Primitive variable regularization to derive novel Hyperbolic Shallow Water Moment Equations

This paper introduces novel Hyperbolic Shallow Water Moment Equations derived through primitive variable regularization, which analytically overcome the hyperbolicity and accuracy limitations of existing models while preserving the correct momentum equation and enabling interpretable steady states.

The Gist

Imagine you are trying to predict how a flood will move down a river. To do this accurately, you need a computer model.

The Old Way: The "Flat Map" Problem
For a long time, scientists used a model called the "Shallow Water Equations." Think of this like looking at a river from a satellite and seeing it as a flat, 2D sheet. It assumes the water moves at the same speed everywhere from the bottom to the surface.

• The Problem: In reality, water near the bottom drags on the riverbed and moves slowly, while water at the surface flows fast. The old model misses this "speed difference," leading to bad predictions about how fast a flood will hit a town or how sediment will move.

The Better Way: The "Layer Cake" Approach
To fix this, scientists developed "Moment Equations." Imagine slicing the water column into layers (like a cake) and tracking the speed of each slice.

• The Catch: When you add these extra layers to your math, the equations become incredibly complex. Sometimes, the math breaks down completely. It's like trying to balance a tower of Jenga blocks; if you add too many, the tower becomes unstable and collapses. In math terms, the system loses hyperbolicity (it becomes unstable and produces nonsense results) and loses the ability to calculate steady states (it can't figure out what a calm, steady river looks like).

The Paper's Solution: The "Primitive" Fix
This paper introduces a new way to build these models that keeps the tower stable and the math solvable. Here is the analogy:

1. The Two Languages: The scientists realized there are two ways to describe the water:

   • Convective Variables (The "Total" View): This describes the total amount of water and momentum in a chunk. It's like describing a car by its total weight and total speed.
   • Primitive Variables (The "Basic" View): This describes the water depth and the speed at specific points. It's like describing the car by its tire pressure and engine RPM.

2. The Mistake: Previous attempts to fix the unstable math tried to "tune" the Total View (Convective). They tried to force the math to work by ignoring the higher layers of the cake. This worked for stability but made the model inaccurate, or it worked for accuracy but the math still broke down in extreme situations.

3. The Breakthrough: The authors decided to switch to the Basic View (Primitive) to do the tuning.

   • The Analogy: Imagine you are trying to fix a wobbly table. Previous engineers tried to sand down the heavy, complex tabletop (Convective) to make it lighter. It helped a little, but the legs were still shaky.
   • The New Idea: These authors went down to the legs (Primitive variables). They realized that if you stabilize the legs first, the whole table becomes rock solid. By doing the math in this "Basic View," they could prove the system would never collapse (it is globally hyperbolic) and could easily calculate what a steady river looks like.

4. The Best of Both Worlds: They created a new model called PMHSWME.

   • It keeps the "legs" (the basic physics of water depth and momentum) exactly as they should be, ensuring high accuracy.
   • It stabilizes the "tabletop" (the complex layers) using the new method.
   • The Result: A model that is as accurate as the complex version but as stable and easy to solve as the simple version.

Why Does This Matter?

• Safety: Better models mean better flood warnings. If you can predict exactly how a wave will behave, you can save lives.
• Speed: Because the math is now stable and has "steady states," computers can solve these problems much faster. This is crucial for real-time disaster prediction.
• Versatility: This method works for any number of layers, meaning it can be as simple or as detailed as needed without breaking.

In a Nutshell:
The authors found that trying to fix the complex math of water flow was like trying to fix a car engine while driving it at 100 mph. They realized they needed to stop, open the hood, and look at the individual parts (the primitive variables) to make the right adjustments. By doing so, they built a new engine that is both powerful (accurate) and reliable (stable).

Technical Summary

1. Problem Statement

Free-surface flows (e.g., tsunamis, flooding) are often modeled using the Shallow Water Equations (SWE), which assume a constant vertical velocity profile. However, this assumption fails in scenarios involving complex physics like vegetation, sediment transport, or bottom friction, where vertical velocity variations are significant.

To address this, Shallow Water Moment Equations (SWME) were developed. These models expand the vertical velocity profile using orthogonal Legendre polynomials, resulting in a system of $N+2$ partial differential equations (PDEs) for water depth ($h$), depth-averaged velocity ($u_m$), and expansion coefficients ($\alpha_i$).

The Core Challenge: Existing SWME models suffer from two critical analytical deficiencies:

1. Lack of Global Hyperbolicity: For orders $N \geq 2$, the original SWME system is only locally hyperbolic. It loses hyperbolicity (leading to numerical instabilities) for small deviations from equilibrium or large higher-order moments.
2. Lack of Analytical Steady States: Existing models do not admit analytical steady states derivable from Rankine–Hugoniot conditions, hindering the development of well-balanced numerical schemes.

Previous attempts to fix these issues involved:

• HSWME: Regularizing in convective variables by setting higher-order coefficients to zero. This ensures global hyperbolicity but loses physical accuracy and still lacks analytical steady states.
• SWLME: Linearizing higher-order moments to gain analytical steady states and hyperbolicity, but at the cost of significant physical accuracy (losing nonlinear momentum effects).

