Past lectures

Abstract

The goal of geometric integration is to derive and analyze numerical methods preserving qualitative and physical properties of

differential systems. In classical mechanics, many phenomenons are described by Hamiltonian ordinary differential equations possessing strong geometric properties (symplecticity of the flow, preservation

of the energy) and their reproduction by numerical methods is not automatically guaranteed. A very important tool in such analysis is given by the backward error analysis theory developed in the nineties stating the following: after discretization of a Hamiltonian ODE by a symplectic time integrator, the discrete numerical solution (almost) coincides with the solution of a modified continuous Hamiltonian system, over extremely long time.

The goal of this course is to establish the same kind of result in

the case of Hamiltonian Partial Differential Equations, with a

particular view towards the discretization of Schrödinger equations arising in computational quantum mechanics. The studying of the corresponding modified Hamiltonian PDE allows to obtain informations on the numerical solution such as a posteriori long time regularity bounds.

The cubic nonlinear Schrödinger equation will serve as guiding example along the whole lecture. We will study stability questions in various situations (small initial data, solitary waves). Moreover resonances issues (numerical and physical) will be addressed, linked with the persistence of the regularity of the solution over long times.