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arxiv: 2408.02665 · v4 · submitted 2024-08-05 · 🧮 math.NA · cs.NA

Structure-preserving approximations of the Serre-Green-Naghdi equations in standard and hyperbolic form

Hendrik Ranocha , Mario Ricchiuto This is my paper

Pith reviewed 2026-05-23 22:03 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Serre-Green-Naghdi equationsstructure-preserving methodsenergy conservationsummation-by-parts operatorswell-balanced schemesdispersive waveshyperbolic approximation
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The pith

Numerical methods for the Serre-Green-Naghdi equations conserve mass and energy using summation-by-parts operators.

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

The paper develops structure-preserving numerical methods for both the classical Serre-Green-Naghdi equations and a hyperbolic approximation that models weakly dispersive free-surface waves. These methods are shown to conserve total water mass and total energy in discrete form. For flat bathymetry the original equations also conserve total momentum, while all variants remain well-balanced at the lake-at-rest equilibrium even with variable topography. The construction works for any discretization that supplies summation-by-parts operators, and the resulting schemes demonstrate accurate long-time wave propagation on coarse meshes.

Core claim

The novel numerical methods conserve both the total water mass and the total energy. In addition, the methods for the original Serre-Green-Naghdi equations conserve the total momentum for flat bathymetry. For variable topography, all the methods proposed are well-balanced for the lake-at-rest state. The schemes are obtained by replacing continuous integration-by-parts identities with their discrete summation-by-parts counterparts, and energy-stable variants follow from consistent high-order artificial viscosity.

What carries the argument

Summation-by-parts operators that replace continuous integration by parts to enforce exact discrete conservation of mass and energy.

If this is right

  • Any discretization family (finite difference, finite element, discontinuous Galerkin, spectral) that supplies summation-by-parts operators can be turned into a mass- and energy-conserving method for the Serre-Green-Naghdi system.
  • The hyperbolic approximation yields fully explicit time stepping while retaining the same conservation properties.
  • Adding consistent artificial viscosity produces energy-stable variants without destroying the underlying conservation structure.
  • Benchmarks confirm that energy preservation yields visibly more accurate dispersive wave solutions on coarse meshes than non-structure-preserving alternatives.

Where Pith is reading between the lines

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

  • The same summation-by-parts construction could be applied to other nonlinear dispersive models whose continuous equations admit similar energy identities.
  • Long-time coastal or ocean simulations that rely on accurate energy budgets may benefit directly from these discretizations.
  • The approach supplies a template for proving conservation in related shallow-water or Boussinesq-type systems once suitable summation-by-parts operators are identified.

Load-bearing premise

That summation-by-parts operators exist in the chosen spatial discretization and that they transfer the continuous conservation identities to the discrete level without extra conditions on the nonlinear elliptic operator.

What would settle it

A numerical run on a periodic or flat-bottom domain in which the computed total energy changes by more than machine round-off after many time steps, or a lake-at-rest state that develops nonzero velocities under the proposed scheme.

Figures

Figures reproduced from arXiv: 2408.02665 by Hendrik Ranocha, Mario Ricchiuto.

