Jean-Paul Chehab

We introduce a Crank-Nicolson scheme to study numerically the long-time behavior of solutions to a one dimensional damped forced nonlinear Schrödinger equation. We prove the existence of a smooth global attractor for these discretized equations. We also provide some numerical evidences of this asymptotical smoothing effect.

In this article, we propose to model the inverse of a given matrix as the state of a proper first order matrix differential equation. The inverse can correspond to a finite value of the independent variable or can be reached as a steady state. In both cases we derive corresponding dynamical systems and establish stability and convergence results. The application of a numerical time marching scheme is then proposed to compute an approximation of the inverse. The study of the underlying schemes can be done by using tools of numerical analysis instead of linear algebra techniques only. With our approach, we recover some known schemes but also introduce new ones. We derive in addition a masked dynamical system for computing sparse inverse approximations. Finally we give numerical results that illustrate the validity of our approach. Mathematical subject classification: 65F10, 65F35, 65L05, 65L12, 65L20, 65N06.