François Bouchut

French National Centre for Scientific Research | CNRS · Institut national des sciences mathématiques et de leurs interactions (INSMI)

We investigate the dynamics and deposits of granular flows and the amplitude of landslide-generated water waves using the HySEA depth-averaged shallow-water numerical model, both at laboratory and field scales. We evaluate the different sources of error by quantitatively comparing the simulations with (i) new laboratory experiments of granular coll...

We are interested in free surface flows where density variations coming e.g. from temperature or salinity differences play a significant role in the hydrodynamic regime. In water, acoustic waves travel much faster than gravity and internal waves, hence the study of models arising from com-pressible fluid mechanics often requires a decoupling betwee...

Depth-averaged models, such as the Savage-Hutter model with Coulomb or Pouliquen friction laws, do not in some cases preserve the physical threshold of motion. In particular, the simulated granular mass can start to flow (or stay at rest) even if the slope angle of its free surface is lower (or higher) than the repose angle of the granular material...

Validation and benchmarking of pyroclastic current (PC) models is required to evaluate their performance and their reliability for hazard assessment. Here, we present results of a benchmarking initiative built to evaluate four models commonly used to assess concentrated PC hazard: SHALTOP, TITAN2D, VolcFlow, and IMEX_SfloW2D. The benchmark focuses...

The Bingham model for viscoplastic materials involves the minimization of a non-differen-tiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: The total variation minimization problem seeks the unique minimizer u ∈ BV(Ω) of bounded variation of the energy 1/2||u − f||_L 2(...

We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with continuous topography with locally integrable derivative. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the...

Depth-averaged thin-layer models are commonly used to model rapid gravity-driven flows such as debris flows or debris avalanches. However, the formal derivation of thin-layer equations for general topographies is not straightforward. The curvature of the topography results in a force that maintains the velocity tangent to the topography. Another cu...

Dilatancy plays a key role in mixtures of grains and fluid but is poorly investigated in dry granular flows. These flows may however dilate by more than $10 \%$ in granular column collapses. We investigate here dilatancy effects in dry flows with a shallow depth-averaged model designed to be further applied to simulate natural landslides. We use a...

This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows. Our starting point is the asymptotic model derived by Bouchut et al. (2016) from a free surface incompressible model with viscoplastic rheology including a Drucker--Prager yield stress. This asymptotic model is based on the...

We present here a multilayer model for shallow grain-fluid mixtures with dilatancy effects. It can be seen as a generalization of the depth-averaged model presented in [Bouchut et al. A two-phase two-layer model for fluidized granular flows with dilatancy effects. J. Fluid Mech., 801:166-221, 2016], that includes dilatancy effects by considering a...

In the first part of this work, we introduce a new relaxation system in order to approximate the solutions to the barotropic Euler equations. We show that the solutions to this two-speed relaxation model can be understood as viscous approximations of the solutions to the barotropic Euler equations under appropriate sub-characteristic conditions. Ou...

We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut, Chalons, Guisset (2019) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This...

We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesia...

Kinetic BGK numerical schemes for the approximation of incompressible Navier-Stokes equations are derived via classical discrete velocity vector BGK approximations, but applied to an inviscid compressible gas dynamics system with small Mach number parameter, according to the approach of Carfora and Natalini (2008). As the Mach number, the grid size...

This paper includes an erratum to (Bouchut et al. (2014) Convection and total variation flow. IMA J. Numer. Anal., 34, 1037-1071.) which deals with a nonlinear hyperbolic scalar conservation law, regularized by the total variation flow operator (or 1-Laplacian), and in which a mistake occurred in the convergence proof of the numerical scheme to the...

The shallow water magnetohydrodynamic system involves several families of physically relevant steady states. In this paper we design a well-balanced numerical scheme for the one-dimensional shallow water magnetohydrodynamic system with topography, that resolves exactly a large range of steady states. Two variants are proposed with slightly differen...

We simulate here the collapse of granular columns immersed in a viscous fluid based on a simplified version of the model developed by [2]. The simulation quite well reproduces the dynamics and deposit of the granular mass as well as the excess pore fluid pressure measured in the laboratory experiments of [10] owing that dilatancy effects and pore p...

We simulate here the emplacement of the debris avalanche generated by the last flank collapse event of Montagne Pelée volcano (30–45 ka), Martinique, Lesser Antilles. Our objective is to assess the maximum distance (i.e., runout) that can be reached by this type of debris avalanche as a function of the volume involved. Numerical simulations are per...