Figure 1
Figure 1. Figure 1: Convergence results using finite difference semidiscretizations with [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Grid convergence to the manufactured solution for the hyperbolic system results using finite difference [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical solutions for the initial condition (147). These reference solutions are all visually converged [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical solutions obtained with second-order finite difference methods with [PITH_FULL_IMAGE:figures/full_fig_p034_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical solutions for the initial condition (148). These reference solutions are all visually converged [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical solutions obtained with second-order finite difference methods with [PITH_FULL_IMAGE:figures/full_fig_p036_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Long-time soliton propagation for the Serre-Green-Naghdi equations. Energy-conserving Fourier [PITH_FULL_IMAGE:figures/full_fig_p040_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Riemann problem using structure-preserving second-order finite differences with ∆ [PITH_FULL_IMAGE:figures/full_fig_p041_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Leading soliton waves obtained from the rectangular initial condition discretized with second-order [PITH_FULL_IMAGE:figures/full_fig_p042_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Favre waves: undular bore sketch, and definition of amplitudes [PITH_FULL_IMAGE:figures/full_fig_p043_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Favre wave: hyperbolic formulation with λ = 500, and grid spacing ∆x = 0.125. Numerical solutions (solid lines) with nonlinearity ε = 0.1 (left), ε = 0.2 (center), and ε = 0.3 (right). Top: energy-conservative fourth-order finite differences with central operators. Bottom: fourth-order finite differences with central oper￾ators and artificial viscosity. The dashed lines show the fully nonlinear potential … view at source ↗
Figure 12
Figure 12. Figure 12: Favre wave: original SGN system, and grid spacing ∆ [PITH_FULL_IMAGE:figures/full_fig_p045_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Favre waves. Solutions of the original SGN system at dimensionless time [PITH_FULL_IMAGE:figures/full_fig_p046_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Favre waves. Solutions of the original SGN system at dimensionless time [PITH_FULL_IMAGE:figures/full_fig_p047_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Favre waves. Evolution of the maximum amplitude obtained from the original SGN system with [PITH_FULL_IMAGE:figures/full_fig_p048_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Favre waves. Evolution of the maximum amplitude obtained from the original SGN system with [PITH_FULL_IMAGE:figures/full_fig_p049_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Favre waves: maximal amplitude amax against Froude number Fr = σ/p gh0. Comparison with experimental data of Favre [36] and Treske [128]. Numerical results with fourth-order structure-preserving finite differences with central operators for the hyperbolic approximation and upwind operators for the original Serre￾Green-Naghdi equations. 11.9 Dingemans experiment In this section, we compare numerical result… view at source ↗
Figure 18
Figure 18. Figure 18: Initial setup and numerical solution at time [PITH_FULL_IMAGE:figures/full_fig_p051_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Experimental data of [24, 25] and total water height [PITH_FULL_IMAGE:figures/full_fig_p052_19.png] view at source ↗
read the original abstract

We develop structure-preserving numerical methods for the Serre-Green-Naghdi equations, a model for weakly dispersive free-surface waves. We consider both the classical form, requiring the inversion of a non-linear elliptic operator, and a hyperbolic approximation of the equations, allowing fully explicit time stepping. Systems for both flat and variable topography are studied. Our novel numerical methods conserve both the total water mass and the total energy. In addition, the methods for the original Serre-Green-Naghdi equations conserve the total momentum for flat bathymetry. For variable topography, all the methods proposed are well-balanced for the lake-at-rest state. We provide a theoretical setting allowing us to construct schemes of any kind (finite difference, finite element, discontinuous Galerkin, spectral, etc.) as long as summation-by-parts operators are available in the chosen setting. Energy-stable variants are proposed by adding a consistent high-order artificial viscosity term. The proposed methods are validated through a large set of benchmarks to verify all the theoretical properties. Whenever possible, comparisons with exact, reference numerical, or experimental data are carried out. The impressive advantage of structure preservation, and in particular energy preservation, to resolve accurately dispersive wave propagation on very coarse meshes is demonstrated by several of the tests.

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

1 major / 2 minor

Summary. The manuscript develops structure-preserving numerical methods for the Serre-Green-Naghdi equations in both their classical form (involving inversion of a nonlinear elliptic operator) and a hyperbolic approximation. Using summation-by-parts (SBP) operators, the methods are designed to conserve mass and energy exactly, conserve momentum for flat bathymetry in the classical case, and remain well-balanced for the lake-at-rest state with variable topography. Energy-stable variants are introduced by adding consistent high-order artificial viscosity. The framework applies to any discretization (finite difference, finite element, discontinuous Galerkin, spectral, etc.) provided SBP operators are available, and the methods are validated on a suite of benchmarks demonstrating advantages for dispersive wave propagation on coarse meshes.

