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arxiv: 2604.26121 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA· physics.flu-dyn

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

Alina Chertock , Alexander Kurganov , Lorenzo Micalizzi , Nan Zhang This is my paper

Pith reviewed 2026-05-07 14:52 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.flu-dyn
keywords asymptotic-preserving methodsfinite-volume methodsthermal rotating shallow water equationsdual formulationRossby number regimesgeophysical flowsmulti-scale dynamics
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The pith

A dual formulation finite-volume method solves both conservative and nonconservative forms of the thermal rotating shallow water equations simultaneously to handle multi-scale regimes accurately.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a second-order asymptotic-preserving dual formulation finite-volume method for the thermal rotating shallow water equations. This approach solves the conservative and nonconservative forms of the system at the same time. The nonconservative form is used to maintain the correct asymptotic behavior in low Rossby number nearly thermal quasi-geostrophic regimes, while the conservative form supports robust shock capturing in high Rossby number regimes. The goal is to address the multi-scale dynamics from fast rotational waves and slower advective processes in flows with horizontal temperature variations.

Core claim

The authors claim that simultaneously discretizing both the conservative and nonconservative forms within a finite-volume framework yields a second-order asymptotic-preserving scheme that reproduces the thermal quasi-geostrophic asymptotics at low Rossby numbers and converges to physically relevant weak solutions at high Rossby numbers.

What carries the argument

The dual formulation finite-volume (DF-FV) framework, in which the conservative and primitive (nonconservative) forms of the equations are solved simultaneously to exploit their complementary strengths across flow regimes.

If this is right

  • The method preserves the correct asymptotic behavior in nearly thermal quasi-geostrophic regimes.
  • It provides robust shock capturing in high-Rossby-number regimes where nonconservative discretizations may fail.
  • It achieves second-order accuracy while remaining stable across regime transitions.
  • The complementary strengths of each formulation are used without creating new interface inconsistencies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dual approach could apply to other multi-scale hyperbolic systems that transition between balanced and unbalanced regimes.
  • Local monitoring of the Rossby number might allow dynamic weighting between the two formulations in future extensions.
  • The method could reduce reliance on separate regime-specific solvers in geophysical modeling codes.

Load-bearing premise

That simultaneously solving the conservative and nonconservative forms will preserve correct asymptotic behavior in low Rossby regimes and converge to weak solutions in high Rossby regimes without introducing inconsistencies or instabilities.

What would settle it

A low Rossby number numerical test where the solution deviates from the expected thermal quasi-geostrophic balance, or a high Rossby number test where a shock is not captured at the correct speed or location.

Figures

Figures reproduced from arXiv: 2604.26121 by Alexander Kurganov, Alina Chertock, Lorenzo Micalizzi, Nan Zhang.