Flows of dense granular materials comprise regions where the material is flowing, and regions where it is static. Describing the dynamics of the interface between these two regions is a key issue to understand the erosion and deposition processes in natural environments. A free interface simplified model for non-averaged thin-layer flows of granula...

This work is devoted to numerical modeling and simulation of granular flows relevant to geophysical flows such as avalanches and debris flows. We consider an incompressible viscoplastic fluid, described by a rheology with pressure-dependent yield stress, in a 2D setting with a free surface. We implement a regularization method to deal with the sing...

We simulate here dry granular flows resulting from the collapse of granular columns on an inclined channel (up to 22 degrees) and compare precisely the results with laboratory experiments. Incompressibility is assumed despite the dilatancy observed in the experiments (up to 10%). The 2-D model is based on the so-called $\mu(I)$ rheology that induce...

Observed avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to ma...

A lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with non-flat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example Godunov, Roe, kinetic...). If th...

We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov-Poisson equation wi...

We propose a two-phase two-thin-layer model for fluidized debris flows that takes into account dilatancy effects, based on the closure relation proposed by Roux & Radjai ( Physics of Dry Granular Media , 1998, Springer, pp. 229–236). This relation implies that the occurrence of dilation or contraction of the granular material depends on whether the...

The shallow water magnetohydrodynamic system describes the thin layer evolution of the solar tachocline. It is obtained from the three dimensional incompressible magnetohydrodynamic system similarly as the classical shallow water system is obtained from the incompressible Navier-Stokes equations. The system is hyperbolic and has two additional wave...

The recently developed theory of Lagrangian flows for transport equations with low regularity coefficients enables to consider non BV vector fields. We apply this theory to prove existence and stability of global Lagrangian solutions to the repulsive Vlasov-Poisson system with only integrable initial distribution function with finite energy. These...

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which...

We propose a unified framework to derive thin-layer reduced models for some shallow free-surface flows driven by gravity. It applies to incompressible homogeneous fluids whose momentum evolves according to Navier-Stokes equations, with stress satisfying a rheology of viscous type (i.e. the standard Newtonian law with a constant viscosity, but also...

The evolution of localised jets and periodic nonlinear waves in rotating shallow water magnetohydrodynamics (rotating SWMHD) and standard rotating shallow water model (RSW) is compared within the framework of translationally-invariant 1.5-dimensional configurations, which are traditionally used in geophysical fluid dynamics for studying geostrophic...

A mechanical and numerical model of dry granular flows is proposed that quantitatively reproduce laboratory experiments of granular column collapse over inclined planes. The rheological parameters are directly derived from the experiments.The so-called \mu(I) rheology is reformulated in the framework of Drucker-Prager plasticity with the yield stre...

We focus on the 6 August 2010 Mount Meager landslide that occurred in Southwest British Columbia, Canada. This 48.5 Mm3 rockslide that rapidly changed into a debris flow was recorded by over 25 broadband seismic stations. We showed that the waveform inversion of the seismic signal making it possible to calculate the time history of the force applie...

This paper proposes a thin layer depth-averaged two-phase model provided by a dissipative energy balance to describe avalanches of solid-fluid mixtures. This model is derived from a 3D two-phase model based on the equations proposed by Jackson [The Dynamics of Fluidized Particles. Cambridges Monographs on Mechanics (2000)] which takes into account...

We study approximations by conforming methods of the solution to the variational inequality $\langle \partial_t u,v-u\rangle + \psi(v) - \psi(u) \ge \langle f,v-u\rangle$, which arises in the context of inviscid incompressible Bingham fluid flows and of the total variation flow problem. We propose a general framework involving total variation funct...

The flow of a Bingham fluid with inertial terms is simplified into a nonlinear hyperbolic scalar conservation law, regularised by the total variation flow operator (or 1-Laplacian). We give an entropy weak formulation, for which we prove the existence and the uniqueness of the solution. The existence result is established using the convergence of a...

Consistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this ‘non-traditional’ approximation captures the increase with height of the solid-body velocity due to planetary rotation. The c...

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an $L^1$ function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the $BV$ theory. We illustrate the related well-posedness theory of Lagr...

We propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the...

