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arxiv: 2606.07155 · v1 · pith:5B4GST2Vnew · submitted 2026-06-05 · 🧮 math.NA · cs.NA

Structure-Preserving Discontinuous Galerkin Methods for Stochastic Shallow Water Equations

Yekaterina Epshteyn , Akil Narayan , Yinqian Yu This is my paper

Pith reviewed 2026-06-27 21:18 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords stochastic shallow water equationsdiscontinuous Galerkin methodsstochastic Galerkinentropy stable methodswell-balanced schemesstructure-preserving discretizationhyperbolic conservation laws
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The pith

A discontinuous Galerkin-stochastic Galerkin method is entropy conservative, entropy stable, and well-balanced for stochastic shallow water equations.

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

The paper constructs a numerical method for stochastic shallow water equations that incorporates uncertainties in initial conditions and bottom topography while preserving essential mathematical structures. It merges a hyperbolicity-preserving stochastic Galerkin approach with entropy-stable discontinuous Galerkin discretizations to achieve entropy conservation or stability and the well-balanced property. This matters for producing reliable long-term simulations of geophysical flows where standard schemes risk instability or loss of physical fidelity under randomness. The construction extends two prior formulations without altering their core guarantees. Numerical tests on multiple problems confirm the method maintains accuracy and the desired properties.

Core claim

Building on the hyperbolicity-preserving stochastic Galerkin formulation for SWE and a stochastic extension of the entropy stable discontinuous Galerkin methods for skew-symmetric SWE, we develop a structure-preserving, entropy conservative, and entropy stable discontinuous Galerkin--stochastic Galerkin method for the stochastic shallow water system, with the well-balanced property.

What carries the argument

The discontinuous Galerkin--stochastic Galerkin (DG-SG) discretization that combines hyperbolicity preservation with entropy stability and well-balance.

If this is right

  • The scheme keeps the stochastic system hyperbolic, so solutions remain well-posed under uncertainty.
  • Entropy conservation or stability prevents artificial growth or decay in the discrete energy.
  • The well-balanced property ensures equilibrium states such as lake-at-rest are captured exactly up to machine precision.
  • The method applies directly to problems with random initial data and random topography without extra stabilization.

Where Pith is reading between the lines

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

  • The same structure-preserving combination could be tested on other stochastic hyperbolic systems such as Euler equations with uncertainty.
  • Because the properties are built into the discretization, the scheme may support longer integration times before accumulated errors appear.
  • Implementation on adaptive meshes or with different polynomial bases would be a direct next check of robustness.

Load-bearing premise

The hyperbolicity-preserving stochastic Galerkin formulation and the stochastic extension of entropy stable DG methods from the prior works remain valid when applied inside the new combined discretization.

What would settle it

A standard lake-at-rest test with stochastic bottom topography in which the computed solution develops spurious oscillations or violates entropy conservation would show the preservation properties do not hold.

Figures

Figures reproduced from arXiv: 2606.07155 by Akil Narayan, Yekaterina Epshteyn, Yinqian Yu.

Figure 1
Figure 1. Figure 1: Relative energy evolution for the accuracy test in Section 5.1. Left: comparison of entropy-conservative (EC) and [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean, standard deviation, quantile region, and augmented relative energy evolution for the near-dry perturbation test [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mean, standard deviation, quantile region of the water surface and discharge for the perturbation-to-lake-at-rest [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Augmented relative energy evolution for the perturbation-to-lake-at-rest problem in Section 5.4. Left: [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean, standard deviation, quantile region of the water surface and discharge plot of Section 5.5. Left: solution with [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Augmented relative energy evolution for the two-dimensional stochastic-input test in Section 5.5. Left: [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
read the original abstract

Shallow water equations (SWE) are fundamental models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. In many practical applications, uncertainties arising from initial conditions and bottom topography must be taken into account, motivating the development of stable and accurate numerical methods for stochastic SWE. Building on the hyperbolicity-preserving stochastic Galerkin formulation for SWE [Dai, Epshteyn, Narayan, 2021 SISC] and a stochastic extension of the entropy stable discontinuous Galerkin methods for skew-symmetric SWE [Fu, 2022 JSC], we develop a structure-preserving, entropy conservative, and entropy stable discontinuous Galerkin--stochastic Galerkin method for the stochastic shallow water system, with the well-balanced property. We demonstrate the accuracy, applicability, and robustness of the proposed structure-preserving algorithms through several numerical experiments.

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 / 2 minor

Summary. The manuscript develops a discontinuous Galerkin–stochastic Galerkin (DG-SG) discretization for the stochastic shallow water equations that is claimed to be structure-preserving, entropy conservative, entropy stable, and well-balanced. The construction combines the hyperbolicity-preserving stochastic Galerkin formulation of Dai et al. (2021) with a stochastic extension of entropy-stable DG methods from Fu (2022), and the properties are asserted to carry over to the combined scheme. Numerical experiments are presented to illustrate accuracy, robustness, and the well-balanced property.

