A fast and stable well-balanced scheme with hydrostatic reconstruc- tion for shallow water flows 

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Second-order extension

Starting from a given first-order method, a common way to obtain a second-order extension is, as it was already mentioned before for the analogy with our hydrostatic reconstruction, to compute the fluxes from limited reconstructed values on both sides of each interface rather than cell-centered values, see [54], [90] or [124]. These new values are classically obtained with three ingredients : prediction of the gradients in each cell, linear extrapolation, and limitation procedure.
In the presence of a source and in the context of well-balanced schemes, this approach needs to be described in more detail. In particular, according to [76], [77], [18], since not only the reconstructed values Ui,r at i+1/2−, and Ui+1,l at i+1/2+ need be defined but also zi,r, zi+1,l, a cell-centered source term Sci must by added to preserve the consistency. We remark that even if zi do not depend on time, the reconstructed values zi,l, zi,r could depend on time via a coupling with Ui in the reconstruction step.
Once these second-order reconstructed values are known, we apply the hydrostatic reconstruction scheme exposed in the previous section at each interface. This gives the second-order well-balanced scheme xi d dt Ui(t) + Fi+1/2 − Fi−1/2 = Si + Sci.

Finite volumes / Kinetic solver

A classical approach for solving hyperbolic systems consists in using finite volume schemes (see [54]) which are defined by the fluxes computed at the control volume interfaces. We show in Section 3.4.2 how the fluxes of the kinetic scheme are deduced from the discretization of the kinetic equation (3.3.7). In this section we do not take into account the topographic source term, i.e. we first consider the homogeneous system of equations (3.2.5) with B = 0. In Section 3.5 we adapt the scheme to the non flat bottom case.

Finite volume formalism

We recall here the general formalism of finite volumes. Let denote the computational domain with boundary 􀀀, which we assume polygonal. Let Th be a triangulation of which vertices are denoted Pi with Si the set of interior nodes and Gi the set of boundary nodes. The dual cells Ci are obtained by joining the centers of mass of the triangles surrounding each vertex Pi. We use the following notations (see Fig. 3.4.1) :
– Ki, set of subscripts of nodes Pj surrounding Pi.
– |Ci|, area of Ci.
– 􀀀ij , boundary edge between the cells Ci and Cj .
– Lij , length of 􀀀ij .
– nij , unit normal to 􀀀ij , outward to Ci (nji=-nij).

Boundary conditions

The treatment of the boundary conditions is presented in details in [23]. Here we just recall some main features about the computation of the boundary flux F(Uni ,Une ,i, ni) appearing in (3.4.3). Notice first that the variable Une ,i can be interpreted as an approximation of the solution in a “fictitious” cell adjacent to the boundary. As before we introduce the local coordinates and define ˆU ne ,i = 􀀀  ,i, qn e,i,n, qn e,i,τ T . Then we can use the local flux vector splitting form associated to the kinetic formulation ˆ F(ˆU n i,ni , ˆU ne ,i) = ˆ F+(ˆU n i,ni) + ˆ F−(ˆU ne ,i).
On the solid wall we prescribe a continuous slip condition – see Section 3.2. In the numerical scheme we prescribe it weakly by defining ˆU ne ,i = 􀀀 hni ,−qn i,n, qn i,τ T . It follows that finally ˆ F(ˆU n i,ni , ˆU ne ,i) = (0, ghni 2 2 , 0)T . (3.4.18)
On the fluid boundaries, the type of the flow and then the number of boundary conditions depend on the Froude number. Here we consider a local Froude number associated  to the normal component of the velocity. For the fluvial cases, we define completely Ue by adding to the given boundary condition, the assumption that the Riemann in- variant related to the outgoing characteristic is constant along this characteristic (see [23]).

Second order extension

The first order scheme defined in Sections 3.4-3.5 can be extended to a “formally” second order one using a MUSCL like extension (see [127]). In Section 3.6.1, we define limited reconstructed variables and in Section 3.6.2, we introduce a “second order” well-balanced scheme.

Second order reconstructions

In the definition of the flux (3.4.7), we replace the piecewise constant values Ui,Uj by more accurate reconstructions deduced from piecewise linear approximations, namely the values Uij ,Uji reconstructed on both sides of the interface. More precisely, we are looking for piecewise linear approximation of the primitive variable ˆW = (h, un, uτ )T , actually the detailed expression of the flux given in (3.4.14) uses the primitive variables. We divide each cell Ci in subtriangles obtained by joining each edge 􀀀ij to the node Pi,

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Table of contents :

