A combined finite volume - finite element scheme for a dispersive shallow water system

International audienceWe propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation. The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces. The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.Several numerical experiments are performed to confirm the relevance of our approach

A combined finite volume - finite element scheme for a dispersive shallow water system

International audienceWe propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation. The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces. The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.Several numerical experiments are performed to confirm the relevance of our approach

Sur les effets non-hydrostatiques en eaux profondes : application à la simulation de tsunamis

International audienceDans ce travail, on s’intéresse à la modélisation de la propagation d’une vague et plus précisément, on aimerait savoir ce qui se passe quand l’hypothèse d'eaux peu profondes n’est plus vérifiée. Souvent des modèles hydrostatiques sont utilisés dans lesquels la pression en un point est définie comme l’accélération de la pesanteur multiplié par la hauteur d’eau au dessus de ce point. Ces modèles simplifiés négligent la partie non-hydrostatique de la pression, définie comme la différence entre la pression hydrostatique et la pression réelle. On compare un modèle non-hydrostatique à un modèle classique de type Saint-Venant. En eaux profondes, des effets non-hydrostatiques apparaissent et on cherche à en décrire l’impact sur l’amplitude et la forme des vagues. Nous présenterons la dérivation du modèle à partir des équations de Navier-Stokes puis nous identifierons une solution analytique. Enfin, nous présenterons des résultats numériques afin de comparer le modèle au modèle de Saint-Venant

Vertically averaged and moment equations: new derivation, efficient numerical solution and comparison with other physical approximations for modeling non-hydrostatic free surface flows

Efficient modeling of flow physics is a prerequisite for a reliable computation of free-surface environmental flows. Non-hydrostatic flows are often present in shallow water environments, making the task challenging. In this work, we use the method of weighted residuals for modeling non-hydrostatic free surface flows in a depth-averaged framework. In particular, we focus on the Vertically Averaged and Moment (VAM) equations model. First, a new derivation of the model is presented using expansions of the field variables in sigma-coordinates with Legendre polynomials basis. Second, an efficient two-step numerical scheme is proposed: the first step corresponds to solving the hyperbolic part with a second-order path-conservative PVM scheme. Then, in a second step, non-hydrostatic terms are corrected by solving a linear Poisson-like system using an iterative method, thereby resulting in an accurate and efficient algorithm. The computational effort is similar to the one required for the well-known Serre-Green-Naghdi (SGN) system, while the results are largely improved. Finally, the physical aspects of the model are compared to the SGN system and a multilayer model, demonstrating that VAM is comparable in physical accuracy to a two-layer model.Funding for open access charge: Universidad de Málaga/CBUA. This work is partially supported by projects RTI2018-096064-B-C2(1-2), PID2020-114688RB-I00, and PID2022-137637NB-C21 funded by Ministry of Science, Innovation and Universities MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”. F. Cantero-Chinchilla was partially supported by the grant IJC2020-042646-I, funded by CIN/AEI/10.13039/501100011033 and by the European Union “NextGenerationEU/PRTR”, through the Spanish Ministry of Science, Innovation and Universities Juan de la Cierva program 2020.

Numerical simulations of a dispersive model approximating free-surface Euler equations

In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – GreenNaghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the LDNH2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projectioncorrection method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the LDNH0 model

A robust and stable numerical scheme for a depth-averaged Euler system

We propose an efficient numerical scheme for the resolution of a non-hydrostatic Saint-Venant type model. The model is a shallow water type approximation of the incompressbile Euler system with free surface and slightly differs from the Green-Naghdi model. The numerical approximation relies on a kinetic interpretation of the model and a projection-correction type scheme. The hyperbolic part of the system is approximated using a kinetic based finite volume solver and the correction step implies to solve an elliptic problem involving the non-hydrostatic part of the pressure. We prove the numerical scheme satisfies properties such as positivity, well-balancing and a fully discrete entropy inequality. The numerical scheme is confronted with various time-dependent analytical solutions. Notice that the numerical procedure remains stable when the water depth tends to zero

Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

International audienceThis work is devoted to the numerical resolution in multidimensional framework of a hierarchy of reduced models of the free surface Euler equations, also called water waves equations.The current paper, the first in a series of two, focuses on a hierarchy of monolayer dispersive models, such is the Serre-Green-Naghdi model.A particular attention is given to the dissipation of the mechanical energy at the discrete level, i.e. to design an entropy-satisfying scheme.To illustrate the accuracy and the robustness of the strategy, several numerical experiments are performed.In particular, the strategy is able to deal with dry areas without particular treatment

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

International audienceIn geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model.In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to model the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity