Boundaries, Numerics, Dispersion – BoND

This project is focused on evolution problems in which dispersion is predominant compared to other phenomena such as diffusion. It is motivated by physical applications in which, in- deed, diffusion is negligible, and the total energy is - to some extent - conserved, and also by numerical issues regarding the approximation, with the least possible amount of numer- ical viscosity, of hyperbolic systems of conservation laws. The model, dispersive equations that are being considered include the Korteweg-de Vries, Nonlinear Schro¨dinger, Kadomtsev– Petviashvili, Kawahara, and the Davey–Stewartson equations, but also the more complicated systems of Euler–Korteweg (for capillary fluids), and of Green–Naghdi (for water waves). All these equations and systems are taken in their most general form, which means that their non- linearities are not predefined, and that integrability arguments should not have a preponderant importance in the proposed work. Furthermore, the project aims at dealing with problems in which boundaries play an important role, be they ‘physical’, fixed boundaries such as walls or pipe extremities in fluid flows, artificial boundaries introduced for numerical purposes, mov- ing boundaries such as shocks in compressible fluids or free surface on top of incompressible fluids submitted to gravity. If the project does not involve any numerical analysis in the usual sense - e.g. proving the convergence of numerical methods - , the numerics is nevertheless ubiquitous, with the will of elaborating innovative numerical schemes and of understanding their qualitative properties. Hence its title.

One of its originalities is that, for various reasons, it is situated at the ‘interface’ between the hyperbolic and dispersive partial differential equations. A first task concerns the analy- sis of continuous, and also discrete, dispersive Initial Boundary Value Problems (IBVP), the latter coming from the discretization of hyperbolic IBVP and requiring a generalization of the Gustafsson-Kreiss-Sundstro¨m stability theory. Within this task, the design and analysis of Artificial Boundary Conditions (ABC) for the models mentioned above is a challenging issue. Another issue is the qualitative analysis of various remarkable solutions of those model equa- tions, which we refer to as dispersive patterns. These include periodic waves, which are special, traveling waves with a usually large number of degrees of freedom, as well as disper- sive shocks, which are complicated, unsteady patterns. Their understanding is closely related by the modulated theory initiated by Whitham in the 1970s. In order to go further, some asymptotic analysis is needed, which is also the case for the analysis of small amplitude wave trains, another topic of interest in this project. Finally, a maybe more applied purpose is to derive new models that take into account multidimensional effects in such spectacular phenomena as tidal bores and roll waves.

Even though it is somewhat hidden here above, if one should retain a single keyword for this project, it should certainly be stability.