|Date:||Mon, November 21, 2011|
|Place:||Research II Lecture Hall|
Abstract: We consider semilinear Hamiltonian evolution equations, for example the semilinear wave equation. When the linear part of the evolution equation is skew symmetric and the nonlinear part is smooth on a suitable scale of Hilbert spaces we show that its flow and an A-stable Runge-Kutta time discretization of it are smooth as maps from an open subset of the highest rung of the scale into the lowest rung, and obtain full order convergence results for the time-discretization. Then we prove fractional order convergence results for the trajectory error of the time-semidiscretization in the case of non-smooth initial data, and show that the energy error, for nonsmooth initial data, has much higher order than the trajectory error. Finally, if time allows, we show that for analytic data symplectic A-stable RK methods conserve a modified energy up to an exponentially small error. This is joint work with Marcel Oliver and Chris Evans (Paris).