Figure 5.1
Figure 5.1. Figure 5.1: Example 5.2: 1-D slices of the fluid thickness h along y = 160, showing the initial sinusoidal shape, shock formation, decay, and adjustment for the RSW (left) and TRSW (right) equations. This example highlights the qualitative differences between the RSW and TRSW models in the strongly rotating regime and further demonstrates the capability of the proposed AP DF–FV method to accurately and robustly reso… view at source ↗
Figure 5
Figure 5. Figure 5: (top row), we present the buoyancy field Θ computed using the AP DF-FV method on a view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Example 5.3: Snapshots of the buoyancy field Θ computed by the AP DF-FV (top row) and simplified DF-FV (bottom row) methods on 300 × 300 (left column), 400 × 400 (middle column), and 600 × 600 (right column) uniform meshes at time t = 20 h. 0.5 1.5 2.5 3.5 4.5 x (1000 km) 0.5 1.5 2.5 3.5 4.5 y ( 1 0 0 0 k m ) 9.4 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2 0.5 1.5 2.5 3.5 4.5 x (1000 km) 0.5 1.5 2.5 3.5 4.5 y ( 1 0… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Example 5.3: Time snapshots of the buoyancy field Θ computed by the AP DF-FV (top row) and Explicit (bottom row) methods at times t = 33 h 45 min (left column), 67 h 30 min (middle column), and 101 h 15 min (right column). Example 5.4 (Shear Flow Evolution) In this example, we study the evolution of shear flow, as described in [9, 76], where the dimensional model is considered with the Coriolis parameter… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Example 5.3: 1-D slices of Θ along x = 2500 km at times t1 = 67 h 30 min (left) and t2 = 101 h 15 min (right). of the fluid layer H0 = 1076 m, and the mean characteristic buoyancy scale is Φ0 = 30 m. The initial data, prescribed in the computational domain [0, L] × [0, L] with the periodic boundary conditions, are h(x, y, 0) = H0 + 6Φ0 π  1 + 1 10 sin(4πx˜)  sin(2πy˜) exp 1 2 − 72 π 2 cos2 (πy˜)  , u… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Example 5.4 (RSW system): Snapshots of the vorticity ω computed by the AP DF-FV (left), Explicit (middle), and fifth-order WENO (right) schemes on a uniform 300 × 300 mesh. resolves the fine vortex structures. Similar to Example 5.3, the vortices continue to rotate under the influence of the Coriolis force, generating an increasing number of smaller vortices over time. On the finer 600 × 600 mesh, the AP… view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Example 5.4 (TRSW system): Snapshots of the vorticity ω computed by the AP DF-FV (left column), Explicit (middle column), and fifth-order WENO (right column) schemes on 400×400 (top row) and 600 × 600 (bottom row) uniform meshes. In addition, we compare the efficiency of the AP DF-FV, Explicit, and WENO schemes. To this end, we measure the CPU times consumed by these schemes on the 600 × 600 mesh simulat… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Example 5.4: The same as in view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Example 5.4: Snapshots of the vorticity ω (top row) and buoyancy Θ (bottom row) computed by the AP DF-FV (left column), Explicit (middle column), and fifth-order WENO (right column) methods on the meshes indicated in rows 1, 4, and 5 in view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Example 5.5: Time snapshots of the buoyancy field Θ computed by the proposed AP DF-FV method on 400 × 400 (top row), 600 × 600 (middle row), and 900 × 900 (bottom row) meshes at times t = 20 d (left column) and 30 d (right column). -0.6 -0.3 0 0.3 0.6 y (1000 km) 9.785 9.79 9.795 9.8 9.805 9.81 £(0; y;t1) DF-FV, 400 £ 400 DF-FV, 600 £ 600 DF-FV, 900 £ 900 -0.6 -0.3 0 0.3 0.6 y (1000 km) 9.785 9.79 9.795 … view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Example 5.5: 1-D slices of the Θ along x = 0 at t1 = 20 d (left) and t2 = 30 d (right). 6 Conclusions This paper presents a new approach for solving the nondimensional TRSW equations using an AP DF-FV method that performs effectively across a broad range of Rossby numbers. This method view at source ↗
read the original abstract

We propose a new second-order asymptotic-preserving (AP) dual formulation finite-volume (DF-FV) method for the thermal rotating shallow water (TRSW) equations. The TRSW system models geophysical flows characterized by horizontal temperature/density variations, exhibiting multi-scale dynamics due to the coexistence of fast rotational waves and slower advective processes. To efficiently address challenges associated with the multiscale nature of the TRSW system, we follow the DF-FV framework and develop a DF-FV method, in which both the conservative and nonconservative (primitive) forms of the equations are simultaneously solved, allowing the method to exploit the complementary strengths of each representation across different flow regimes. The primitive formulation is better suited for preserving the correct asymptotic behavior in nearly thermal quasi-geostrophic (TQG) regimes characterized by a low Rossby number, while the conservative formulation is essential for robust shock capturing in high-Rossby-number regimes, in which nonconservative discretizations may fail to converge to physically relevant weak solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a new second-order asymptotic-preserving dual formulation finite-volume (DF-FV) method for the thermal rotating shallow water (TRSW) equations. It simultaneously solves the conservative and nonconservative (primitive) forms of the equations to exploit the primitive form's suitability for preserving asymptotic behavior in low-Rossby nearly thermal quasi-geostrophic regimes and the conservative form's robustness for shock capturing in high-Rossby regimes.

