Maximum principle for the concentration of pollutant

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Table of contents

1 Introduction
2 A fast and stable well-balanced scheme with hydrostatic reconstruction 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 kinetic 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