This work is devoted to the numerical simulation of Shallow Water Equations involving dry areas, a moving shoreline and in the context of mesh adaptation. The space and time discretization using the Runge-Kutta Discontinuous Galerkin approach is applied to nonlinear hyperbolic Shallow Water Equations. Problems with dry areas are challenging for such methods. To counter this issue, special treatment is applied around the shoreline. This work compares three treatments, one based on Slope Modification, one based on p-adaptation and the last one based on eXtended Finite Element methods and mesh adaptation.

This work examines the physics of free-surface flow and groundwater flow within a coupled model. Coupled models for such phenomena are not clearly justified, and there is a lack of precision in the derivation of such models. The primary objective of this work is to derive a coupled model of the shallow water equations (SWE) and Richards’ equation (RE) using asymptotic considerations. The numerical coupling approach chosen for the unified model will be described as a parallel coupling. Additionally, numerical considerations regarding how to solve this model using the discontinuous Galerkin (DG) methods will be provided. Furthermore, the exchange of information between the two models, which are time-synchronized, will be explained. The solution of RE coupled with SWE, following the described procedure and implemented using the DG formulation, is integrated into RIVAGE (an in-house numerical code based on the DG method). This implementation is then tested on a numerical problem and validated against an experimental benchmark.

This study introduces a model based on Richards' equation to describe variably-saturated beach groundwater flow. The surface wave propagation is computed by the phase-resolving non-hydrostatic SWASH code. The SWASH data are used to make a suitable dynamic boundary condition at the beach face to force Richards' equation. The latter is solved by a weighted discontinuous Galerkin method together with adaptive mesh refinement. The model is validated by comparison with a laboratory experiment of a transient water table recharge problem. Then, the BARDEX II prototype-scale experiment is considered to assess the model abilities for beach groundwater dynamics. The barrier beach is studied for three cases with different lagoon levels. Steady-state results with no-wave conditions show excellent agreement. Transient waves simulations are evaluated in terms of pressure heads, saturations, water table position and groundwater velocities for time-averaged, swash-resolving and spectral analysis. Results bring interesting insights about beach groundwater modelling by comparison with the experimental data as well as a Darcy's equation-based model. A first investigation is carried out to assess the groundwater effect on the bed sediment dynamics through the modification of sediment relative weight.

Numerical solution of Richards' equation remains challenging to get robust, accurate and cost-effective results, particularly for moving sharp wetting fronts. An adaptive strategy for both space and time is proposed to deal with 2D sharp wetting fronts associated with varying and possibly vanishing diffusivity caused by nonlinearity, heterogeneity and anisotropy. Adaptive time stepping makes nonlinear convergence reliable and backward difference formula provides high-order time scheme. Adaptive mesh refinement tracks wetting fronts with an a posteriori error indicator. The novelty of this paper consists in using this technique in combination with a weighted discontinuous Galerkin framework to better approximate steep wetting fronts by a discontinuity. The potential of the overall approach is shown through various examples including analytical and laboratory benchmarks and simulation of full-scale multi-materials dam wetting experiment.

Numerical solution of Richards' equation remains challenging to get robust, accurate and cost-effective results, particularly for moving sharp wetting fronts. An adaptive strategy for both space and time is proposed to deal with 2D sharp wetting fronts associated with varying and possibly vanishing diffusivity caused by nonlinearity, heterogeneity and anisotropy. Adaptive time stepping makes nonlinear convergence reliable and backward difference formula provides high-order time scheme. Adaptive mesh refinement tracks wetting fronts with an a posteriori error indicator. The novelty of this paper consists in using this technique in combination with a weighted discontinuous Galerkin framework to better approximate steep wetting fronts by a discontinuity. The potential of the overall approach is shown through various examples including analytical and laboratory benchmarks and simulation of full-scale multi-materials dam wetting experiment.

A groundwater model is developed to simulate flows under the swash zone of sandy beach. Variably saturated beach is described by Richards equation, which is solved by the Rivage code thanks to a discontinuous Galerkin method. The SWASH code is used to simulate wave propagation in nearshore waters and compute suitable boundary conditions at the beachface to force Richards equation. An idealized case is considered to investigate the beach groundwater response to the action of swash cycles associated with long IG-like waves: infiltration/exfiltration, water table, hydraulic head and pore velocities are outlined. A good qualitative agreement is found with observations from experimental studies in the literature.

Beside a validation step of numerical models regarding academic tsunami test cases, one of the topic of the TANDEM project aims to qualify long distance propagation models on real tsunami events.In this context, the Saint-Venant model is used to simulate the 2011 Tohoku tsunami in Japan.The numerical resolution is based on a well-balanced finite volumes solver on unstructured mesh.The results are compared to available near and far shore in-situ data.The numerical accuracy is improved by the use of an adaptive mesh refinement method.The relevancy of a non-dispersive model for this event is then discussed.

In this work, a parallel finite volume scheme on unstructured meshes is applied to fluid flow for multidimensional hyperbolic system of conservation laws. It is based on a block-based adaptive mesh refinement strategy which allows quick meshing and easy parallelisation. As a continuation and as an extension of a previous work, the useful numerical density of entropy production is used as mesh refinement criterion combined with a local time-stepping method to preserve the computational time. Then, we numerically investigate its efficiency through several test cases with a confrontation with exact solution or experimental data.

Into the frame of the TANDEM project (Tsunamis in the Atlantic and the English ChaNnel: Definition of the Effects through numerical Modeling) a first step of the study aims for Principia to qualify different 3D CFD (Computational Fluid Dynamics) models for the simulation of tsunamis. The EOLE code solves a 3D bi-fluid flow on multi-structures meshes coupled with a free surface tracking VOF model, whereas the EOLENS code is based on an interface capturing method using unstructured meshes. This paper presents some simulations / measurements comparisons for academic test cases of wave propagation and run-up. On the whole, the results are satisfactory showing the high potential of both codes to simulate tsunamis impact.

RESUME . Dans le cadre du projet TANDEM -Tsunamis en Atlantique et MaNche : Definition des Effets par Modelisation- une premiere phase de l'etude a pour but (pour Principia) de qualifier differents modeles CFD (Computational Fluid Dynamics) 3D pour la simulation de tsunamis. Un premier modele (code EOLE) resout un ecoulement bi-fluide 3D couple avec une methode de reconstruction de type VOF, sur des maillages structures multi-blocs. Un autre modele (code EOLENS) base sur une approche de capture d'interface, utilise des maillages non-structures. Cet article presente, pour differents cas tests academiques de propagation de vagues et de run-up en zone cotiere, des comparaisons entre simulations et donnees experimentales. Les resultats obtenus sur ces cas-tests sont dans l'ensemble satisfaisants et demontrent tout le potentiel de ces deux codes pour simuler en situation reelle l'impact cotier 3D d'un tsunami.

We use an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). It is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. We numerically investigate the efficiency of the scheme through several 3D dam break test cases.