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A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model
Philippe Bonneton, Florent Chazel, David Lannes, Fabien Marche, Marion Tissier
TL;DR
The paper addresses numerical simulation of large-amplitude shallow-water waves while retaining dispersive effects and handling breaking and dry areas. It reformulates the Green-Naghdi equations and introduces a hybrid finite-volume/finite-difference splitting method. Extensive validations show successful description of shoaling, wave breaking, run-up, and back-wash.
Problem
Large-amplitude coastal waves require a model that retains dispersive effects while addressing wave breaking and vanishing water depth.
Method
The paper combines a high-order finite-volume treatment of the hyperbolic component with finite differences for the dispersive component in a splitting scheme.
Results
The approach successfully describes highly nonlinear wave propagation and shoaling, wave breaking, run-up, and back-wash in comparisons with analytical, experimental, and reference results.
Takeaways & Limitations
The shock-capturing finite-volume-based approach appears promising for studying dispersive shallow-water waves in coastal areas.
Takeaways & Limitations
For problematic time-step regions, choices with α < 1 can induce high-frequency instabilities, while the optimal α depends on δt.
Abstract
from arXiv · showhide
The fully nonlinear and weakly dispersive Green-Naghdi model for shallow water waves of large amplitude is studied. The original model is first recast under a new formulation more suitable for numerical resolution. An hybrid finite volume and finite difference splitting approach is then proposed. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a classical finite difference approach. Extensive numerical validations are then performed in one horizontal dimension, relying both on analytical solutions and experimental data. The results show that our approach gives a good account of all the processes of wave transformation in coastal areas: shoaling, wave breaking and run-up.
1. Introduction
The paper targets large-amplitude coastal waves by combining dispersive Green-Naghdi modeling with numerical treatment of breaking and vanishing water depth. It introduces a split hybrid scheme and validates it through analytical and numerical tests.
- Motivation: Large-amplitude coastal applications require a fully nonlinear model because the small-amplitude assumption is too restrictive.
- Motivation: Green-Naghdi equations describe dispersive waves up to breaking, but become unusable beyond the breaking point without modifications.
- Objectives: The method aims to preserve dispersive effects while handling wave breaking and vanishing water depth.
- Numerical approach: A second-order splitting scheme separates the hyperbolic NSWE component from the dispersive component.The hyperbolic operator uses a high-order, robust, well-balanced finite-volume relaxation method; the dispersive operator uses finite differences.
- Numerical approach: Dispersive computation is skipped near detected shocks, where third-order derivatives become singular and NSWE describe wave dynamics near breaking.
- Validation: The paper validates the model with solitary waves, periodic waves, wall reflection, and numerical tests of coastal wave transformation.
2. The physical model
The paper reformulates the Green–Naghdi equations to avoid third-order derivatives and introduces a parameterized family with improved frequency dispersion. The resulting formulation supports numerical treatment of nonlinear shallow-water waves while retaining the model’s dispersive structure.
- Physical setting: The Green–Naghdi model describes surface elevation and depth-averaged velocity using dimensionless parameters for nonlinearity, shallowness, and topography.The water depth is h = h0 + ζ − b, with ε, µ, and β representing nonlinearity, shallowness, and bottom variations.
- Reformulation: An equivalent formulation replaces the third-order derivative terms with expressions involving at most second-order derivatives.This reformulation is of practical interest because it avoids numerically stiff third-order derivatives, especially for steep waves.
- Conservative variables: The reformulated equations can also be expressed in conservative variables h and hV for numerical computation.The resulting system is then transformed again to eliminate third-order derivatives before discretization.
- Improved dispersion: Weighted optimization over kh0 ∈ [0; 4] selects the global value α = 1.159 for general wave conditions.The paper states that this value is especially suited to irregular waves and regular waves over uneven bottoms, while monochromatic flat-bottom waves may require a local optimum.
3. Numerical methods
The numerical method exploits the reformulated Green–Naghdi system through a splitting approach that separates hyperbolic and dispersive dynamics. Each component is discretized with a method suited to its numerical structure.
- Numerical methods: The proposed method combines a high-order finite-volume discretization for the hyperbolic component with high-order finite differences for the dispersive component.The finite-volume method is described as efficient, robust, and Godunov-like, while finite differences handle the dispersive terms.
