Abstract
In this Ph.D., we investigate two main research problems: (i) the design of stabilization patches for higher-order discontinuous-Galerkin (DG) methods applied to highly nonlinear free-surface shallow-water flows, (ii) the construction of a new numerical approximation strategy for the simulation of nonlinear interactions between waves in a free-surface shallow flow and a partly immersed floating object. The stabilization methods developed in the first research line are used in the second part of this work.High-order discontinuous-Galerkin (DG) methods generally suffer from a lack of nonlinear stability in the presence of singularities in the solution. Such singularities may be of various kinds, involving discontinuities, rapidly varying gradients or the occurence of dry areas in the particular case of free-surface flows. In the first part of this work, we introduce two new stabilization methods based on the use of Finite-Volume Subcells in order to alleviate these robustness issues. The first method relies on an a priori limitation of the DG scheme, together with the use of a TVB slope-limiter and a PL. The second one is built upon an a posteriori correction strategy, allowing to surgically detect the incriminated local subcells, together with the robustness properties of the corresponding lowest-order Finite-Volume scheme. This last strategy allows to ensure the nonlinear stability of the DG scheme in the vicinity of discontinuities, as well as the positivity of the discrete water-height, while preserving the subcell resolution of the initial scheme. This second strategy is also preliminary investigated in the two dimensional horizontal case. An extensive set of test-cases assess the validity of this approach.In the second part, we introduce a new numerical strategy designed for the modeling and simulation of nonlinear interactions between surface waves in shallow-water and a partially immersed surface piercing object. At the continuous level, the flow located in the textit{exterior} domain is globally modeled with the nonlinear hyperbolic shallow-water equations, while the description of the flow beneath the object reduces to a nonlinear ordinary differential equation. The coupling between the flow and the object is formulated as a free-boundary problem, associated with the computation of the time evolution of the spatial locations of the air-water-body interface. At the discrete level, the proposed formulation relies on an arbitrary-order discontinuous Galerkin approximation, which is stabilized with the a posteriori Local Subcell Correction method through low-order finite volume scheme introduced in the first part. The time evolution of the air-water-body interface is computed from an Arbitrary-Lagrangian-Eulerian (ALE) description and a suitable smooth mapping between the original frame and the current configuration. For any order of polynomial approximation, the resulting algorithm is shown to: (1) preserves the Discrete Geometric Conservation Law, (2) ensures the preservation of the water-height positivity at the subcell level, (3) preserves the class of motionless steady states (well-balancing), possibly with the occurrence of a partially immersed object.Several numerical computations and test-cases are presented, highlighting that the proposed numerical model(1) effectively allows to model all types of wave / object interactions, (2) efficiently provides the time-evolution of the air-water-body contact points and accordingly redefine the new mesh-grid thanks to ALE method (3) accurately handles strong flow singularities without any robustness issues, (4) retains the highly accurate subcell resolution of discontinuous Galerkin schemes.