Fabien Marche

A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model

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, which could be adapted to many physical models that are dispersive corrections of hyperbolic systems. 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.

Numerical Simulation of Strongly Nonlinear and Dispersive Waves Using a Green---Naghdi Model

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.

A Positive Preserving High Order VFRoe Scheme for Shallow Water Equations: A Class of Relaxation Schemes

The VFRoe scheme has been recently introduced by Buffard, Gallouet, and Herard [Comput. Fluids, 29 (2000), pp. 813-847] to approximate the solutions of the shallow water equations. One of the main interests of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFRoe methods perform approximate solutions in good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. Following the ideas introduced by Jin and Xin [Comm. Pure Appl. Math., 45 (1995), pp. 235-276], a relevant relaxation method is derived. The interest of this relaxation scheme is twofold. In the first hand, the relaxation scheme is shown to coincide with the considered VFRoe scheme. In the second hand, the robustness of the relaxation scheme is established, and thus the nonnegativity of the water height obtained involving the VFRoe approach is ensured. Following the same idea, a family of relaxation schemes is exhibited. Next, robust high order slope limiter methods, known as MUSCL reconstructions, are proposed. The final scheme is obtained when considering the hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure.

On the well-balanced numerical discretization of shallow water equations on unstructured meshes

We consider in this work a finite volume numerical approximation of weak solutions of the shallow water equations with varying topography, on unstructured meshes. Relying on an alternative formulation of the shallow water equations that involves the free surface as a conservative variable, instead of the water height, we introduce a simple discretization of the bed slope source term, together with some suitable conservative variables reconstructions. The resulting scheme is automatically consistent and well-balanced, for any given consistent numerical flux for the homogeneous system. We obtain a very simple formulation, which do not need to be modified when second order accuracy MUSCL reconstructions are adopted. Additionally, the positivity of the water height is preserved under a relevant stability condition, as soon as the numerical flux for the associated homogeneous system does. Numerical assessments, involving dry areas and complex geometry are performed.

A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations

We introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green-Naghdi equations over varying topography. These new Green-Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces.For these new models, we develop a high order, well balanced, and robust numerical code relying on a hybrid finite volume and finite difference splitting approach. 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 finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through an SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth.Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations.

A Numerical Scheme for a Viscous Shallow Water Model with Friction

We consider a particular viscous shallow water model with topography and friction laws, formally derived by asymptotic expansion from the three-dimensional free surface Navier-Stokes equations. Emphasize is put on the numerical study: the viscous system is regarded as an hyperbolic system with source terms and discretized using a second order finite volume method. New steady states solutions for open channel flows are introduced for the whole model with viscous and friction terms. The proposed numerical scheme is validated against these new benchmarks.

A Hybrid High-Order method for the Cahn-Hilliard problem in mixed form

In this work we develop a fully implicit Hybrid High-Order algorithm for the Cahn--Hilliard problem in mixed form. The space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces. The proposed method has several advantageous features: (i) It supports fairly general meshes possibly containing polyhedral elements and nonmatching interfaces; (ii) it allows arbitrary approximation orders; and (iii) it has a moderate computational cost thanks to the possibility of locally eliminating element-based unknowns by static condensation. We perform a detailed stability and convergence study, proving optimal convergence rates in energy-like norms. Numerical validation is also provided using some of the most common tests in the literature.

Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes

The following work is devoted to the construction and validation of a numerical scheme for the 2D shallow water system on unstructured meshes, supplemented by topography and friction source terms. Approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws, involving the water free surface and some fractions of water, accounting for the topography variations. The discretization of the friction source terms relies on the use of a modified Riemann solver for the flux computation. The resulting scheme is then corrected in order to achieve an asymptotic regime preservation. A MUSCL reconstruction is also performed to increase the space order of accuracy. The overall numerical approach is shown to be consistent, well-balanced and to satisfy a domain invariant principle. These results are assessed through several benchmark tests, involving complex geometry and varying bathymetry. In the presence of dry areas, special interest is given to the wave front speed computation, highlighting the stability of the method, even when implementing the asymptotic preserving correction.

Modeling Rapid Flood Propagation Over Natural Terrains Using a Well-Balanced Scheme

AbstractThe consequences of rapid and extreme flooding events, such as tsunamis, riverine flooding, and dam breaks show the necessity of developing efficient and accurate tools for studying these flow fields and devising appropriate mitigation plans for threatened sites. Two-dimensional simulations of these flows can provide information about the temporal evolution of water depth and velocities, but the accurate prediction of the arrival time of floods and the extent of inundated areas still poses a significant challenge for numerical models of rapid flows over rough and variable topographies. Careful numerical treatments are required to reproduce the sudden changes in velocities and water depths, evolving under strong nonlinear conditions that often lead to breaking waves or bores. In addition, new controlled experiments of flood propagation in complex geometries are also needed to provide data for testing the models and evaluating their performance in more realistic conditions. This work implements a ro...

A POSITIVE WELL‐BALANCED VFROE‐NCV SCHEME FOR NON‐HOMOGENEOUS SHALLOW‐WATER EQUATIONS

In the present work, we propose a new interpretation of the VFRoe—ncv scheme for the shallow‐water equations. We propose suitable modifications to ensure that the scheme is always well defined (at the discrepancy with its initial derivation). We enforce relevant condition to preserve the positivity of the water depth and thus we ensure the robustness of the proposed numerical procedure.