We consider a family of systems of Boussinesq type due to Bona, Chen and Saut approximating the Euler equations of surface water wave theory, and modeling two-way nonlinear dispersive long wave propagation. We review recent progress on the theory of well-posedness of initial- and initial-boundary-value problems for these systems in two space dimensions. We approximate the systems by fully discrete numerical schemes using Galerkin - finite element methods for the spatial discretization, and analyze the stability and convergence of these schemes. The numerical methods are used as exploratory tools in a series of numerical experiments simulating various complex two-dimensional flows. We also study, by numerical means, interactions of solitary-wave solutions of these systems in one space dimension, including head-on and overtaking collisions, and interactions of solitary waves with the boundaries.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

In the mountain regions snow avalanches represent a major natural hazard for both life and property. In the present study we focus mainly on two aspects of the avalanche simulation problem. Namely, the first part of the talk is devoted to some two-fluid models of a powder-snow avalanche. After a brief review of existing approaches we present a model which has a property to be consistent in the kinetic energy. Numerical results on an avalanche flow around an obstacle are presented.
In the second part, we investigate the obstacle deformation process under an avalanche impact. We use a weak fluid-structure interaction approach which is valid in the case of infinitesimal deformations. It is often the case in many practical situations when the loading process does not lead to drastic consequences. We present several numerical simulations which allow to determine the stress distribution inside the obstacle. This information is crucial for material damage estimation. Presented techniques and results have potentially important applications in vulnerability assessment of protecting structures.
This is a joint work with D. Dutykh and D. Bresch

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

Thin fluid films can have surprising behaviors depending on the boundary conditions enforced, the energy input, and the specific Reynolds number of the fluid motion. Here we study the equations of motion for a thin fluid film with a free boundary and its other interface in contact with a solid wall. Although shear dissipation increases for thinner layers and the motion can generally be described in the limit as viscous, inertial modes can always be excited for a sufficiently high input of energy. We derive the minimal set of equations containing inertial effects in this strongly dissipative regime.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

Abstract: The problems of wave impact, wave breaking and other violent phenomena necessitate taking into account the compressibility of the air-water mixture. To meet these practical needs, F. Dias, D. Dutykh and J.-M. Ghidaglia proposed recently a simple single velocity, single energy two-phase model [Dias et al., 2009]. Properties and performance of this so-called four-equations model have already been discussed in the literature.
However, this model [Dias et al., 2009] was presented as it is without any derivation or justification procedure. The main goal of this communication is to fill in this gap. Namely, we derive the four-equations model from the complete six-equations two-phase system [Ishii, 1975, Ghidaglia et al., 2001, Rovarch, 2006] as a result of a special velocity and energy relaxation procedure. Mathematically this goal is achieved with the Chapman-Enskog type asymptotic expansion. Finally, the incompressible limit is discussed and some numerical results on compressible lock-exchange type flows are presented.
This is a joint work with Y. Meyapin and D. Dutykh.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

We apply and analyse some finite volume schemes to several Boussinesq type systems of water wave theory. A comparison with other numerical methods such as pseudo-spectral, standard Galerkin and discontinuous Galerkin is made. Special attention is given to the run-up of long waves on a plane beach. Various algorithms are considered. Validation by experimental data is presented for the head-on collision of solitary waves, wet dam break problem and the run-up of non-breaking and breaking solitary waves on a plane beach.
This is a joint work with D. Dutykh and Th. Katsaounis.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

For the simulation of fully nonlinear surface gravity waves, a fast, accurate and robust numerical scheme is presented. The method is based on a boundary integral formulation,rewritten in a convenient form, together with a pseudo-spectral spatial scheme and a high-order temporal one. Various applications are presented.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

We present a class of relaxation schemes for the shallow water equations. These schemes are based on classical relaxation models for conservation laws. We consider finite volume as well as finite element spatial discretizations combined with TVD Runge-Kutta time stepping mechanisms. Numerical results are presented for several benchmark test problems.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

The Partial Differential Equations (PDE) which naturally arise in Fluid Mechanics problems admit transformations conserving the whole set of solutions. They form the so-called symmetry group of the PDE. Usually, this group contains some important physical properties of the system expressed in the language of symmetries. It appears natural to expect numerical methods to preserve at least some of symmetry transformations of the continuous system. In this talk, we present various approaches for the construction of such invariant schemes. Some comparisons are made and good performance of invariant schemes is highlighted.
This is a joint work with A. Hamdouni.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

Deep-water capillary-gravity waves on the surface of a three-dimensional fluid exhibit a very interesting range of behavior - including lump or wave-packet solitary waves - and are also numerically challenging. We shall describe some of the models we have put forth and the numerical methods used to compute solutions. We present computations of the dynamics of waves showing interesting inelastic solitary wave collisions and, in some models, computations pointing to a wave-breaking singularity.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

After recalling briefly its derivation from the two-layer system with a rigid lid, we present theoretical and numerical results on a model for large amplitude internal waves which in some sense extends to internal waves the classical Saint-Venant ("Shallow water") system for surface waves. This model turns out to be nonlocal in two horizontal dimensions due to the rigid lid assumption.
This is based on a joint work with Philippe Guyenne and David Lannes.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)

Nowadays, there exist different mathematical formulations to study the dynamics of fully nonlinear free-surface waves. On the other hand, there have been a growing interest in understanding the wind effects on different nonlinear surface waves phenomena, like Benjamin-Feir instability and the formation of extreme "freak" waves. As we shall see, no definitive conlusions about the wind effects in fully nonlinear simulations have been reached, owing to the complexity of the wind-waves interactions problem. In this study, we use a Higher Order Spectral (HOS) method to simulate numerically the nonlinear evolution of gravity waves in the presence of a turbulent airflow above the waves. After a description of the physics involved in the problem of wind-waves interactions, we present various approaches to introduce aerodynamic drag forces in DNS of surgace gravity waves. It will be assumed that the wave motions are "weakly viscous". Thus, we can use a quasi-potential approximation to incorporate weak dissipation effects in our fully nonlinear simulations. Although the coupling of the wave with the airflow, via the pressure surface field, is usually based on the linear approach for the wind-wave interaction problem, we will present methods to go beyong the linear theory of wind-wave interactions. These models of wind effects are such that nonlinearity in the waves can affect the forcing during the course of the evolution.

• Event: "Numerical methods for complex fluid flows - Effective algorithms and physical modeling" (2009)