Section: New Results

Miscellaneous

This section gathers results from members of the team that are not directly related to the core of the scientific program of the team.

In [12], I. Violet-Lacroix and co-authors consider the derivation of continuous and fully discrete artificial boundary conditions for the linearized Korteweg-de-Vries equation. They are provided for two different numerical schemes. The boundary conditions being nonlocal with respect to time variable, they propose fast evaluations of discrete convolutions. Various numerical tests are presented to show the effectiveness of the constructed artificial boundary conditions.

A semi-discrete in time Crank-Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation in the whole space $(ℝ$ is considered by C. Calgaro and co-authors in [28]. They prove that such semi-discrete equation provides a discrete infinite dimensional dynamical in $H^{\alpha }\left(ℝ\right)$ that possesses a global attractor. They show also that if the external force is in a suitable weighted Lebesgue space then this global attractor has a finite fractal dimension.

In [35], F. Nabet considers a finite-volume approximation, based on a two point flux approximation, for the Cahn-Hilliard equation with dynamic boundary conditions. An error estimate for the fully-discrete scheme on a possibly smooth non-polygonal domain is proved and numerical simulations which validate the theoretical result are given.