Optimization of the method
The method presented in this Phd thesis involves a significant computational cost to solve a real problem. It can be optimized by constructing an efficient preconditioner for the iterative method presented in Chapter 4. This requires studying the Sturm Liouville operator and finding a way to approach the inverse of the elliptic operator at the discrete level. In terms of accuracy, the interest of using a finite element method is that it allows us to increase the order of the method by constructing new pairs of approximation spaces for the mixed problem in velocity-pressure. The method including a breaking wave criterion has been presented for the one-dimensional case and can be easily extended to two-dimensional case, which will be done in a forthcoming development.
Coupling models and methods
Another possible investigation consists in solving the elliptic problem only for a selected period and domain. This implies coupling different models, a hydrostatic multilayer model on one region and a dispersive model for another region. This is a tricky task and depends on the case being simulated, however, it could be useful to simulate a real phenomenon over a long period of time and a significant domain size.
Multi layer version
The multilayer model described in  has been derived by discretizing the Euler model in the vertical direction. The resulting model presents good properties. It admits an energy balance and involves a total pressure for each layer composed with a hydrostatic and a non-hydrostatic part as for the one layer model (1.41). Therefore, the method presented in this work for the DAE model (1.34)-(1.38) can be extended to solve the multi-layer model. It implies solving the elliptic equation for each layer taking into consideration the exchange terms at the interfaces of the layers. As stated in paragraph 1.2, dispersive models and their numerical simulation can improve our knowledge of coastal oceanography modeling and they can have applications in the development of sustainable energies. As we expect a different behavior regarding the micro-algae if we use a multi-layer model with dispersion, we plan to use a multi-layer dispersive model with an extended method to predict and optimize the trajectories of micro-algae in a raceway.
A depth-averaged Euler system
Several strategies are possible for the derivation of shallow water type models extending the Saint Venant system. A usual process is to assume potential flows and an extensive literature exists concerning these models [29, 48, 105, 4, 5, 66]. An asymptotic expansion, going one step further than the classical Saint-Venant system is also possible [79, 44, 141] but such an approach does not always lead to properly defined unique closure relations. In this paper, we start from a non-hydrostatic model derived and studied in , where the closure relations are obtained by a minimal energy constraint.
The non-hydrostatic model we intend to discretize in this paper has several interesting properties:
the model formulation only involves first order partial derivatives and appears as a depth averaged version of the Euler system.
the proposed model is similar to the well-known Green-Naghdi model  but keeps a natural expression of the topography source term.
Kinetic interpretation of the depth-averaged Euler system
Since we take into account the non-hydrostatic effects of the pressure, the microscopic vertical velocity of the particles has to be considered and we now construct the new density of particles M(x; t; ; ) defined by a Gibbs equilibrium: the microscopic density of particles present at time t, abscissa x and with microscopic horizontal velocity and microscopic vertical velocity is given by M(x; t; ; ) = H c u c ( w) .
The semi-discrete (in space) scheme
To approximate the solution X = (H;Hu;Hw)T of the system (2.51), we use a combined finite volume/finite element framework. We assume that the computational domain is discretized with I nodes xi, i = 1; : : : ; I. We denote Ci the cell (xi1=2; xi+1=2) of length xi = xi+1=2 xi1=2 with xi+1=2 = (xi + xi+1)=2. We denote Xi = (Hi; qx;i; qz;i)T with Xi 1 xi Z Ci X(x; t)dx.
Fully discrete entropy inequality
We have precised in paragraph 2.4.3 a general scheme for the resolution of the nonhydrostatic model. In this paragraph we study the properties of the proposed scheme in the context of one particular solver for the hyperbolic part, namely the kinetic solver, since it allows to ensure stability properties among which are entropy inequalities (semidiscrete and fully discrete) . The presentation in the sequel follows closely that of .
Looking for a kinetic interpretation of the HR scheme, we would like to write down a kinetic scheme for Eq. (2.45) such that the associated macroscopic scheme is exactly (2.90)-(2.91) with the definitions (2.92)-(2.94).
We drop the superscript n and keep superscripts n+1 and n+1=2. We denote Mi = M(Hi; ui; ), Mi+1=2 = M(Hi+1=2; ui; ), Mi+1=2+ = M(Hi+1=2+; ui+1; ), fn+1=2 i = fn+1=2 i () where M is defined by (2.39) and we consider the scheme fn+1=2 i = Mi i frm[o]<0Mi+1=2+ + 1>0Mi+1=2 + Mi+1=2.
Table of contents :
1.1 General issues
1.2 Applications and motivations
1.3 Geophysical models
1.4 Numerical methods
1.5 Outline of the main contributions of this PhD thesis
1.6 Outline of the conclusion
1.7 Future work
2 A robust and stable numerical scheme for a depth-averaged Euler system
2.2 A depth-averaged Euler system
2.3 Kinetic description
2.4 Numerical scheme
2.5 Fully discrete entropy inequality
2.6 Analytical solutions
2.7 Numerical simulations
3 A combined finite volume/ finite element method for a 1D dispersive Shallow Water system
3.2 The projection scheme for the non-hydrostatic model
3.3 Numerical approximation
3.4 Analytical solutions
3.5 Numerical results
4 A numerical method for a two-dimensional dispersive shallow water system on unstructured grids
4.2 The model
4.3 Time and space discretizations
4.4 The mixed problem
4.5 Finite element approximations for the mixed problem
4.6 Numerical algorithm
4.7 Validation with analytical solutions
4.8 Numerical results
5 Supplementary results
5.2 Stationary solutions for the one-dimensional problem
5.3 Another analytical solution
5.4 Dispersion relation
5.5 Breaking wave
A A coupled Exner/ Stokes model
A.2 The model
A.3 Variational formulation
A.4 Numerical tests