Equilibrium Stability Analysis of Hyperbolic Shallow Water Moment Equations

TL;DR

This paper investigates the stability of equilibrium states in hyperbolic shallow water moment equations, revealing conditions for stability and potential instabilities depending on velocity profiles, with numerical validation.

Contribution

It provides a detailed stability analysis of hyperbolic shallow water moment equations, identifying conditions for equilibrium stability based on different physical velocity profiles.

Findings

Water-at-rest and constant-velocity equilibria are structurally stable.

Bottom-at-rest equilibrium may lead to instabilities depending on velocity profiles.

Numerical experiments confirm relaxation towards stable equilibria.

Abstract

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 towards the respective equilibrium manifolds is investigated numerically for different models.