3.1. The splitting scheme
The splitting scheme decomposes each time step into hyperbolic and dispersive solution operators. Its mixed discretization requires consistent conversion between cell-averaged and nodal unknowns, and the implementation is restricted to one-dimensional waves.
- Operator splitting: The solution operator is advanced with a second-order splitting that separates the hyperbolic operator S1 from the dispersive operator S2.S1 solves the nonlinear shallow-water equations, while S2 solves the remaining dispersive part.
- Scope: The numerical implementation described in this section considers only one-dimensional surface waves.The implementation for two-dimensional surface waves is left for future work.
- Spatial discretization: S1 is computed with a finite-volume method, whereas S2 is computed with finite differences.This mixed approach requires both cell-averaged and nodal representations of each unknown.
- Variable conversion: Fourth-order Taylor relations convert between cell averages and nodal values while preserving the scheme’s global spatial order.The conversion formulas preserve the rest steady state and are accurate up to O(δx4) terms.
3.2. Spacial discretization of the hyperbolic component S1(·)
The hyperbolic component uses a robust, high-order, well-balanced finite-volume scheme designed to control dissipation, dispersion, shocks, positivity, and topographic source terms. Hydrostatic reconstruction preserves motionless steady states and water-height positivity.
- Design goals: The scheme targets stable surf and swash simulations while controlling numerical dissipation and dispersion and preserving motionless steady states.It is designed for wave–topography interactions in the presence of source terms.
- Base scheme: A low-dissipation, well-balanced extension of a robust relaxation finite-volume scheme provides the hyperbolic discretization.The underlying relaxation flux is used for the numerical validations.
- High-order reconstruction: High-order interface reconstructions reduce numerical dissipation and dispersion, while a three-entry limiter maintains non-oscillatory and positive solutions near shocks.The limiter is intended to retain the highest possible order away from extrema and discontinuities.
- Well-balancing: Hydrostatic reconstruction supplies a well-balanced treatment of the topography source term.The resulting scheme preserves motionless steady states and water-height positivity.
- Well-balancing: High-order reconstruction must be applied to both conservative variables and surface elevation to achieve well-balancing together with high-order accuracy.The surface-elevation reconstruction is therefore part of the steady-state-preserving discretization.
3.3. Spacial discretization of the dispersive component S2(·)
The dispersive component S2 is solved in one dimension with a classical finite-difference method, using explicitly defined operators, fourth-order spatial derivatives, and boundary conditions from §3.5.
- The one-dimensional system associated with S2 is solved using a classical finite-difference technique.
- The operators T and Q1 are given explicitly for the dispersive system.
- Spatial derivatives are discretized with fourth-order formulae.
- Boundary conditions for the dispersive discretization are imposed using the method presented in §3.5.
3.4. Time discretization and dispersive properties
The scheme uses explicit fourth-order Runge-Kutta time integration and analyzes the splitting through its semi-discrete dispersion relation. The analysis shows second-order temporal accuracy in dispersion, purely dispersive splitting errors, and no induced linear instability, while motivating time-step-dependent optimization of α.
- Time discretization: Both S1 and S2 are integrated in time with an explicit classical fourth-order Runge-Kutta scheme.
- Dispersive properties: The semi-discretized analysis compares the splitting's linear phase and group velocities with the reference Green-Naghdi velocities.
- Dispersive properties: The semi-discretized dispersion relation approaches the exact Green-Naghdi relation at order 2 in δt.
- Dispersive properties: The O(δt^2) splitting error is always real, so the numerical errors are dispersive and do not induce linear instability.
- Optimization of α: Lax-Wendroff-inspired correction is possible, but δt-optimized α provides better results with a simpler method.
- Optimization of α: The global α = 1.159 is no longer optimal after the additional O(δt^2) term, motivating α values that depend on δt.
- Optimization of α: Figure 3 reports global and local optimal α values against eδt, with local curves parameterized by kh0.
- Optimization of α: For eδt ∈[0.26; 0.42], stable optimization requires α = 1 because computed values below 1 induce high-frequency instabilities.
3.5. Boundary conditions
Boundary conditions are implemented with ghost cells for the hyperbolic part and reflected stencil coefficients for the dispersive part, preserving regular discretization structures at boundaries. Dry-cell stabilization imposes threshold depth and zero velocity.