Significance. If the preservation properties are rigorously established, the work would supply a theoretically grounded numerical framework for stochastic hyperbolic conservation laws with source terms, which is relevant for uncertainty quantification in geophysical flows. The explicit retention of entropy stability and well-balancing in a stochastic Galerkin setting would be a useful technical contribution beyond the two cited predecessor papers.

major comments (2)
  1. [§3] §3 (Method formulation): The central claim that hyperbolicity preservation and entropy stability transfer from the Dai et al. (2021) SG formulation and the Fu (2022) entropy-stable DG scheme to the combined DG-SG discretization is not supported by an explicit compatibility argument. The stochastic projection and the spatial DG numerical flux interact through the stochastic flux and bottom-topography source terms; without a new estimate showing that the combined operator remains entropy conservative (or stable) and that the hyperbolicity region is invariant under the joint discretization, the extension rests on an unverified assumption.
  2. [§4] §4 (Well-balanced property): The well-balanced property is asserted for the stochastic setting, but the proof sketch does not address how the stochastic Galerkin basis affects the exact cancellation between the flux and the stochastic source term when the bottom topography is itself stochastic. The argument appears to reuse the deterministic cancellation without a stochastic analogue.
minor comments (2)
  1. Notation for the stochastic basis functions and the SG projection operator is introduced without a dedicated preliminary subsection; a short table summarizing the symbols would improve readability.
  2. Figure captions for the numerical experiments should explicitly state the polynomial degree in both space and stochastic dimensions and the number of SG modes used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the theoretical foundations of the DG-SG method. We will revise the manuscript to supply the requested explicit arguments.

read point-by-point responses

  1. Referee: [§3] §3 (Method formulation): The central claim that hyperbolicity preservation and entropy stability transfer from the Dai et al. (2021) SG formulation and the Fu (2022) entropy-stable DG scheme to the combined DG-SG discretization is not supported by an explicit compatibility argument. The stochastic projection and the spatial DG numerical flux interact through the stochastic flux and bottom-topography source terms; without a new estimate showing that the combined operator remains entropy conservative (or stable) and that the hyperbolicity region is invariant under the joint discretization, the extension rests on an unverified assumption.

    Authors: We agree that an explicit compatibility argument is required. The manuscript currently asserts transfer of the properties from the cited works without a joint analysis of the stochastic projection with the DG fluxes and source terms. In revision we will add a dedicated subsection deriving the entropy identity for the combined operator and proving invariance of the hyperbolicity region under the joint discretization. revision: yes

  2. Referee: [§4] §4 (Well-balanced property): The well-balanced property is asserted for the stochastic setting, but the proof sketch does not address how the stochastic Galerkin basis affects the exact cancellation between the flux and the stochastic source term when the bottom topography is itself stochastic. The argument appears to reuse the deterministic cancellation without a stochastic analogue.

    Authors: The referee is correct that the well-balanced argument must be extended to stochastic topography. The present proof sketch reuses the deterministic cancellation and does not treat the SG expansion of the bottom. We will revise the proof to verify exact cancellation of the projected flux and source terms in the SG coefficient space when the velocity vanishes and the water height satisfies the stochastic equilibrium relation. revision: yes

Circularity Check

1 steps flagged

Central claim builds on self-cited prior frameworks without exhibited new compatibility proof

specific steps
  1. self citation load bearing [Abstract]
    "Building on the hyperbolicity-preserving stochastic Galerkin formulation for SWE [Dai, Epshteyn, Narayan, 2021 SISC] and a stochastic extension of the entropy stable discontinuous Galerkin methods for skew-symmetric SWE [Fu, 2022 JSC], we develop a structure-preserving, entropy conservative, and entropy stable discontinuous Galerkin--stochastic Galerkin method for the stochastic shallow water system, with the well-balanced property."

    The paper's central premise (development of a structure-preserving method retaining hyperbolicity, entropy conservation/stability, and well-balancedness) is justified by direct appeal to the cited prior results. The first citation overlaps in authors (Epshteyn, Narayan), and the abstract supplies no indication of a new compatibility argument or modified flux that would independently establish the properties for the combined DG-SG discretization.

full rationale

The abstract explicitly positions the new DG-SG method as an extension of two prior works, one of which shares two authors with the present paper. This creates a moderate self-citation load-bearing dependence for the claimed preservation of hyperbolicity, entropy stability, and well-balanced properties. No equations or derivations in the provided text reduce by construction to the inputs, and the extension itself may contain independent content, so the circularity remains partial rather than total. No other patterns (self-definitional, fitted predictions, etc.) are visible from the given material.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Full manuscript text unavailable; ledger populated only from abstract references to prior formulations.

axioms (2)
  • domain assumption The hyperbolicity-preserving stochastic Galerkin formulation for SWE holds as established in the cited 2021 work.
    Invoked to justify the base stochastic formulation before the new DG extension.
  • domain assumption The stochastic extension of entropy stable DG methods for skew-symmetric SWE holds as established in the cited 2022 work.
    Invoked to justify the entropy stability component before the new structure-preserving combination.

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Reference graph

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