We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems such as the Navier Stokes equations. Efficiency of the scheme is discussed. We also draw some research perspectives.

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A nonconservative hyperbolic depth-averaged model for granular-fluid mixtures: application to debris flows and landslides.

We present a two-phased model for simulating shallow free-surface environmental flows composed of granular-fluid mixtures, such as debris flows, landslides and volcanic lahars. The model equations are depth-averaged and related to the shallow water equations, yet include hyperbolic equations for the fluid-pore pressure and solid volume fraction. The model accounts for the important feedback between the divergence of the solid phase (dilation rate) and the resulting effect on the pore pressure, which affects the buoyancy and stress of the mixture. By including these effects, the model can be used for flow modeling as well as determination of stability and initiation of quasi-static mixtures near steady-state balance. The model can therefore bridge the gap between flow models and slope stability models, which have historically been distinct. The resulting set of equations are a non-strictly hyperbolic system with nonconservative products and a source term. While hyperbolicity is always maintained, the important balance between gradient terms and the source term must be preserved numerically. We will present the mathematical model, numerical issues, and experimental data for comparison and validation.

Well-balanced schemes for linear models of Boltzmann equation: the legacy of Case, Cercignani and Siewert

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We review some recent numerical methods for hyperbolic equations with singular (discontinuous or measure-valued) coeffcients. Such problems arise in wave propagation through interfaces or barriers, or nonlinear waves through singular geometries. The connection between the well-balanced schemes for shallow-water equations with discontinuous bottom topography and the Hamiltonian preserving schemes for Liouville equations with discontinuous Hamiltonians is illustrated. Various developments of numerical methods with applications in high frequency waves through interfaces, and in multi-scale coupling between classical and quantum mechanics, are discussed.

Because of the scales — in space and time — of the considered problems in geophysics, there exists a demand for models of reduced complexity such as Shallow Water type models but able to represent complex flows (dam breaks, floodings, avalanches,...). We propose several models approximating the free surface Navier-Stokes equations and extending the classical Saint-Venant system. For each of them, the derivation and a numerical scheme are given. The discrete model is then confronted with with analytical solutions and/or exprimental data associated with complex flows. The numerical schemes we propose rely on a kinetic interpretation of each of the models, the microscopic description allowing to prove necessary stability conditions of the schemes. Several of the proposed models have been implemented in industrial codes.

In fluid mechanics, the dambreak problem refers to the gravity-driven flow of a fixed volume of fluid along a solid boundary. The problem is not only of great importance to ensuring safety of manmade and natural dams, but also to understanding the dynamics of time-dependent free-surface flows. Considerable attention has been paid to this fundamental problem since Ritter worked out the earliest solution to the dambreak problem for inviscid fluids in 1892. In particular, many experiments were carried out to measure the flow depth profiles and front position for various settings. Until very recently, most measurements have concerned bulk flow variables such as the flow depth or the front velocity. Today, image processing makes it possible to visualize not only what occurs at the flow boundary (e.g., velocity profiles at the flume sidewall or free surface), but also inside the flow, allowing for a better understanding of the relations between stress and deformation rate inside moving fluids. Combining different visualization techniques (iso-index fluids, laser fluorescence, particle tracking, etc.), we were able to measure the velocity profiles within the tip region as well as the body of fluid avalanches. In addition to these measurements (taken far from the sidewall), we also measured the flow depth profiles, front position, and particle concentrations when the flow involved dilute or moderately concentrated suspensions of particles (measuring the concentration in highly concentrated suspensions is still a challenge). Various settings were investigated: one- and two-phase flows (homogeneous fluids, particle suspensions), linear and nonlinear rheologies (Newtonian fluids, viscoplastic fluids, suspensions), laminar and turbulent regime, fixed and mobile beds, flows down an inclined flume or along a sloping bed. The talk reviews some of the analytical results available for studying the dambreak problem. Emphasis is then given to a thorough comparison between experimental data, analytical results, and numerical simulations (obtained using the shallow ``water'' equations). In particular, we show that the front has a specific behavior, due to a large extent to the pronounced curvature of the free surface that causes the shear stress to significantly increase (relative to the behavior in the body), but surprisingly enough, the details of this front behavior are not essential to determining the flow behavior of the fluid avalanche.

Abstract : There is a large amount of experimental work dealing with dry granular flows (such as sand, glass beads, small rocks...) supporting the so called μ(I) rheology made popular by the GDR MiDi. This rheology states that the ratio of the tangential to the normal constraints behaves as a Coulomb like friction depending on the Inertial number (this number is the product of the grain size by the shear of the velocity divided by the square root of pressure divided by the grain density). Hence, we propose the implementation of this non newtonian rheology in a Navier Stokes Solver (the Gerris Flow Solver uses a finite-volume approach with the Volume-of-Fluid (VOF) method to describe variable-density two-phase flows). The rheology is tested on the collapse of granular columns and quantitative comparisons with numerical simulations from Contact Dynamics and from Saint-Venant Savage Hutter (finite-volume) are performed. This gives a complete vision of granular flow from the grain scale to the continuum theory via the 1D thin layer approximation.

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