- Hyperbolic part: The hyperbolic part uses ghost cells with prescribed relations for periodic, homogeneous Neumann, and homogeneous Dirichlet boundaries.
- Dispersive part: The dispersive part handles boundary stencils by periodic, even, or odd reflection for periodic, Neumann, or Dirichlet conditions.
- Dispersive part: Reflection avoids decentered boundary formulae while maintaining a regular structure in the discretized model.
- Solid walls and dry areas: For solid walls, homogeneous Neumann conditions are imposed on h and h̄, while homogeneous Dirichlet conditions are imposed on hu and h̄u.
- Solid walls and dry areas: If h falls below a threshold ǫ, the scheme sets h = ǫ and v = 0 to ensure stability.
3.6. Wave breaking
Wave breaking is handled by locally switching from the Green-Naghdi equations to the nonlinear shallow water equations. The dispersive step is skipped near detected breaking fronts, so dissipation is represented by shock energy dissipation.
- Breaking treatment: The method switches locally from the Green-Naghdi equations to the NSWE by skipping the dispersive step when a wave is ready to break.
- Breaking treatment: At breaking wave fronts, only the hyperbolic part is solved, and wave-breaking dissipation is represented by shock energy dissipation.
- Breaking detection: The energy flux density is close to zero in regular regions and peaks as shocks appear, enabling localization of eventual breaking fronts.
- Breaking detection: The dispersive step is skipped only at the wave fronts identified at each time step.
4. Numerical validation
Numerical validation covers solitary-wave convergence, dispersive periodic waves, wall reflection, and nonlinear shoaling over beaches. The scheme achieves second-order convergence, agrees with reference and experimental data, and reproduces key coastal transformation processes.
- 4.1. Propagation of a solitary wave: The solitary-wave tests show convergence slopes of 1.91 for a/h0 = 0.05 and 1.83 for a/h0 = 0.2, consistent with second-order splitting.Errors are evaluated for free-surface elevation and averaged velocity against analytical solutions.
- 4.2. Propagation of periodic and regular nonlinear waves: For periodic waves over a flat bottom, α = 1.153 agrees excellently with the reference after 25 periods, while α = 1.16 remains broadly accurate but less satisfying.The test uses very dispersive, weakly nonlinear waves and compares water height and averaged velocity with stream-function reference solutions.
- 4.2. Propagation of periodic and regular nonlinear waves: The periodic-wave test demonstrates that the model handles intermediate or deep-water waves and benefits from optimizing α through the semi-discrete dispersion relation.The conclusion is supported by comparisons after 25 periods for two locally optimized parameter values.
- 4.3. Reflection of a solitary wave at a wall: Wall-reflection simulations accurately match experimental time series for solitary waves with two amplitudes, including the incident and reflected peaks.The comparison is performed near the wall at x = 17.75 m and remains accurate for the more complex nonlinear propagation case.
- 4.4. Nonlinear shoaling of solitary waves propagating over a beach: Beach-shoaling comparisons show good agreement in wave amplitude and shape, while the model reproduces shoaling, breaking, run-up, run-down, and backwash-bore formation without additional treatment.The less nonlinear case has significant errors, partly attributed to the experimental surface not being fully at rest before propagation.
5. Conclusion
The study reformulates the fully nonlinear, weakly dispersive Green–Naghdi model and combines finite-volume and finite-difference discretizations in a splitting approach. The scheme is validated against analytical and experimental references for coastal wave transformation, while its future development includes dispersion optimization, breaking-wave sensors, and two-dimensional extension.
- The original fully nonlinear Green–Naghdi model is reformulated into a form more suitable for numerical resolution.
- The proposed hybrid method combines a high-order finite-volume discretization for hyperbolic effects with high-order finite differences for dispersion.
- The semi-discrete scheme approaches the Green–Naghdi dispersion relation at order two.
- The time-step-dependent choice of α provides better results than the standard model-based choice by accounting for time-discretization dispersion.
- Extensive validation successfully describes nonlinear wave propagation, shoaling, breaking, run-up, and backwash using analytical solutions, experimental data, and reference solutions.
- Future work includes dispersion-optimized models, numerical sensors for breaking-wave detection, and two-dimensional extension of the scheme.