Introduction to the special issue on numerical methods and applications for waves in coastal environments

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On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids

International audienceThis work aims to supplement the realization and validation of a higher-order well-balanced unstructured finite volume (FV) scheme, that has been relatively recently presented, for numerically simulating weakly non-linear weakly dispersive water waves over varying bathymetries. We in-vestigate and develop solution strategies for the sparse linear system that appears during this FV discretisation of a set of extended Boussinesq-type equations on unstructured meshes. The resultant linear system of equations must be solved at each discrete time step as to recover the actual velocity field of the flow and advance in time. The system’s coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the undisturbed water depth and the mesh topology. To this end, we investigate the application of different well-known iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. The iiterative methods considered are the GMRES and the BiCGSTAB, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is finally proposed which combines the use of the BiCGSTAB method with the ILUT preconditioner and the Reverse Cuthill–McKee reordering

On wave breaking for Boussinesq-type models

International audienceWe consider the issue of wave breaking closure for Boussinesq type models, and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical set-up, and in particular to mesh size. For relatively classical choices of weakly dispersive propagation models, we compare two closure strategies. The first is the hybrid method consisting in suppressing the dispersive terms in breaking regions, as initially suggested by Tonelli and Petti in 2009. The second is an eddy viscosity approach based on the solution of a a turbulent kinetic energy. The formulation follows early work by O. Nwogu in the 90’s, and some more recent developments by Zhang and co-workers (Ocean Mod. 2014), adapting it to be consistent with the wave breaking detection used here. We perform a study of the behavior of the two closures for different mesh sizes, with attention to the possibility of obtaining grid independent results. Based on a classical shallow water theory, we also suggest some monitors to quantify the different contributions to the dissipation mechanism, differentiating those associated to the scheme from those of the partial differential equation. These quantities are used to analyze the dynamics of dissipation in some classical benchmarks, and its dependence on the mesh size. Our main results show that numerical dissipation contributes very little to the the results obtained when using eddy viscosity method. This closure shows little sensitivity to the grid, and may lend itself to the development and use of non-dissipative/energy conserving numerical methods. The opposite is observed for the hybrid approach, for which numerical dissipation plays a key role, and unfortunately is sensitive to the size of the mesh. In particular, when working, the two approaches investigated provide results which are in the same ball range and which agree with what is usually reported in literature. With the hybrid method, however, the inception of instabilities is observed at mesh sizes which vary from case to case, and depend on the propagation model. These results are comforted by numerical computations on a large number of classical benchmarks

Sensitivity analysis for two wave breaking models used by the Green-Naghdi equation

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Low dispersion finite volume/element discretization of the enhanced Green-Naghdi equations for wave propagation, breaking and runup on unstructured meshes

International audienceWe study a hybrid approach combining a FV and FE method to solve a fully nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard P 1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in H(div) spaces. However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in H 1. We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates. Finally, the full mode, coupling a wave breaking closure previously developed by the authors, is thoroughly tested on standard benchmarks using unstructured grids with sizes comparable or coarser than those usually proposed in literature

Sur la solution numerique des systems creuses et linéaires émergents de la discretization volume finis des modeles 2D de type Boussinesq

This work supplements the realization and validation of a higher-order well balanced finite volume (FV) scheme developed for numerically simulating, on triangular meshes, weakly non-linear weakly dispersive water waves over varying bathymetries. The scheme has been recently presented by Kazolea et al. \textit{(Coastal Eng. 69:42-66, 2012 and J. Comp. Phys. 271:281-305, 2014)}. More precisely, we investigate and develop solution strategies for the sparse linear system that occurs during this FV discretisation of a set of Boussinesq-type equations on unstructured meshes. The resultant system of equations must be solved at each time step as to recover the actual velocity field of the flow. The system's coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the un-disturbed water depth and the mesh topology. This work investigates the application of different iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. Two different iterative methods, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is proposed.Ce travail concerne la réalisation et la validation d’un schéma Volumes Finis d’ordre élevé pour la simulation des vagues en régime faiblement non-linéaire et faiblement dispersife sur bathymétries variables. Le schéma implémenté est celui proposé récemment par Kazolea et al. (Coastal Eng 69:. 42-66, 2012 et J. Phys Comp 271:.. 281-305, 2014). Plus précisément, nous étudions et développons des stratégies de solution pour le système linéaire creux qui se produit au cours de la discrétisation des équations de Boussinesq sur maillagesnon structurés. Le système d’équations résultant doit être résolu à chaque pas de temps pour récupérer la vitesse. La matrice du système est creuse, non symétrique et souvent mal conditionné. Ses caractéristiques sont affectées par des quantités physiques tels que la profondeur de l’ eau au repos et la topologie du maillage. Ce travail étudie l’ application de différentes techniques itératives, avec et sans l’ utilisation de pré conditionneurs et de ré-numérotation, pour la solution de ce système linéaire creux. Deux méthodes itératives différentes, troistechniques de pré conditionnement, y compris les différents factorisations ILU et deux techniques de ré ordonnancement différentes sont mises en œuvre et évaluées. Une stratégie optimale, en termes d’efficacité de calcul et de robustesse, est proposé

Parameter sensitivity for wave breaking closures in Boussinesq-type models

We consider the issue of wave-breaking closure for the well known Green-Naghdi model and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical setup. More precisely and based on we used two closure strategies for modelling wave-breaking of a solitary wave over a slope. The first one is the hybrid method consisting of suppressing the dispersive terms in a breaking region and the second one is an eddy viscosity approach based on the solution of a turbulent kinetic energy model. The two closures use the same conditions for the triggering of the breaking mechanisms. Both the triggering conditions and the breaking models themselves use case depended / ad/hoc parameters which are affecting the numerical solution wile changing. The scope of this work is to make use of sensitivity indices computed by means of Analysis of Variance (ANOVA) to provide the sensitivity of wave breaking simulation to the variation of parameters such as the mesh size and the breaking parameters specific to each breaking model. The sensitivity analysis is performed using the UQlab framework for Uncertainty Quantification

Hybrid finite-volume/finite-element simulations of fully-nonlinear/weakly dispersive wave propagation, breaking, and runup on unstructured grids

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A hybrid FV/FD scheme for a novel conservative form of extended Boussinesq equations for waves in porous media

International audienceThis paper introduces a conservative form of the extended Boussinesq equations for waves in porous media. This model can be used in both porous and non-porous media since it does not requires any boundary condition at the interface between the porous and non-porous media. A hybrid Finite Volume/Finite Difference (FV/FD) scheme technique is used to solve the conservative form of the extended Boussinesq equations for waves in porous media. For the hyperbolic part of the governing equations, the FV formulation is applied with a Riemann solver of Roe approximation. Whereas, the dispersive and porosity terms are discretized by using FD. The model is validated with experimental data for solitary waves interacting with porous structures and a porous dam break of a one-dimensional flow