On the rotational invariance and hyperbolicity of shallow water moment equations in two dimensions

TL;DR

This paper extends and analyzes a 2D shallow water moment model, demonstrating its rotational invariance and hyperbolicity, and proposes modifications for a globally hyperbolic and more general closure relations.

Contribution

It proves rotational invariance of the 2D moment model using two approaches and establishes hyperbolicity, including modifications for a globally hyperbolic model with general closure relations.

Findings

The 2D moment model is rotationally invariant.

The model is hyperbolic in most cases, weakly hyperbolic in some edge cases.

A modified model achieves global hyperbolicity.

Abstract

In this paper, we investigate the two-dimensional extension of a recently introduced set of shallow water models based on a regularized moment expansion of the incompressible Navier-Stokes equations \cite{kowalski2017moment,koellermeier2020analysis}. We show the rotational invariance of the proposed moment models with two different approaches. The first proof involves the split of the coefficient matrix into the conservative and non-conservative parts and proves the rotational invariance for each part, while the second one relies on the special block structure of the coefficient matrices. With the aid of rotational invariance, the analysis of the hyperbolicity for the moment model in 2D is reduced to the real diagonalizability of the coefficient matrix in 1D. Then we analyze the real diagonalizability by deriving the analytical form of the characteristic polynomial. We find that the moment model in 2D is hyperbolic in most cases and weakly hyperbolic in a degenerate edge case. With a simple modification to the coefficient matrices, we fix this weakly hyperbolicity and propose a new global hyperbolic model. Furthermore, we extend the model to include a more general class of closure relations than the original model and establish that this set of general closure relations retains both rotational invariance and hyperbolicity.