Goal: The paper aims to derive a new hierarchy of models that simultaneously satisfies four criteria:

1. Preserves mass and momentum balance.
2. Is globally hyperbolic.
3. Admits analytical steady states.
4. Maintains high accuracy relative to the unregularized SWME.

2. Methodology

The authors propose a novel Primitive Variable Regularization strategy, contrasting with previous methods that regularized in convective variables.

Key Steps:

1. Variable Transformation: The system is transformed between primitive variables ($U_p = [h, u_m, \alpha_1, \dots, \alpha_N]$) and convective variables ($U_c = [h, hu_m, h\alpha_1, \dots, h\alpha_N]$) using an invertible transformation matrix $T$.
2. Regularization in Primitive Space: Instead of setting higher-order coefficients ($\alpha_i = 0$ for $i \geq 2$) directly in the convective system matrix (as done in HSWME), the authors perform the regularization in the primitive system matrix ($A_p$).
   • They evaluate the primitive system matrix around linear velocity profiles (setting $\alpha_i = 0$ for $i \geq 2$ in the moment equations).
3. Transformation Back: The regularized primitive system matrix is transformed back to convective variables. Crucially, due to the non-linear nature of the transformation, the higher-order coefficients $\alpha_i$ reappear in the convective system matrix, preserving coupling terms that were lost in previous methods.
4. Hybrid Approach (PMHSWME): The authors combine the primitive regularization with the strategy of keeping the momentum balance equation unchanged (a feature of the SWLME/MHSWME approaches). This results in a model where only the higher-order moment equations are regularized, while the mass and momentum equations remain exact.

3. Key Contributions

A. Derivation of New Models

The paper derives two new model hierarchies:

1. PHSWME (Primitive Hyperbolic Shallow Water Moment Equations): A model regularized entirely in primitive variables.
2. PMHSWME (Primitive Moment Hyperbolic Shallow Water Moment Equations): A hybrid model that regularizes the primitive moment equations but keeps the original momentum equation unchanged.

B. Analytical Proofs

• Global Hyperbolicity: The authors prove that both PHSWME and PMHSWME are globally hyperbolic for any order $N$. The eigenvalues are shown to be real and distinct, derived via characteristic polynomials involving derivatives of Legendre polynomials ($P'_{N+1}$).
• Analytical Steady States: Unlike previous models, the new systems allow for the derivation of analytical steady states. The steady states depend on the Froude number and dimensionless moment numbers, generalizing the standard SWE hydraulic jump conditions.

C. Theoretical Insight

The paper demonstrates that the choice of variable space for regularization is critical. Regularizing in convective variables (HSWME/MHSWME) leads to local hyperbolicity or infeasible steady states. Regularizing in primitive variables allows for a similarity transformation that preserves the hyperbolic structure while recovering necessary coupling terms in the convective form.

4. Results

Numerical Validation (Dam-Break Test Case)

The authors tested the models against a standard dam-break problem with a linear initial velocity profile, comparing them to the original SWME and a high-resolution reference solver.

• Accuracy:
  • SWLME: Performed the worst, showing large errors due to excessive linearization.
  • HSWME: Performed poorly for water height ($h$) and mean velocity ($u_m$) because it altered the momentum equation.
  • PHSWME: Improved over HSWME but still showed errors in $h$ and $u_m$ due to changes in the momentum coupling.
  • PMHSWME: Achieved the highest accuracy for all variables ($h, u_m, \alpha_1, \alpha_2$). By preserving the exact momentum equation while regularizing only the higher-order moments, it minimized deviation from the original SWME.
• Stability: The new models (PHSWME and PMHSWME) remained stable, whereas the original SWME became unstable for higher orders ($N \geq 5$) in the test case.

Comparative Analysis

A summary table (Table 1 in the paper) highlights that PMHSWME is the only model satisfying all desired properties:

• Global Hyperbolicity: Yes
• Preserved Momentum Equation: Yes
• Analytical Steady States: Yes
• Nonlinear Moment Equations: Yes

5. Significance

This work resolves a long-standing trade-off in reduced-order modeling of free-surface flows.

1. Theoretical Breakthrough: It establishes that primitive variable regularization is a superior strategy for deriving hyperbolic moment systems compared to convective regularization.
2. Practical Impact: The PMHSWME model provides a robust, globally hyperbolic framework that is mathematically tractable (analytical steady states) and physically accurate. This enables the development of well-balanced numerical schemes capable of handling complex free-surface flows with high fidelity, essential for accurate hazard prediction (tsunamis, floods) and long-term risk assessment.
3. Future Directions: The paper opens the door for structure-preserving numerical schemes, multi-dimensional extensions, and the inclusion of complex friction terms, as the analytical foundation (hyperbolicity and steady states) is now secure.

In conclusion, the paper successfully derives a new class of shallow water moment equations that overcome the limitations of previous models by shifting the regularization paradigm from convective to primitive variables, resulting in the PMHSWME model which is globally hyperbolic, analytically solvable for steady states, and highly accurate.