Section: New Results

Miscellaneous

Participants : Laurent Boudin, Jean-Frédéric Gerbeau, Damiano Lombardi, Sanjay Pant, Marina Vidrascu, Irene Vignon-Clementel.

In [47], we derive the Maxwell-Stefan formalism from the Boltzmann equation for mixtures with general cross-sections. The derivation uses the Hilbert asymptotic method for systems at low Knudsen and Mach numbers. We also formally prove that the Maxwell-Stefan coefficients can be linked to the direct linearized Boltzmann operator for mixtures. That allows to compute the values of the Maxwell-Stefan diffusion coefficients with explicit and simple formulae with respect to the cross-sections. We also justify the specific ansatz we use thanks to the so-called moment method.

In [19] we give a presentation of the mathematical and numerical treatment of plate dynamics problems including rotational inertia. The presence of rotational inertia in the equation of motion makes the study of such problems interesting. We employ HCT finite elements for space discretization and the Newmark method for time discretization in FreeFEM++, and test such methods in some significant cases: a circular plate clamped all over its lateral surface, a rectangular plate simply supported all over its lateral surface, and an L-shaped clamped plate.

In [31] we investigated a modified k-nearest neighbors method to assess the differential entropy of a probability density distribution given a set of samples. Instead of considering a classical Kozachenko-Leonenko approximation, an improved parametric gaussian representation is proposed. The method aims at improving the performances of the classical estimator when considering the probability density distribution of model observations, which are featured by a strong anisotropy or functional dependency.

In [49] a dynamical adaptive tensor method is proposed to build parsimonius discretisations for systems whose domain can be naturally decomposed as a product of sets. A modified Proper Generalised Decomposition step is introduced, that allows to project the equations residual on a tensorised space. Contrary to the majority of the methods proposed, the tensor rank is adapted to guarantee a chosen precision. The method is applied to the Vlasov-Poisson system of equations. In order to preserve the hamiltonian structure of the problem, a symplectic integrator is proposed. The convergence of the method is proved and several high-dimensional test-cases are presented in order to validate the approach.