Significance. If the conservation properties hold rigorously, the work provides a general, discretization-agnostic framework for structure-preserving schemes on dispersive free-surface models. The explicit handling of both standard and hyperbolic forms, the well-balancing property, and the demonstrated accuracy on coarse meshes would be useful for long-time simulations of nonlinear waves. The availability of machine-checkable or parameter-free derivations is not claimed, but the SBP-based construction is a clear strength if the elliptic-operator cancellation is fully justified.

major comments (1)
  1. [§3 (discrete energy identity)] §3 (or the section presenting the discrete energy identity): the assertion that the inner product of the solution with the residual of the nonlinear elliptic operator vanishes identically solely from the SBP property needs explicit verification. The continuous energy conservation relies on integration by parts that cancels the elliptic term; the discrete analogue requires that this cancellation survives the chosen discretization of the elliptic operator (including any approximation or boundary treatment). Without a detailed proof or a statement of the precise compatibility conditions (e.g., exact commutation with the discrete divergence), the exact energy conservation claim remains conditional.
minor comments (2)
  1. [Abstract / §1] The abstract states that comparisons with exact, reference numerical, or experimental data are carried out 'whenever possible,' but the manuscript would benefit from a short table or list in the introduction or results section indicating which benchmarks admit exact solutions versus numerical/experimental references.
  2. [§2] Notation for the hyperbolic approximation (e.g., the relaxation parameter or the auxiliary variables) should be introduced once with a clear forward reference to the section where the hyperbolic system is derived.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on the discrete energy identity. We address the single major comment below.

read point-by-point responses

  1. Referee: §3 (or the section presenting the discrete energy identity): the assertion that the inner product of the solution with the residual of the nonlinear elliptic operator vanishes identically solely from the SBP property needs explicit verification. The continuous energy conservation relies on integration by parts that cancels the elliptic term; the discrete analogue requires that this cancellation survives the chosen discretization of the elliptic operator (including any approximation or boundary treatment). Without a detailed proof or a statement of the precise compatibility conditions (e.g., exact commutation with the discrete divergence), the exact energy conservation claim remains conditional.

    Authors: We agree that the manuscript would benefit from an explicit verification of the cancellation. While the construction in §3 is designed so that the SBP property directly implies the required inner-product identity (by mimicking the continuous integration-by-parts step for the chosen discretization of the elliptic operator), we acknowledge that the precise compatibility conditions on boundary terms and commutation with the discrete divergence operator are not spelled out in full detail. In the revised manuscript we will insert a short lemma (or expanded paragraph) in §3 that states the exact assumptions on the SBP operators and the elliptic discretization under which the inner product vanishes identically, together with a one-page derivation of the discrete energy identity. This addition will make the claim unconditional and machine-checkable from the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity; conservation derived from SBP properties on continuous equations

full rationale

The paper constructs structure-preserving discretizations for the Serre-Green-Naghdi system (both classical and hyperbolic forms) by applying summation-by-parts operators that replicate the continuous integration-by-parts identities for mass, energy, and momentum. This yields the claimed conservation statements directly from the PDE structure without fitted parameters, self-referential predictions, or load-bearing self-citations. The theoretical setting is stated as general for any SBP-equipped discretization (FD, FE, DG, spectral) and is validated on benchmarks against external data. No step reduces by construction to its own inputs; the elliptic-operator cancellation is asserted to follow from the SBP inner-product property alone, which is an independent algebraic fact rather than a redefinition. The reader's score of 2.0 is consistent with this self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard mathematical properties of summation-by-parts operators and the continuous conservation laws of the Serre-Green-Naghdi system; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract description.

axioms (1)
  • standard math Summation-by-parts operators satisfy a discrete analog of integration by parts that enables exact conservation when applied to the Serre-Green-Naghdi system.
    Invoked to construct schemes that inherit mass, energy, and momentum preservation from the continuous level.

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