Section: New Results

Geometric numerical integration and Schrödinger equations

Participant : Erwan Faou.

The goal of geometric numerical integration is the simulation of evolution equations by preserving their geometric properties over long times. This question is of particular importance in the case of Hamiltonian partial differential equations typically arising in many application fields such as quantum mechanics or wave propagations phenomena. This implies many important dynamical features such as energy preservation and conservation of adiabatic invariants over long times. In this setting, a natural question is to know how and to which extent the reproduction of such long time qualitative behavior is ensured by numerical schemes.

Starting from numerical examples, these notes [37] try to provide a detailed analysis in the case of the Schrödinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. This text analyzes the possible stability and instability phenomena induced by space and time discretization, and provides rigorous mathematical explanations for them.