Section: New Results

Other contributions

Corrosion models

The Diffusion Poisson Coupled Model [47] is a model of iron based alloy in a nuclear waste repository. It describes the growth of an oxide layer in this framework. The system is made of a Poisson equation on the electrostatic potential and convection-diffusion equations on the densities f charge carriers (electrons, ferric cations and oxygen vacancies), supplemented with coupled Robin boundary conditions. The DPCM model also takes into account the growth of the oxide host lattice and its dissolution, leading to moving boundary equations. In [12] , C. Chainais-Hillairet and I. Lacroix-Violet consider a simplified version of this model, where only two charge carriers are taken into account and where there is no evolution of the layer thickness. They prove the existence of a steady-state solution to this model. More recently, C. Chainais-Hillairet and I. Lacroix-Violet have also obtained an existence result for the time-dependent simplified model. This result will be soon submitted for publication.

In [4] , C. Chainais-Hillairet and coworkers have studied some numerical methods for the approximation of the DPCM model. The choice of the numerical methods is justified by a stability analysis and by the study of their numerical performance. These methods have been implemented in the code CALIPSO developed at ANDRA. Numerical experiments with real-life data show the efficiency of the developed methods.

Transparent boundary conditions for the Helmholtz equation

C. Besse and I. Violet start a collaboration with S. Fliss (Poems), J. Coatleven and K. Ramdani (Corida) to build transparent boundary conditions for the Helmholtz equation. They propose in [6] a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media. They obtain a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via Floquet-Bloch transform, a family of elementary strip problems and a family of stationary Riccati equations. The DtD operator is the solution of an affine operator valued equation which can be reformulated as a non standard integral equation.

Analysis of subcycling techniques

Several physics situations involve phenomena which occur on very different time scales. A popular option to integrate the equations in time in this context is to use sub-cycling techniques, which allow to weaken the stability constraints. Several questions are still open for the asymptotic behavior of such methods, e.g. the preservation of equilibrium states. New results about the asymptotic orders if such methods have been derived on toy-model problems which allow a better understanding of these methods and their preservation of equilibrium states [40] .

Phase transitions

We analyzed numerically a forward-backward diffusion equation with a cubic- like diffusion function, –emerging in the framework of phase transitions modelling– and its “entropy” formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we proposed schemes for both the second and the third order equations, we discussed the analytical properties of their semi-discrete counter- parts and we compared the numerical results in the case of initial data of Riemann type, showing strengths and flaws of the two approaches, the main emphasis being put on the propagation of transition interfaces. This is a joint work with C. Mascia (Univ. La Sapienza) [25] .

Modelling of the biological populations

We worked on two problems of biological populations: the understanding of the occurrence of collective behavior for large populations and the extinction probabilities in some population dynamics.

Several approaches are used in the modelling of collective behavior models for large populations of fish : we obtained results at the particle and kinetic levels for a model involving self-propulsion, friction and an attractive/repulsive potential. By introducing a new dimensionless setting, we identified five parameters that govern the possible asymptotic states for this system (clumps, spheres, dispersion, mills, rigid-body rotation, flocks) and performed a numerical analysis on the 3D particle-setting. Also, we described the kinetic system derived as the limit from the particle model as N tends to infinity; and we proposed, in 1D, a numerical scheme for the simulations, and performed a numerical analysis devoted to trying to recover asymptotically patterns similar to those emerging for the equivalent particle systems, when particles originally evolved on a circle. this is a joint work with J. Rosado (UCLA) and F. Vecil (Univ. Valencia) [43] .

The extinction probabilities of a flower population may be modelled by an imhomogeneous random walk on the positive quadrant. On the one hand, introducing the generating function, that solves a PDE, we computed an explicit solution. On the other hand, we compared stochastic and deterministic resolutions of the random walk. This is a joint work with K. Raschel (Univ. Tours), V. C. Tran (Univ. Lille 1) [26] .