Search

Abstract:: The aim of this paper is to present a numerical scheme to compute Saint-Venant equations with a source term, due to the bottom topography, in a one-dimensional framework, which satisfies the following theoretical properties: it preserves the steady state of still water, satisfies an entropy inequality, preserves the non-negativity of the height of water and remains stable with a discontinuous bottom. This is achieved by means of a kinetic approach to the system, which is the departing point of the method developed here. In this context, we use a natural description of the microscopic behavior of the system to define numerical fluxes at the interfaces of an unstructured mesh. We also use the concept of cell-centered conservative quantities (as usual in the finite volume method) and upwind interfacial sources as advocated by several authors. We show, analytically and also by means of numerical results, that the above properties are satisfied.

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non-linear diffusion process. In this way we derive new L^∞-estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non‐linear diffusion process. In this way we derive new L∞‐estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.

Numerical modeling of debris avalanche is presented here. The model uses the long‐wave approximation based on the small aspect ratio of debris avalanches as in classical Saint Venant model of shallow water. Depth‐averaged equations using this approximation are derived in a reference frame linked to the topography. Debris avalanche is treated here as a single‐phase, dry granular flow with Coulomb‐type behavior. The numerical finite volume method uses a kinetic scheme based on the description of the microscopic behavior of the system to define numerical fluxes at the interfaces of a finite element mesh. The main advantage of this method is to preserve the height positivity. The originality of the presented scheme stands in the introduction of a Dirac distribution of particles at the microscopic scale in order to describe the stopping of a granular mass when the driving forces are under the Coulomb threshold. Comparisons with analytical solutions for dam break problems and experimental results show the efficiency of the model in dealing with significant discontinuities and reproducing the flowing and stopping phase of granular avalanches. The ability of the model to describe debris avalanche behavior is illustrated here by schematic numerical simulation of an avalanche over simplified topography. Coulomb‐type behavior with constant and variable friction angle is compared in the framework of this simple example. Numerical tests show that such an approach not only provides insights into the flowing and stopping stage of the granular mass but allows observation of interesting behavior such as the existence of a fluid‐like zone behind a stopped solid‐like granular mass in specific situations, suggesting the presence of horizontal surfaces in the deposited mass.