An Asymptotic-Preserving Dual Formulation Finite-Volume Method for the Thermal Rotating Shallow Water Equations

This paper introduces a second-order asymptotic-preserving dual formulation finite-volume method that simultaneously solves the conservative and nonconservative forms of the thermal rotating shallow water equations to effectively capture both shock dynamics in high-Rossby-number regimes and correct asymptotic behavior in low-Rossby-number quasi-geostrophic regimes.

The Gist

Imagine the Earth's atmosphere and oceans as a giant, swirling dance floor. Sometimes, the dancers (air and water) move slowly and gracefully, following a strict rhythm set by the Earth's rotation. Other times, they move chaotically, crashing into each other and forming sudden, sharp waves.

This paper introduces a new, super-smart computer program designed to simulate these dances. The challenge is that the dance changes speed depending on the "Rossby number" (a fancy term for how much the Earth's spin influences the flow).

• Fast Dance (High Rossby Number): The dancers move quickly, creating sharp waves and shocks. To simulate this, you need a method that treats the dancers as a solid crowd that can crash.
• Slow Dance (Low Rossby Number): The dancers move in a slow, balanced waltz. To simulate this, you need a method that treats them as individuals following a strict, invisible rhythm.

The Problem:
Old computer programs were like one-size-fits-all shoes. If you tried to use a "shock-capturing" shoe (good for fast crashes) on the slow waltz, the simulation would become incredibly slow and expensive because it would try to calculate every tiny step of the slow dance. If you used a "slow-waltz" shoe on the fast crashes, the simulation would fall apart and give the wrong results.

The Solution: The "Dual-Formulation" Method
The authors created a new method called an Asymptotic-Preserving Dual Formulation Finite-Volume (DF-FV) method. Here is how it works, using simple analogies:

1. The "Dual" Approach: Two Sets of Glasses

Instead of choosing just one way to look at the problem, this method wears two pairs of glasses at the same time:

• Glasses A (The Conservative View): This looks at the flow as a "conservation of mass and momentum." It's great at handling crashes and sharp edges (shocks) without breaking.
• Glasses B (The Primitive View): This looks at the flow based on velocity and pressure. It's excellent at maintaining the slow, balanced rhythm of the Earth's rotation.

The computer solves the equations for both views simultaneously. It's like having a security guard (Conservative) watching for crashes and a choreographer (Primitive) watching the rhythm, both reporting to the same director.

2. The "Splitting" Trick: Separating the Fast from the Slow

The equations governing these flows have two types of parts:

• Stiff (Fast) parts: These are the rapid vibrations caused by the Earth's spin. They are hard to calculate because they happen so fast.
• Non-stiff (Slow) parts: These are the slower movements of the water or air.

The authors invented a special "hyperbolic splitting" to separate these two.

• The Analogy: Imagine a car with a very sensitive engine (the stiff part) and a heavy body (the non-stiff part). Instead of trying to drive the whole car with one foot, they treat the engine with a "semi-implicit" brake (a smart, stable calculation that doesn't require tiny time steps) and the body with a standard "explicit" gas pedal.
• The Result: The computer doesn't get stuck trying to calculate the tiny, fast vibrations. It skips over them efficiently, allowing the simulation to run fast even when the Earth's rotation is very strong.

3. The "Post-Processing" Glue

At the end of every time step, the computer takes the results from both "glasses" and blends them together using a special switch (a switching function).

• If the flow is fast (High Rossby): The switch turns on the "Conservative" view, ensuring the simulation captures sharp waves correctly.
• If the flow is slow (Low Rossby): The switch turns on the "Primitive" view, ensuring the simulation captures the slow, balanced waltz correctly.
• The Magic: This blending happens automatically. The user doesn't have to tell the computer which regime they are in; the method figures it out and switches gears seamlessly.

Why is this a big deal?

• It's Universal: It works equally well for fast, chaotic storms and slow, giant ocean currents. You don't need different software for different weather conditions.
• It's Efficient: Old methods would slow down to a crawl when simulating slow, balanced flows. This new method stays fast because it uses a "semi-implicit" trick to handle the fast vibrations without needing a super-computer.
• It's Accurate: The authors tested it with various scenarios, from swirling vortex pairs to waves decaying over time. In every test, their method matched the "gold standard" reference solutions but did so much faster and without the "jittery" errors (oscillations) that plague other methods.

In Summary:
The authors built a universal simulator for Earth's fluid flows. By wearing two pairs of glasses at once and using a smart "splitting" technique to handle fast and slow movements differently, they created a tool that is both fast and accurate, no matter how the Earth's rotation influences the dance.

Technical Summary

1. Problem Statement

The paper addresses the numerical challenges associated with the Thermal Rotating Shallow Water (TRSW) equations, a fundamental model for geophysical flows (atmospheric and oceanic) that incorporates horizontal temperature/density variations and the Coriolis force.

• Multiscale Dynamics: The TRSW system exhibits dynamics across a wide spectrum of Rossby numbers ($Ro$).
  • High-$Ro$ regimes: Dynamics are dominated by compressible, wave-like phenomena and shocks. Accurate simulation requires conservative formulations to ensure correct shock speeds and convergence to weak solutions.
  • Low-$Ro$ regimes: Rotation strongly constrains the flow, leading to Thermal Quasi-Geostrophic (TQG) dynamics. Here, the system becomes stiff due to fast inertial oscillations. Standard explicit schemes become prohibitively expensive (requiring tiny time steps), while non-conservative (primitive) formulations are better suited for preserving the correct asymptotic behavior and divergence-free constraints.
• The Challenge: Existing methods struggle to handle both regimes simultaneously. Explicit schemes fail in low-$Ro$ due to stiffness; fully implicit schemes are computationally costly and may converge to incorrect limits; and standard asymptotic-preserving (AP) schemes often lack robustness in capturing shocks in high-$Ro$ regimes.

