-velocity, and bottom-at-rest velocity profiles. The stability analysis then shows that the structural stability conditions are fulfilled for the water-at-rest equilibrium and the constant-velocity equilibrium. However, the bottom-at-rest equilibrium can lead to instable modes depending on the velocity profile. Relaxation toward the respective equilibrium manifolds is investigated numerically for different models.

在本文中，我们分析了双曲型浅水矩方程平衡流形的稳定性。浅水矩方程描述了在垂直方向上变化的复杂速度剖面的浅水流动，并且这些模型可被视为标准浅水方程的扩展。平衡稳定性是平衡律的一个重要性质，它决定了平衡流形附近解的线性稳定性，并且被视为稳定数值解的必要条件。在对模型的双曲结构进行分析之后，我们基于右侧摩擦项的三种不同极限确定了三个不同的稳定流形，它们在物理上分别对应于静止水、恒定速度和底部静止的速度剖面。稳定性分析随后表明，对于静止水平衡和恒定速度平衡，结构稳定性条件得到满足。然而，底部静止平衡可能会根据速度剖面导致不稳定模式。针对不同的模型，对向相应平衡流形的松弛进行了数值研究。