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

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.

Figures (3)

• Figure 1: Simulation of a radial dam break at $t=0.1$. (1) Blue solid line denotes the shallow water equation (SWE); (2) orange solid line denotes the hyperbolic shallow water moment equation (HSWME) with $N=1$; (3) green solid line denotes the HSWME with $N=2$; (4) red dashed line denotes the $\beta$-HSWME with $N=2$; (5) purple dashed dot line denotes the example HSWME with $N=2$; (6) brown dot line denotes the globally HSWME with $N=2$.
• Figure 2: Simulation of a radial dam break at $t=0.1$ using globally HSWME. Left: the water height in 1D cut along $y=0.5$, $x=0.5$, and $y=x$. Here $r=\sqrt{(x-0.5)^2 + (y-0.5)^2}$. Right: the water height in 3D plot.
• Figure 3: Simulation of the smooth wave at $t=1$. (1) Blue solid line denotes the shallow water equation (SWE); (2) orange solid line denotes the hyperbolic shallow water moment equation (HSWME) with $N=1$; (3) green solid line denotes the HSWME with $N=2$; (4) red dashed line denotes the $\beta$-HSWME with $N=2$; (5) purple dashed dot line denotes the example HSWME with $N=2$; (6) brown dot line denotes the globally HSWME with $N=2$.

Theorems & Definitions (54)

• Definition 3.1: rotational invariance
• Remark 3.1
• Proposition 3.1
• Proof 1
• Proposition 3.2
• Proof 2
• Lemma 3.1: rotational invariance for conservative part
• Proof 3
• Lemma 3.2: rotational invariance for non-conservative part
• Proof 4