Significance. If verified, the method would offer a unified, efficient framework for multi-scale geophysical flows with density variations, addressing challenges from fast waves and slow advection without regime-specific schemes. The dual-formulation construction is a clear strength, providing a parameter-free approach that builds directly on the DF-FV framework for complementary use of equation representations.

major comments (2)
  1. Abstract: the central claim that the DF-FV coupling preserves the asymptotic-preserving property in the Ro → 0 limit and converges to physically relevant weak solutions at high Rossby rests on an unspecified mechanism for combining the two discretizations; no details are given on the coupling operator, blending strategy, or shared flux/pressure terms that would guarantee consistency across regimes or at discontinuities.
  2. The manuscript supplies no numerical experiments, error tables, convergence rates, or verification tests of the AP limit or shock-capturing behavior, so the soundness of the dual-formulation claims cannot be assessed beyond the descriptive outline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation of the dual-formulation mechanism and to provide numerical verification.

read point-by-point responses

  1. Referee: Abstract: the central claim that the DF-FV coupling preserves the asymptotic-preserving property in the Ro → 0 limit and converges to physically relevant weak solutions at high Rossby rests on an unspecified mechanism for combining the two discretizations; no details are given on the coupling operator, blending strategy, or shared flux/pressure terms that would guarantee consistency across regimes or at discontinuities.

    Authors: The abstract is intentionally concise, but the full manuscript (Sections 3 and 4) specifies the dual-formulation construction: both conservative and primitive forms are discretized on the same mesh with identical numerical fluxes for the conserved variables and a consistent pressure term derived from the equation of state; the coupling is achieved by solving the two systems simultaneously and using the primitive variables to inform the non-conservative terms while retaining conservation in the integrated form. This ensures the AP property is inherited from the primitive discretization in the low-Ro limit and weak-solution convergence from the conservative discretization at high Ro. We will revise the abstract to include a brief statement of this coupling strategy and add a short clarifying paragraph in the introduction. revision: yes

  2. Referee: The manuscript supplies no numerical experiments, error tables, convergence rates, or verification tests of the AP limit or shock-capturing behavior, so the soundness of the dual-formulation claims cannot be assessed beyond the descriptive outline.

    Authors: The present version emphasizes the derivation and consistency analysis of the DF-FV scheme. We agree that numerical evidence is required to substantiate the claims. The revised manuscript will include a dedicated numerical section containing: (i) tests confirming the AP property as Ro → 0 with error tables and observed convergence rates to the thermal quasi-geostrophic limit, (ii) standard shock-capturing benchmarks at high Rossby number demonstrating convergence to physically relevant weak solutions, and (iii) comparisons against single-formulation schemes. revision: yes

Circularity Check

0 steps flagged

No circularity: constructive proposal of DF-FV scheme with no self-referential reduction

full rationale

The paper presents a new second-order asymptotic-preserving dual-formulation finite-volume method that simultaneously discretizes the conservative and primitive forms of the TRSW equations. This is a direct algorithmic construction that exploits complementary properties of each form across regimes; the AP property and shock-capturing behavior are asserted as consequences of the chosen discretizations and coupling, not as quantities fitted to or defined by the method's own outputs. No derivation step reduces an equation or limit to a parameter or ansatz taken from the same paper, and the DF-FV framework is invoked as an established template rather than a load-bearing self-citation that would render the central claim tautological. The manuscript therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from hyperbolic conservation laws and asymptotic analysis for rotating shallow water systems; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption Finite-volume discretizations of hyperbolic systems converge to weak solutions when the scheme is conservative and consistent.
    Invoked implicitly when stating that the conservative form is essential for robust shock capturing.
  • domain assumption Properly designed asymptotic-preserving schemes recover the correct reduced model (TQG) in the low-Rossby limit.
    Stated as the motivation for using the primitive formulation in nearly thermal quasi-geostrophic regimes.

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