2. Methodology

The authors propose a second-order Asymptotic-Preserving (AP) Dual Formulation Finite-Volume (DF-FV) method. The core innovation is the simultaneous evolution of both the conservative and nonconservative (primitive) forms of the equations, coupled via a post-processing step.

A. Primitive System Formulation (Low-$Ro$ Focus)

To handle the stiff low-$Ro$ regime efficiently, the authors develop an AP scheme for the primitive variables $(u, v, \phi, \theta, q)$, where $q$ is the potential vorticity.

• Hyperbolic Splitting: The system is split into nonstiff (slow) and stiff (fast) components based on the Rossby number $\varepsilon$.
• Semi-Implicit (SI) Time Discretization:
  • Nonstiff terms are treated explicitly using a Path-Conservative Central-Upwind (PCCU) scheme to handle nonconservative products and shocks.
  • Stiff terms are treated semi-implicitly using central differences.
  • Key Advantage: This approach requires solving only linear well-posed elliptic problems (Poisson-like equations) at each time step, avoiding expensive nonlinear solvers.
• AP Property: The scheme is proven to converge to the correct TQG limit as $\varepsilon \to 0$ without mesh refinement or time-step reduction, provided initial data is well-prepared.

B. Conservative System Formulation (High-$Ro$ Focus)

For high-$Ro$ regimes where shocks dominate, the authors employ a standard Central-Upwind (CU) scheme on the conservative variables $(h, hu, hv, h\Theta)$. This ensures the correct Rankine-Hugoniot conditions are met for shock capturing.

C. Dual Formulation and Post-Processing

The method evolves both systems simultaneously. A switching function $\phi(\varepsilon)$ is used to blend the solutions at each time step:
$$V_{final} = (1 - \phi(\varepsilon)) V(U) + \phi(\varepsilon) V_{primitive}$$

• **High-$Ro$($\varepsilon \sim 1$):**$\phi \approx 0$. The solution relies on the conservative system ($V(U)$) to capture shocks accurately.
• **Low-$Ro$($\varepsilon \to 0$):**$\phi \approx 1$. The solution relies on the primitive AP system to capture TQG dynamics and avoid stiffness.
• This coupling ensures the method is uniformly accurate across all Rossby numbers, preserving shock speeds in high-$Ro$ and asymptotic limits in low-$Ro$.

3. Key Contributions

1. Novel DF-FV Framework: Extension of the dual formulation concept (previously used for RSW and Euler equations) to the Thermal RSW equations, addressing the added complexity of buoyancy and thermal coupling.
2. Efficient AP Primitive Scheme: Development of a semi-implicit scheme for the primitive TRSW system that avoids nonlinear solvers by solving only linear elliptic equations, significantly reducing computational cost compared to fully implicit methods.
3. Robust Coupling Mechanism: A post-processing strategy that seamlessly transitions between conservative and primitive formulations, preventing spurious oscillations in high-$Ro$ regimes while maintaining asymptotic correctness in low-$Ro$ regimes.
4. Theoretical Proof: Rigorous proof of the AP property, demonstrating that the numerical solution converges to the TQG system as the Rossby number vanishes.

4. Numerical Results

The method was validated through five numerical experiments:

• Accuracy Test: Demonstrated second-order convergence in space and time for various Rossby numbers ($\varepsilon = 1$ to $10^{-6}$), confirming uniform accuracy.
• Freely Decaying Wavetrain: Showed the method's ability to capture nonlinear wave adjustment and shock formation in rotating regimes, outperforming standard explicit schemes in the TQG limit.
• Vortex Pair Interaction: Validated the necessity of the full PCCU discretization (including nonconservative jump terms). Simplified versions led to spurious oscillations and instability, whereas the proposed method remained stable and accurate.
• Shear Flow Evolution (TQG Regime): In the low-$Ro$ regime ($\varepsilon \approx 0.028$), the AP DF-FV method resolved fine vortex structures significantly better than explicit schemes (which suffered from $O(\varepsilon^{-1})$ dissipation) and was more computationally efficient than high-order WENO schemes while maintaining comparable accuracy.
• Anticyclonic Propagation ($\beta$-Plane): Successfully captured complex geophysical features like $\beta$-drift and secondary circulations with strong mesh convergence.

5. Significance

This work provides a unified numerical framework for geophysical fluid dynamics that overcomes the traditional trade-off between shock capturing and asymptotic preservation.

• Efficiency: It offers a computational cost comparable to explicit schemes in low-$Ro$ regimes (where explicit schemes fail) and avoids the high cost of fully implicit solvers.
• Robustness: It eliminates the need for regime-specific solvers, allowing a single code to simulate flows ranging from fast, shock-dominated waves to slow, balanced geostrophic flows.
• Applicability: The method is highly relevant for climate modeling, weather prediction, and oceanography, where simulations must span multiple scales and regimes without manual intervention or parameter tuning.