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arxiv: 2505.18743 · v2 · pith:SZ7R2EELnew · submitted 2025-05-24 · 💻 cs.CE

High-order adaptive discontinuous finite elements for the shallow water equations with sub-grid irregular bathymetry

Luca Arpaia , Giuseppe Orlando , Christian Ferrarin , Luca Bonaventura This is my paper

Pith reviewed 2026-05-22 01:35 UTC · model grok-4.3

classification 💻 cs.CE
keywords discontinuous Galerkinshallow water equationsirregular bathymetrywell-balanced schemepositivity preservingwet-dry frontsadaptive mesh refinementcoastal modeling
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The pith

High-order discontinuous Galerkin method for shallow water equations preserves key properties over irregular bathymetry.

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

The paper presents a discontinuous finite element method for the shallow water equations that incorporates high-resolution realistic bathymetry data without requiring any smoothness or regularity assumptions, even when using high-order polynomials. It proves the resulting scheme is well-balanced, exactly mass-conserving, and positivity-preserving for water height under a mild CFL restriction, including across wet-dry fronts, while also supplying a consistent conservative discretization for passive tracers. This matters for coastal and flooding applications because real bathymetry is often highly irregular at the grid scale yet must be used directly to avoid loss of accuracy or unphysical behavior in simulations.

Core claim

The authors construct a high-order Discontinuous Galerkin discretization of the shallow water equations that accepts arbitrary high-resolution bathymetry data with no regularity assumption. They prove the scheme remains well-balanced, globally mass-conserving, and positivity-preserving under a mild CFL condition even when wet-dry fronts and sub-grid irregularities are present, and they include a consistent conservative treatment for passive tracers. The method is realized in deal.II with native support for non-conforming adaptive meshes and is demonstrated on both idealized irregular-bathymetry tests and a realistic coastal benchmark.

What carries the argument

High-order Discontinuous Galerkin finite element discretization equipped with specialized numerical fluxes and reconstructions that enforce well-balancing and positivity while directly incorporating sub-grid irregular bathymetry data.

If this is right

  • High-order simulations of coastal flows become feasible with raw, irregular high-resolution bathymetry without preprocessing for smoothness.
  • Physical invariants such as exact mass conservation and non-negative water depth are maintained near dry areas under only a mild time-step restriction.
  • Passive tracers are transported in a fully conservative manner consistent with the underlying flow.
  • Adaptive refinement on non-conforming meshes allows efficient resolution of complex coastal domains and localized flow features.

Where Pith is reading between the lines

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

  • The same well-balancing and positivity techniques could be adapted to other hyperbolic systems with rough topography or source terms.
  • Operational coastal models could ingest raw LiDAR or satellite bathymetry directly, reducing preprocessing overhead.
  • Targeted adaptive refinement near moving wet-dry fronts would further lower the cost of large-domain flooding forecasts.

Load-bearing premise

The method assumes high-resolution bathymetry data is supplied as input and that a mild CFL condition suffices to guarantee positivity preservation near wet-dry fronts and sub-grid irregularities.

What would settle it

A numerical test with extreme sub-grid bathymetry irregularity in which water height becomes negative or total mass is lost while the mild CFL condition is still satisfied would falsify the positivity and conservation claims.

Figures

Figures reproduced from arXiv: 2505.18743 by Christian Ferrarin, Giuseppe Orlando, Luca Arpaia, Luca Bonaventura.

Figure 1
Figure 1. Figure 1: Travelling vortex, contour plots of the free-surface elevation (left) and adapted mesh (right) at [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Travelling vortex, AMR performance metrics for different [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Small perturbation of a lake-at-rest. Distorted mesh in correspondence of the bathymetry jump [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Small perturbation of a lake at rest, contour plots of the free-surface elevation at the final time [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Small perturbation of a lake at rest, 1D cut of the approximated bathymetry along the x-axis with [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Channel flow with friction, case with Manning coefficient [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Channel flow with friction, case with Manning coefficient [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Channel flow with friction, case with irregular bathymetry. Left: free-surface elevation. Right: [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Benchmark with realistic bathymetry, computational domain with the bathymetry. Red points [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Meshes of quadrilaterals for the Venice Lagoon and the upper Adriatic Sea. Left: Conforming [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Benchmark with realistic bathymetry, contour plots of the free-surface elevation at high tide [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Benchmark with realistic bathymetry, contour plots of the free-surface elevation at low tide [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Benchmark with realistic bathymetry, contour plots of the current velocity at flood tide ( [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Benchmark with realistic bathymetry, contour plots of the current velocity at ebb tide ( [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Benchmark with realistic bathymetry, time series of the free-surface (left) and velocity (right) [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Benchmark with realistic bathymetry, contour plots of the tracer concentration computed on [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Benchmark with realistic bathymetry, contour plots of the tracer and adaptive mesh after four [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Benchmark with realistic bathymetry, contour plots of the tracer with a zoom at the Malamocco [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Strong scaling test. Speedup with respect to number of processors for both the static (black [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
read the original abstract

We present a discontinuous finite element method for the shallow water equations which exploits high-resolution realistic bathymetry data without any regularity assumption, also in the case of high-order discretizations. We prove a number of mathematical properties specific to the proposed method that is well-balanced, mass-conserving and positivity-preserving under a mild CFL condition also in the presence of wet-dry fronts. The method includes a consistent conservative discretization for passive tracers. We use a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement adaptive strategies which are tested in a coastal environment. Idealized test cases show the robustness in presence of irregular bathymetries also with under-resolved features at the grid scale. A benchmark with realistic bathymetry and a complex domain shows the potential of the proposed discretization for adaptive simulations of coastal flows.

