We consider the Saint-Venant system for shallow water flows, with non-flat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, 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 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 semi-discrete entropy inequality.

Abstract.The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.Mathematics Subject Classification. 35Q30, 35Q35, 76D05.

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We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions

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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 this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We prove also that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term.2000 Mathematics Subject Classification. 65M12, 74S10, 76M12, 35L65.

SUMMARYWe present a multilayer Saint-Venant system for the simulation of 3D free surface flows with friction and viscosity effects. A vertical discretization of a Navier-Stokes system approximation deduced from a precise analysis of the shallow water assumption leads to a set of coupled Saint-Venant-type systems. The idea is to obtain an accurate description of the vertical profile of the horizontal velocity while preserving the robustness and the computational efficiency of the usual Saint-Venant system.For each time-dependent layer, a Saint-Venant-type system is solved on the same 2D mesh by a kinetic solver using a finite volume framework. The free surface is directly deduced from the sum of layers water depth.We validate the model with some numerical academic and realistic examples. We present comparisons with simulations computed with the hydrostatic Navier-Stokes solver of the Telemac-3D code developed by Electricité de France.