Normal forms and geometric numerical integration of Hamiltonian PDEs

E. Faou, INRIA, Rennes, France

Wednesday, April 1
at 16.15, HG E 1.2

We present some recent developments in the understanding of the long-time behavior of geometric integrators applied to Hamiltonian PDEs. We first consider the case of splitting methods applied to a wide classe of semi-linear Hamiltonian PDEs. Using normal form techniques, we show the long-time preservation of the regularity of the numerical solution, provided the initial value is small in Sobolev norm and under a generic non resonance assumption on the step size This is a joint work with B. Gr\'ebert and E. Paturel (University of Nantes). We then show how the use of implicit-explicit integrators can allow to avoid resonance issues, and yields backward error analysis results similar to the finite dimensional case when applied to the linear Schr\"odinger equation (joint work with A. Debussche).