1 Pr´esentation des travaux et Principaux r´esultats 
1.1 Introduction
1.2 Syst`eme de Saint-Venant et Volumes finis
1.2.1 Pr´esentation et domaine de validit´e
1.2.2 Hyperbolicit´e, Stabilit´e, Equilibres
1.2.3 Les volumes finis
1.3 Un sch´ema ´equilibre positif d’ordre 2 pour le syst`eme de Saint-Venant avec termes sources sur maillages non structur´es : Analyse et mise en oeuvre num´erique
1.3.1 Syst`eme de Saint-Venant homog`ene : Etat de l’art
1.3.2 Syst`eme de Saint-Venant homog`ene : Les sch´emas cin´etiques
1.3.3 D´ecentrement des sources aux interfaces : Etat de l’art
1.3.4 D´ecentrement des sources aux interfaces : La reconstruction hydrostatique
1.3.5 Le sch´ema d’ordre 2
1.4 Un sch´ema `a deux pas de temps pour le syst`eme Saint-Venant / transport : Analyse et mise en oeuvre num´erique
1.4.1 Etat de l’art
1.4.2 Le sch´ema `a deux pas de temps
1.5 Lois de conservation scalaires avec flux discontinu : Un th´eor`eme d’unicit´e
1.5.1 Position du probl`eme
1.5.2 Etat de l’art
1.5.3 Les entropies de Kruzkov partiellement adapt´ees
1.5.4 Le th´eor`eme d’unicit´e
1.6 Un mod`ele Saint-Venant multicouche : D´erivation et Analyse du mod`ele, Mise en oeuvre num´erique
1.6.1 Un mod`ele interm´ediaire entre Saint-Venant et Navier-Stokes .
1.6.2 Etat de l’art
1.6.3 D´erivation et analyse du syst`eme multicouche
1.6.4 Etude num´erique du syst`eme multicouche
1.7 Conclusions et Perspectives
2 A fast and stable well-balanced scheme with hydrostatic reconstruc- tion for shallow water flows 
2.1 Introduction
2.2 Well-balanced scheme with hydrostatic reconstruction
2.2.1 Semi-discrete scheme
2.2.2 Fully discrete scheme and CFL condition
2.3 Second-order extension
2.4 Numerical results
2.4.1 1d assessments
2.4.2 2d assessments
3 A second order well-balanced positivity preserving scheme for the Saint-Venant system on unstructured grids 
3.1 Introduction
3.2 The Saint-Venant system
3.2.1 Equations
3.2.2 Properties of the system
3.3 Kinetic representation
3.4 Finite volumes / Kinetic solver
3.4.1 Finite volume formalism
3.4.2 Kinetic solver
3.4.3 Numerical implementation
3.4.4 Upwind kinetic scheme
3.4.5 Boundary conditions
3.4.6 Properties of the scheme
3.5 Well-balanced scheme
3.6 Second order extension
3.6.1 Second order reconstructions
3.6.2 Second order well-balanced scheme
3.7 Numerical results
3.8 Conclusion and outlook
4 Transport of pollutant in shallow water flows : A two times step ki- netic method 
4.1 Introduction
4.2 Equations
4.3 The kinetic scheme
4.3.1 Kinetic interpretation of the shallow water equations
4.3.2 The kinetic scheme
4.3.3 Preservation of the equilibria
4.4 Properties of the scheme
4.4.1 Positivity of the water height
4.4.2 Positivity of the concentration of pollutant
4.4.3 Maximum principle for the concentration of pollutant
4.5 Larger time steps for the pollutant
4.5.1 Motivation
4.5.2 Algorithm
4.5.3 Consistency, conservativity, positivity, maximum principle and preservation of equilibria
4.6 Numerical results
4.6.1 Transport of pollutant in a flat bottom channel with constant discharge
4.6.2 Dam break
4.6.3 Peak in the concentration of pollutant
4.6.4 Emission of pollutant in a non flat bottom channel
4.6.5 With a non uniform mesh
4.7 Extension to the 2D case
4.7.1 A 2D dam break problem
4.7.2 Emission of pollutant in a realistic river
4.8 Conclusion
5 Uniqueness for a scalar conservation law with discontinuous flux via adapted entropies 
5.1 Introduction
5.2 Hypothesis on the flux
5.3 Partially Adapted Kruzkov Entropies
5.4 Uniqueness Theorem
5.5 Application : Discontinuous convex flux
6 A multilayer Saint-Venant model 
6.1 Introduction
6.2 Navier-Stokes equations and hydrostatic approximations
6.2.1 A viscous hydrostatic model
6.2.2 A classical hydrostatic model
6.3 The Multilayer Saint-Venant System
6.3.1 The Multilayer Saint-Venant model
6.3.2 Properties of the Multilayer Saint-Venant System
6.3.3 Non conservativity and non hyperbolicity of the Multilayer Saint- Venant System
6.3.4 Conservative Form of the Multilayer Saint-Venant Model
6.4 The discrete multilayer scheme
6.4.1 The finite volume solver
6.4.2 The implicit computation
6.4.3 Properties of the discrete multilayer kinetic scheme
6.5 Numerical assessment : a dam break problem
6.5.1 The zero friction case
6.5.2 Comparison with monolayer Saint-Venant models
6.5.3 Multilayer aspect of the model
6.5.4 Comparisons with Navier-Stokes velocity profiles
6.5.5 Computational cost
6.5.6 Influence of the number of layers
6.5.7 Some other friction coefficients
6.5.8 Robustness of the scheme
6.6 Conclusion and perspectives

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