A good understanding of waves in shallow water, typically in coastal regions, is important for several environmental and societal issues: submersion risks, protection of harbors, erosion, offshore structures, wave energies, etc. The goal of this serie of lectures is to show how efficient asymptotic models can be derived from the full fluid equations (Navier-Stokes and Euler) and to point out several modelling, numerical and mathematical challenges that one still has to understand in order to describe correctly and efficiently such complex phenomena as wave breaking, overtopping, wave-structures interactions, etc.

I Derivation of several shallow water models

We will show how to derive several shallow water models (nonlinear shallow water equations, Boussinesq and Serre-Green-Naghdi systems) from the free surface Euler equations. We will consider first the case of an idealized configuration where no breaking waves are involved, where the water height does not vanish (no beach!), and where the flow is irrotational – this is the only configuration for which a rigorous justification of the asymptotic models can be justified.

II Brief analysis of these models.

We will briefly comment the mathematical structure of these equations, with a particular focus on the properties that are of interest for their numerical implementation. We will also discuss how these models behave in when the water height vanishes, since they are typically used in such configurations (see the lecture by P. Bonneton).

III Vorticity and turbulent effects

We will propose a generalization of the derivation of the main shallow water models in the presence of vorticity, and show that the standard irrotational shallow water models must be coupled with an equation for a ”turbulent” tensor. We will also make the link with a modelling of wave breaking proposed by Gavrilyuk and Richard in which wave breaking is taken into account as a source term in this additional equation.

IV Floating objects.

This last section will be devoted to the description of a new approach to describe the interaction of waves in shallow water with floating objects, which leads to several interesting mathematical and numerical issues.[-]

A good understanding of waves in shallow water, typically in coastal regions, is important for several environmental and societal issues: submersion risks, protection of harbors, erosion, offshore structures, wave energies, etc.

The goal of this serie of lectures is to show how efficient asymptotic models can be derived from the full fluid equations (Navier-Stokes and Euler) and to point out several modelling, numerical and mathematical challenges that one still has to understand in order to describe correctly and efficiently such complex phenomena as wave breaking, overtopping, wave-structures interactions, etc.

I Derivation of several shallow water models

We will show how to derive several shallow water models (nonlinear shallow water equations, Boussinesq and Serre-Green-Naghdi systems) from the free surface Euler equations. We will consider first the case of an idealized configuration where no breaking waves are involved, where the water height does not vanish (no beach!), and where the flow is irrotational – this is the only configuration for which a rigorous justification of the asymptotic models can be justified.

II Brief analysis of these models.

We will briefly comment the mathematical structure of these equations, with a particular focus on the properties that are of interest for their numerical implementation. We will also discuss how these models behave in when the water height vanishes, since they are typically used in such configurations (see the lecture by P. Bonneton).

III Vorticity and turbulent effects.

We will propose a generalization of the derivation of the main shallow water models in the presence of vorticity, and show that the standard irrotational shallow water models must be coupled with an equation for a ”turbulent” tensor. We will also make the link with a modelling of wave breaking proposed by Gavrilyuk and Richard in which wave breaking is taken into account as a source term in this additional equation.

IV Floating objects.

This last section will be devoted to the description of a new approach to describe the interaction of waves in shallow water with floating objects, which leads to several interesting mathematical and numerical issues.[-]

Optimal transport, a mathematical theory which developed out of a problem raised by Gaspard Monge in the 18th century and of the reformulation that Leonid Kantorovich gave of it in the 20th century in connection with linear programming, is now a very lively branch of mathematics at the intersection of analysis, PDEs, probability, optimization and many applications, ranging from fluid mechanics to economics, from differential geometry to data sciences. In this short course we will have a very basic introduction to this field. The first lecture (2h) will be mainly devoted to the problem itself: given two distributions of mass, find the optimal displacement transforming the first one into the second (studying existence of such an optimal solution and its main properties). The second one (2h) will be devoted to the distance on mass distributions (probability measures) induced by the optimal cost, looking at topological questions (which is the induced topology?) as well as metric ones (which curves of measures are Lipschitz continuous for such a distance? what can we say about their speed, and about geodesic curves?) in connection with very natural PDEs such as the continuity equation deriving from mass conservation.[-]

Optimal transport, a mathematical theory which developed out of a problem raised by Gaspard Monge in the 18th century and of the reformulation that Leonid Kantorovich gave of it in the 20th century in connection with linear programming, is now a very lively branch of mathematics at the intersection of analysis, PDEs, probability, optimization and many applications, ranging from fluid mechanics to economics, from differential geometry to data sciences. In this short course we will have a very basic introduction to this field. The first lecture (2h) will be mainly devoted to the problem itself: given two distributions of mass, find the optimal displacement transforming the first one into the second (studying existence of such an optimal solution and its main properties). The second one (2h) will be devoted to the distance on mass distributions (probability measures) induced by the optimal cost, looking at topological questions (which is the induced topology?) as well as metric ones (which curves of measures are Lipschitz continuous for such a distance? what can we say about their speed, and about geodesic curves?) in connection with very natural PDEs such as the continuity equation deriving from mass conservation.[-]

We combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on the fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

Keywords: generalized multiscale finite element method - nonlinear PDEs - heterogeneous porous media - discrete empirical interpolation - proper orthogonal decomposition[-]

A good understanding of waves in shallow water, typically in coastal regions, is important for several environmental and societal issues: submersion risks, protection of harbors, erosion, offshore structures, wave energies, etc.

The goal of this serie of lectures is to show how efficient asymptotic models can be derived from the full fluid equations (Navier-Stokes and Euler) and to point out several modelling, numerical and mathematical challenges that one still has to understand in order to describe correctly and efficiently such complex phenomena as wave breaking, overtopping, wave-structures interactions, etc.

I Derivation of several shallow water models

We will show how to derive several shallow water models (nonlinear shallow water equations, Boussinesq and Serre-Green-Naghdi systems) from the free surface Euler equations. We will consider first the case of an idealized configuration where no breaking waves are involved, where the water height does not vanish (no beach!), and where the flow is irrotational – this is the only configuration for which a rigorous justification of the asymptotic models can be justified.

II Brief analysis of these models.

We will briefly comment the mathematical structure of these equations, with a particular focus on the properties that are of interest for their numerical implementation. We will also discuss how these models behave in when the water height vanishes, since they are typically used in such configurations (see the lecture by P. Bonneton).

III Vorticity and turbulent effects.

We will propose a generalization of the derivation of the main shallow water models in the presence of vorticity, and show that the standard irrotational shallow water models must be coupled with an equation for a ”turbulent” tensor. We will also make the link with a modelling of wave breaking proposed by Gavrilyuk and Richard in which wave breaking is taken into account as a source term in this additional equation.

IV Floating objects.

This last section will be devoted to the description of a new approach to describe the interaction of waves in shallow water with floating objects, which leads to several interesting mathematical and numerical issues.[-]