Primitive variable regularization to derive novel Hyperbolic Shallow Water Moment Equations

TL;DR

The paper tackles the lack of global hyperbolicity and analytic steady states in traditional Shallow Water Moment Equations (SWME). It introduces two new hierarchies, PHSWME and PMHSWME, by performing hyperbolic regularization in primitive variables, with PMHSWME keeping the momentum equation unchanged. The authors prove global hyperbolicity, derive analytical steady states, and validate improved accuracy via a dam-break test, highlighting the importance of preserving the momentum equation. PMHSWME achieves global hyperbolicity while retaining nonlinear moment coupling, providing a robust reduced-order model for free-surface flows and enabling extensions to higher dimensions and richer physics.

Abstract

Shallow Water Moment Equations are reduced-order models for free-surface flows that employ a vertical velocity expansion and derive additional so-called moment equations for the expansion coefficients. Among desirable analytical properties for such systems of equations are hyperbolicity, accuracy, correct momentum equation, and interpretable steady states. In this paper, we show analytically that existing models fail at different of these properties and we derive new models overcoming the disadvantages. This is made possible by performing a hyperbolic regularization not in the convective variables (as done in the existing models) but in the primitive variables. Via analytical transformations between the convective and primitive system, we can prove hyperbolicity and compute analytical steady states of the new models. Simulating a dam-break test case, we demonstrate the accuracy of the new models and show that it is essential for accuracy to preserve the momentum equation.

Figures (6)

• Figure 1: Hyperbolicity regions of MHSWME for $N=2$ (left) depending on scaled $\alpha_1, \alpha_2$ and $N=3$ (right) depending on scaled $\alpha_1, \alpha_2, \alpha_3$. MHSWME shows large hyperbolicity regions that do not decrease in size with increasing $N$.
• Figure 2: Dam break test case relative errors w.r.t. SWME with the same number of moments $N=2,3,4$ for HSWME (red), SWLME (green), MHSWME (blue), PHSWME (purple), PMHSWME (olive). Water height $h$ (top left), mean velocity (top right), linear coefficient $\alpha_1$ (bottom left), and quadratic coefficient $\alpha_2$ (bottom right). Existing models like HSWME and SWLME perform bad for most variables while the new model like PMHSWME shows best results for most variables.
• Figure 3: Dam break test case water height $h$ solutions for SWME (top left), HSWME (top right), SWLME (middle left), MHSWME (middle right), PSWME (bottom left), PMSWME (bottom right), for $N=1,2,3,4,5$. Note that all models are equivalent for $N=1$, so this is only shown in the SWME plot. SWME5 is unstable and left out.
• Figure 4: Dam break test case average velocity $u_m$ solutions for SWME (top left), HSWME (top right), SWLME (middle left), MHSWME (middle right), PSWME (bottom left), PMSWME (bottom right), for $N=1,2,3,4,5$. Note that all models are equivalent for $N=1$, so this is only shown in the SWME plot. SWME5 is unstable and left out.
• Figure 5: Dam break test case linear coefficient $\alpha_1$ solutions for SWME (top left), HSWME (top right), SWLME (middle left), MHSWME (middle right), PSWME (bottom left), PMSWME (bottom right), for $N=1,2,3,4,5$. Note that all models are equivalent for $N=1$, so this is only shown in the SWME plot. SWME5 is unstable and left out.

Theorems & Definitions (25)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 1
• Theorem 1
• proof
• Lemma 4