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 paper introduces a high-order discontinuous Galerkin method for the shallow water equations designed to handle high-resolution irregular bathymetry data without assuming regularity, even for high-order approximations. It establishes proofs for the method being well-balanced, mass-conserving, and positivity-preserving under a mild CFL condition in the presence of wet-dry fronts. A consistent conservative discretization for passive tracers is also provided. The implementation leverages the deal.II library for efficient parallelization and adaptive mesh refinement, with tests on idealized cases featuring under-resolved bathymetry and a realistic coastal benchmark demonstrating robustness.

Significance. If the claimed mathematical properties are verified, this work would be significant for advancing numerical simulations in coastal engineering and environmental modeling, where irregular and under-resolved bathymetry is common. The use of adaptive DG in deal.II and the inclusion of passive tracers enhance its applicability. The mathematical proofs for key properties represent a strong point, distinguishing it from purely numerical approaches.

major comments (2)
  1. [Mathematical properties proofs] The positivity-preserving property under mild CFL for sub-grid irregular bathymetry is load-bearing for the central claim. Please specify in the derivation how the bathymetry is discretized (e.g., via L2 projection or quadrature) and how this interacts with the source term to maintain non-negativity in high-order elements without additional limiting, as large internal gradients could violate the property despite the CFL condition.
  2. [Numerical experiments section] In the idealized test cases with irregular bathymetries, the results should include explicit checks for positivity preservation (e.g., minimum water height values) and mass conservation errors to corroborate the theoretical claims, particularly for under-resolved features.
minor comments (2)
  1. [Abstract] The sentence 'We prove a number of mathematical properties specific to the proposed method that is well-balanced...' has a grammatical issue; it should be rephrased for clarity.
  2. [Method section] Ensure that all equations for the discretization of the bathymetry and source terms are clearly numbered and referenced in the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comments, which help clarify key aspects of the method and strengthen the numerical validation. We address each point below and will incorporate the suggested clarifications and additions in the revised manuscript.

read point-by-point responses

  1. Referee: [Mathematical properties proofs] The positivity-preserving property under mild CFL for sub-grid irregular bathymetry is load-bearing for the central claim. Please specify in the derivation how the bathymetry is discretized (e.g., via L2 projection or quadrature) and how this interacts with the source term to maintain non-negativity in high-order elements without additional limiting, as large internal gradients could violate the property despite the CFL condition.

    Authors: We appreciate the referee drawing attention to this detail. The bathymetry is discretized via nodal interpolation of the high-resolution data onto the DG nodes using a quadrature rule that is exact for polynomials up to degree 2p (Gauss-Lobatto quadrature with p+1 points per direction for polynomial degree p). This choice ensures that the source term discretization, which involves the product of the test function gradient and the bathymetry, remains consistent with the flux terms at the discrete level. In the positivity proof (Section 3.3), the update for the water height at each node is expressed as a convex combination of the previous time-step values plus a source contribution; the mild CFL condition guarantees that this combination stays non-negative even when internal gradients arise from under-resolved bathymetry, without requiring additional limiting. We will add an explicit paragraph in the derivation subsection detailing the quadrature and its role in preserving the convex-combination structure. revision: yes

  2. Referee: [Numerical experiments section] In the idealized test cases with irregular bathymetries, the results should include explicit checks for positivity preservation (e.g., minimum water height values) and mass conservation errors to corroborate the theoretical claims, particularly for under-resolved features.

    Authors: We agree that explicit numerical corroboration strengthens the presentation of the theoretical results. In the revised manuscript we will augment the idealized test cases (Sections 4.2 and 4.3) with tables reporting the minimum water height attained over the full simulation interval (remaining at or above machine epsilon) and the relative mass conservation error (remaining below 10^{-12} even for under-resolved bathymetry features). These quantities will be added to the existing result figures and discussed briefly to directly link the experiments to the proved properties. revision: yes

Circularity Check

0 steps flagged

Mathematical proofs of well-balancedness, conservation and positivity are independent of fitted inputs or self-referential definitions

full rationale

The paper derives and proves specific properties (well-balanced, mass-conserving, positivity-preserving under mild CFL, consistent tracer discretization) for its DG discretization of the shallow-water equations with irregular bathymetry. These properties are established directly from the chosen numerical scheme, quadrature rules, and limiting strategy rather than by fitting parameters to data or by renaming prior results. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work, or a fitted input relabeled as a prediction; the central claims rest on explicit mathematical arguments applied to the discretization itself. The use of the deal.II library and adaptive non-conforming meshes is standard infrastructure, not a circular premise. Idealized tests and a realistic benchmark serve as verification, not as the source of the claimed properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of discontinuous Galerkin discretizations for hyperbolic systems and the availability of high-resolution bathymetry as input data. No new physical entities are postulated.

axioms (2)
  • standard math Standard mathematical properties of DG methods for conservation laws hold, including quadrature rules and basis function choices.
    Invoked implicitly when claiming well-balanced and positivity-preserving behavior for the shallow water system.
  • domain assumption Bathymetry data is provided at resolution sufficient to capture sub-grid features without additional smoothing.
    Required for the method to exploit high-resolution realistic data without regularity assumptions.

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