The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long periods of time. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long periods of time. In this setting, a natural question is how and to which extent the reproduction of such longtime qualitative behavior can be ensured by numerical schemes. Starting from numerical examples, these notes provide a detailed analysis of the Schrödinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations are provided for them. The book grew out of a graduatelevel course and is of interest to researchers and students seeking an introduction to the subject matter. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in geometric numerical integration, symplectic integrators, backward error analysis, and Schrödinger equations. Table of Contents  Introduction
 Finitedimensional backward error analysis
 Infinitedimensional and semidiscrete Hamiltonian flow
 Convergence results
 Modified energy in the linear case
 Modified energy in the semilinear case
 Introduction to longtime analysis
 Bibliography
 Index
