[Paper Review] Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model

Florent Chazel, David Lannes+1 more|arXiv (Cornell University)|Apr 20, 2010

Nonlinear Waves and Solitons15 references4 citations

TL;DR

This paper proposes a three-parameter Green-Naghdi model with enhanced frequency dispersion properties to accurately simulate strongly nonlinear and dispersive wave propagation over submerged topography. Using a hybrid finite-volume and finite-difference splitting scheme, the model outperforms the standard one-parameter formulation—particularly in capturing high-harmonic content—demonstrating superior agreement with experimental data at gauges behind a submerged bar where dispersive effects dominate.

ABSTRACT

We investigate here the ability of a Green-Naghdi model to reproduce strongly nonlinear and dispersive wave propagation. We test in particular the behavior of the new hybrid finite-volume and finite-difference splitting approach recently developed by the authors and collaborators on the challenging benchmark of waves propagating over a submerged bar. Such a configuration requires a model with very good dispersive properties, because of the high-order harmonics generated by topography-induced nonlinear interactions. We thus depart from the aforementioned work and choose to use a new Green-Naghdi system with improved frequency dispersion characteristics. The absence of dry areas also allows us to improve the treatment of the hyperbolic part of the equations. This leads to very satisfying results for the demanding benchmarks under consideration.

Motivation & Objective

• To develop a Green-Naghdi model with improved frequency dispersion characteristics for strongly nonlinear and dispersive wave problems.
• To address the limitations of existing models in accurately capturing high-order harmonics generated by nonlinear interactions over complex topography.
• To enhance numerical stability and accuracy in simulating wave propagation over submerged bars, especially in regions where high harmonics are fully released.
• To optimize the model for uneven bottom topography by introducing additional adjustable parameters beyond the standard α parameter.
• To validate the new model against experimental benchmarks involving wave shoaling, steepening, and dispersive wave generation behind a submerged bar.

Proposed method

• Derive a new family of Green-Naghdi equations with three adjustable parameters (α, θ, γ) that improve linear dispersion relations compared to the classical one-parameter formulation.
• Use a splitting scheme that separates hyperbolic (nonlinear shallow water) and dispersive components, solving them sequentially with second-order accuracy.
• Apply a hybrid numerical approach: finite-volume method for the hyperbolic part and finite-difference method for the dispersive terms, enabling robust treatment of dry areas.
• Optimize the three-parameter model using linear shoaling conditions to match experimental dispersion behavior, particularly for high wavenumbers (kh₀ ≈ 4).
• Implement absorbing boundary conditions at the inflow and sponge layers at the outflow to minimize wave reflections and ensure stable wave generation.
• Use the optimized three-parameter model (α=1, θ=0.207, γ=0.071) for simulations, comparing results with the standard one-parameter model (α=1.159) and experimental data.

Experimental results

Research questions

• RQ1 Can a three-parameter Green-Naghdi model significantly improve the accuracy of wave dispersion properties compared to the classical one-parameter formulation?
• RQ2 How well does the new model capture the generation and evolution of high-order harmonics during wave propagation over a submerged bar?
• RQ3 Does optimizing the model for linear shoaling behavior enhance its performance in predicting wave profiles behind a submerged bar where dispersive effects are dominant?
• RQ4 Can the hybrid finite-volume/finite-difference splitting scheme maintain accuracy and stability in the presence of strong nonlinearity and variable bathymetry?
• RQ5 To what extent does the improved dispersion of the three-parameter model reduce errors in wave profile prediction at gauges located far behind the submerged bar?

Key findings

• The three-parameter Green-Naghdi model (α=1, θ=0.207, γ=0.071) shows significantly better agreement with experimental data at gauge #11, located behind the submerged bar, where high-harmonic content is fully released.
• The one-parameter model (α=1.159) fails to accurately reproduce the wave profile at gauge #11, under- or over-estimating amplitudes despite good performance at earlier gauges.
• At gauge #9 (just behind the bar), the three-parameter model provides a slightly better match to measurements than the one-parameter model, indicating improved resolution of dispersive effects.
• The model optimized for flat bottoms (equation 2.12) yields less accurate results than the three-parameter model optimized for uneven bottoms, indicating that linear shoaling optimization is essential for this benchmark.
• The improved dispersion properties of the three-parameter model allow it to accurately capture wave profiles up to kh₀ ≈ 4, where standard Boussinesq-type models typically fail.
• The hybrid numerical scheme enables stable and accurate simulation of wave propagation over the entire domain, including regions with strong nonlinearity and dispersive effects, without spurious reflections or instabilities.