This Ph.D. thesis aims to develop numerical models for two-dimensional and three-dimensional shallow water systems over mobile beds. In order to accomplish the goal of this dissertation, the following sub-projects are defined and completed. 1: The first sub-project consists in developing a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet–dry fronts that is a complex yet practically important problem. In this task, the central-upwind scheme is extended to simulation of bed erosion and sediment transport, a modified shallow water system is adopted to improve the model, a wetting and drying scheme is proposed for tracking wet-dry interfaces and stably predict the bed erosion near wet-dry area. The shallow water, sediment transport and bed evolution equations are coupled in the governing system. The proposed model can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The additional terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate in dealing with various challenging cases. 2: The second sub-project consists in developing a novel numerical scheme for a coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The resulting 5*5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A spatially second-order and temporally third-order central-upwind scheme has been derived to discretize the conservative hyperbolic sub-system. However, such schemes need a correct evaluation of local wave speeds to avoid instabilities. To address such an issue, a mathematical result by the Lagrange theorem is used in the proposed scheme. Consequently, a computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. In addition, a special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. 3: The third sub-project consists in designing a novel scheme to estimate bed-load fluxes which can produce more accurate results than the previously reported coupled model. Using a pair of local wave speeds different from those used for the flow, a novel wave estimator in conjunction with the central upwind method is proposed and successfully applied to the coupled water-sediment system involving a rapid bed-erosion process. It was demonstrated that, in comparison with the decoupled model, applying the proposed novel scheme to approximate the bed-load fluxes can successfully avoid the numerical oscillations caused by simple and less stable schemes, e.g. simple upwind methods; in comparison with the coupled model using same flux-estimator for both hydrodynamic and morphological systems, the proposed numerical scheme successfully prevents excessive numerical diffusion for prediction of bed evolution. Consequently, the proposed scheme has advantages in terms of accuracy which are shown in several numerical tests. In addition, analytical expressions have been provided for calculating the eigenvalues of the coupled shallow-water-Exner system, which greatly enhances the efficiency of the proposed method. 4: The fourth sub-project consists in developing a three-dimensional numerical model for the simulation of unsteady non-hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics-based scheme which simulates sub- and super-critical flows. Three-dimensional velocity components are considered in a collocated arrangement with a sigma coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term. The unstructured grid in the horizontal direction and the sigma coordinate in the vertical direction facilitate the use of the model in complicated geometries. 5: The fifth sub-project consists in developing a well-balanced three-dimensional shallow water model which is able to simulate shock waves over dry bed. Due to the hydrostatic simplification of the vertical momentum equation, the governing system of equations is not hyperbolic and can not be solved using standard hyperbolic solvers. That is, one can not use a high-order Godunov-type scheme to compute all fluxes through cell-interfaces. This may cause the model to fail in simulations of some unsteady-flows with discontinuities, e.g., dam-break flows and floods. To overcome this difficulty, a novel numerical scheme for the three-dimensional shallow water equations is proposed using a relaxation approach in order to convert the system to a hyperbolic one. Thus, a high-order Godunov-type central-upwind scheme based on the finite volume method can be applied to approximate the numerical fluxes. The proposed model can also preserve the ``lake at rest'' state and positivity of water depth over irregular bottom topographies based on special reconstruction of the corresponding parameters. 6: The sixth sub-project consists in extending the result of the fifth sub-project to development of a three-dimensional numerical model for shallow water flows over mobile beds, which is able to simulate morphological evolutions under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations using a finite volume method on prismatic cells in sigma coordinates based on the scheme prposed in sub-project 5. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. The performance of the proposed model has been demonstrated by several laboratory experiments of dam-break flows over mobile beds.

On étudie dans le cadre de la thèse une famille de schémas numériques permettant de résoudre les équations de Saint-Venant. Ces schémas utilisent une décomposition d'opérateur de type Lagrange-projection afin de séparer les ondes de gravité et les ondes de transport. Un traitement implicite du système acoustique (relié aux ondes de gravité) permet aux schémas de rester stable avec de grands pas de temps. La correction des flux de pression rend possible l'obtention d'une solution approchée précise quel que soit le régime d'écoulement vis-à-vis du nombre de Froude. Une attention toute particulière est portée sur le traitement du terme source qui permet la prise en compte de l'influence de la topographie. On obtient notamment la propriété dite équilibre permettant de conserver exactement certains états stationnaires, appelés état du "lac au repos". Des versions 1D et 2D sur maillages non-structurés de ces méthodes ont été étudiées et implémentées dans un cadre volumes finis. Enfin, une extension vers des méthodes ordres élevés Galerkin discontinue a été proposée en 1D avec des limiteurs classiques ainsi que combinée avec une boucle MOOD de limitation a posteriori. / In this thesis we study a family of numerical schemes solving the shallow water equations system. These schemes use a Lagrange-projection like splitting operator technique in order to separate the gravity waves and the transport waves. An implicit-explicit treatment of the acoustic system (linked to the gravity waves) allows the schemes to stay stable with large time step. The correction of the pressure fluxes enables the obtain of a precise approximation solution whatever the regime flow is with respect to the Froude number. A particular attention has been paid over the source term treatment which permits to take the topography into account. We especially obtain the so-called well-balanced property giving the exact conservation of some steady states, namely the "lake at rest" state. 1D and 2D versions of this methods have been studied and implemented in the finite volumes framework. Finally, a high order discontinuous Galerkin extension has been proposed in 1D with classical limiters along with a combined MOOD loop a posteriori limiting strategy.