François Bouchut

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

On the use of Saint Venant equations to simulate the spreading of a granular mass

[1]xa0Cliff collapse is an active geomorphological process acting at the surface of the Earth and telluric planets. Recent laboratory studies have investigated the collapse of an initially cylindrical granular mass along a rough horizontal plane for different initial aspect ratios a = Hi/Ri, where Hi and Ri are the initial height and radius, respectively. A numerical simulation of these experiments is performed using a minimal depth-integrated model based on a long-wave approximation. A dimensional analysis of the equations shows that such a model exhibits the scaling laws observed experimentally. Generic solutions are independent of gravity and depend only on the initial aspect ratio a and an effective friction angle. In terms of dynamics, the numerical simulations are consistent with the experiments for a ≤ 1. The experimentally observed saturation of the final height of the deposit, when normalized with respect to the initial radius of the cylinder, is accurately reproduced numerically. Analysis of the results sheds light on the correlation between the area overrun by the granular mass and its initial potential energy. The extent of the deposit, the final height, and the arrest time of the front can be directly estimated from the “generic solution” of the model for terrestrial and extraterrestrial avalanches. The effective friction, a parameter classically used to describe the mobility of gravitational flows, is shown to depend on the initial aspect ratio a. This dependence should be taken into account when interpreting the high mobility of large volume events.

Numerical modeling of avalanches based on Saint Venant equations using a kinetic scheme

[1] 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.

A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows

Abstract We introduce a new model for shallow water flows with non-flat bottom. A prototype is the Saint Venant equation for rivers and coastal areas, which is valid for small slopes. An improved model, due to Savage–Hutter, is valid for small slope variations. We introduce a new model which relaxes all restrictions on the topography. Moreover it satisfies the properties (i) to provide an energy dissipation inequality, (ii) to be an exact hydrostatic solution of Euler equations. The difficulty we overcome here is the normal dependence of the velocity field, that we are able to establish exactly. Applications we have in mind concern, in particular, computational aspects of flows of granular material (for example in debris avalanches) where such models are especially relevant. To cite this article: F.xa0Bouchut et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Numerical modeling of self-channeling granular flows and of their levee-channel deposits

When not laterally confined in valleys, pyroclastic flows create their own channel along the slope by selecting a given flowing width. Furthermore, the lobe-shaped deposits display a very specific morphology with high parallel lateral levees. A numerical model based on Saint Venant equations and the empirical variable friction coefficient proposed by Pouliquen and Forterre (2002) is used to simulate unconfined granular flow over an inclined plane with a constant supply. Numerical simulations successfully reproduce the self-channeling of the granular lobe and the levee-channel morphology in the deposits without having to take into account mixture concepts or polydispersity. Numerical simulations suggest that the quasi-static shoulders bordering the flow are created behind the front of the granular material by the rotation of the velocity field due to the balance between gravity, the two-dimensional pressure gradient, and friction. For a simplified hydrostatic model, competition between the decreasing friction coefficient and increasing surface gradient as the thickness decreases seems to play a key role in the dynamics of unconfined flows. The description of the other disregarded components of the stress tensor would be expected to change the balance of forces. The fronts shape appears to be constant during propagation. The width of the flowing channel and the velocity of the material within it are almost steady and uniform. Numerical results suggest that measurement of the width and thickness of the central channel morphology in deposits in the field provides an estimate of the velocity and thickness during emplacement.

Avalanche mobility induced by the presence of an erodible bed and associated entrainment.

The partial fluidization model developed by Aranson and Tsimring (2002) is used to simulate the spreading of a 2D circular cap of granular material over an erodible bed made of the same material. Numerical results show that the presence of even a very thin layer of granular material lying on the solid bed strongly increases the mobility of granular flows. Furthermore, as the thickness of the granular layer increases, the dynamics of the flowing mass changes from a decelerating avalanche to a traveling wave. Numerical simulation suggest that surges are generated if enough mass is entrained, increasing the energy of the flowing material and balancing the energy lost by friction.

Numerical Approximations of Pressureless and Isothermal Gas Dynamics

We study several schemes of first- or second-order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations. The pressureless gas system is weakly hyperbolic, giving rise to the formation of density concentrations known as delta-shocks. For the isothermal gas system, the infinite speed of expansion into vacuum leads to zero timestep in the Godunov method based on exact Riemann solver. The schemes we consider are able to bypass these difficulties. They are proved to satisfy positiveness of density and discrete entropy inequalities, to capture the delta-shocks, and to treat data with vacuum.

Renormalized Solutions to the Vlasov Equation with Coefficients of Bounded Variation

Abstract: We prove that weak bounded solutions to the Vlasov equation with BV coefficients have the renormalization property, and we show that when the renormalization property holds for a general transport equation, it also holds for only Lipschitz nonlinearities.

Numerical modeling of landquakes

The Thurwieser landslide that occurred in Italy in 2004 and the seismic waves it generated have been simulated and compared to the seismic signal recorded a few tens of kilometers from the landslide source (i.e., landquake). The main features of the low frequency seismic signal are reproduced by the simulation. Topography effects on the flowing mass have a major impact on the generated seismic signal whereas they weakly affect low-frequency wave propagation. Simulation of the seismic signal makes it possible to discriminate between possible alternative scenarios for flow dynamics and to provide first estimates of the rheological parameters during the flow. As landquakes are continuously recorded by seismic networks, our results provide a new way to collect data on the dynamics and rheology of natural flows.

Classical Solutions and the Glassey-Strauss Theorem for the 3D Vlasov-Maxwell System

R. Glassey and W. Strauss have proved in Arch. Rational Mech. Anal. 92 (1986), 59{90] that C 1 solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simpliies their proof.