The rock-ice avalanche that occurred in 2005 on Mount Steller, Alaska and the resulting long period seismic waves have been simulated for different avalanche scenarios (i.e., flow histories), with and without erosion processes taken into account. This 40-60 Mm(3) avalanche traveled about 10 km down the slope, mainly on top of a glacier, eroding a s...

We undertake a detailed study of inertial instability of the barotropic Bickley jet and its nonlinear saturation in the 2 layer rotating shallow water (RSW) model on the f -plane, and compare it with the classical barotropic and baroclinic instabilities. We start with analytical and numerical investigation of the linear stability problem under hypo...

Landslides dynamics prediction remains difficult in spite of a considerable number of studies. The runout distance is widely used in analysis of landslide dynamics and in the calibration of the rheological parameters involved in numerical modeling. However, the unknown impact of the significant uncertainty in the shape of the initial released mass...

We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time-step and the ratio of the space step to the time-step tend to zero. We then consider a second...

online xx xx xxxx We derive a two-layer rotating shallow-water model for a moist atmosphere with water vapor condensation and related diabatic heating. Moist convection is represented by additional mass exchanges between the layers, which are determined from the moist enthalpy conservation principle, and related to the precipitation. Various bounda...

We present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that...

This paper is the second part of the work on linear and nonlinear stability of buoyancy-driven coastal currents. Part 1, concerning a passive lower layer, was presented in the companion paper Gula & Zeitlin (J. Fluid Mech., vol. 659, 2010, p. 69). In this part, we use a fully baroclinic two-layer model, with active lower layer. We revisit the linea...

Recent gullies on Mars are suspected to be the result of liquid-water-bearing flows. A formation from wet flows has been challenged by studies invoking granular (dry) flows. Our study focuses on the sinuous shapes observed for some of the recent Martian gullies. Sinuous gullies are found in locations and slopes (of 10°-15°) similar to straight gull...

1] 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...

In the first part of this work Bouchut etal. (J Comput Phys 108:7–41, 2007) we introduced an approximate Riemann solver for one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we con...

The numerical resolution of the multi-layer shallow water system encounters two additional difficulties with respect to the one-layer system. The first is that the system involves noncon-servative terms, and the second is that it is not always hyperbolic. A splitting scheme has been proposed by Bouchut and Morales, that enables to ensure a discrete...

We consider the Saint-Venant system for shallow water flows with nonflat bottom. In past years, efficient well-balanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic-well-balanced sch...

In this work we establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. The semi-discrete approach is less restrictive than the fully-discrete case and allows to grant some other good properties for numerical schemes. First, conditions are established in an abstract framework for simple Riemann s...

We describe a shallow-water type atmospheric model which includes the transport of moisture as well as related precipitation and convection effects. The model combines hydrodynamic nonlinearity of the standard shallow-water model with the intrinsic nonlinearity due to the precipitation threshold. It allows for both theoretical treatment by the meth...

The 1999 Chi-Chi earthquake triggered the catastrophic Tsaoling landslide in Taiwan. The geomorphological change measured from the data of the 1989 and 2000 aerial photos reveals that the scar and deposit volumes are about 0.126 km3 and 0.15 km3 respectively. The debris material ran over a distance of 1.6 km with 500 m descent in elevation. In this...

We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum bala...

In this paper, we present a new two-layer model of Savage–Hutter type to study submarine avalanches. A layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid (e.g. water). The model is derived in a system of local coordinates following a non-erodible bottom and takes into account its cur...

In this work, we study the modeling of one-dimensional avalanche flows made of a moving layer over a static base, where the interface between the two can be time dependent. We propose a general model, obtained by looking for an approximate solution with constant velocity profile to the incompressible Euler equations. This model has an energy dissip...

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical...

We present the results of fully nonlinear numerical simulations of the geostrophic adjustment of a pressure front over topography, represented by an escarpment with a linear slope. The results of earlier simulations in the linear regime are confirmed and new essentially nonlinear effects are found. Topography influences both fast and slow component...

For the International Forum on Landslide Disaster Management 2007 framework, our team performed several numerical simulations on both theoretical and natural cases of granular flows. The objective was to figure out the ability and the limits of our numerical model in terms of reproduction and prediction. Our benchmarking exercises show that for alm...

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...

We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisf...

This chapter is devoted to the description of recently developed efficient tools for the numerical resolution of shallow water models, and in particular for the case of Coriolis force. We try to give a general introduction to the finite volume approach, but nevertheless our presentation is especially focused on the approach that has been developed...

1] 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 propos...