| The lectures give an introduction to numerical methods that pre-serve geometric properties of the ?ow of a di?erential equation: sym-plectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving ?rst integrals, etc. The main issue is long-time integration, which can be understood with the help of backward error analysis for ordinary di?erential equations, and with modulated Fourier expansions for problems with highly oscillatory so-lutions. First lecture. The ideas of “geometric numerical integration” are presented. Main emphasis is put on symplectic methods applied to Hamiltonian di?erential equations. In particular, symplectic Runge– Kutta methods, splitting methods, and variational integrators are dis-cussed. Second lecture. The long-time behaviour of numerical integrators is explained by using backward error analysis. This is done by consid-ering a modi?ed di?erential equation and a modi?ed Hamiltonian (for symplectic methods). A recent application of backward error analy-sis provides insight into the near-preservation of energy for the Boris algorithm in charged particle dynamics. Third lecture. For problems, where high oscillations originate from a linear part in the di?erential equation (e.g., Fermi-Pasta-Ulam-type problems), trigonometric time integrators are considered. Besides the energy, which is exactly conserved, the oscillatory energy is nearly con-served (adiabatic invariant). Concerning the long-time behaviour, the technique of modulated Fourier expansions gives much insight. Most of the material for these lectures are taken from the monograph “Geometric Numerical Integration” (see below). Further references are: K. Feng, M.-Z. Qin. Symplectic geometric algorithms for Hamiltonian systems. Zhejiang Science and Technology Publishing House, Hangzhou, 2010. Translated and revised from the Chinese original. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Di?erential Equations. 2nd edition. Springer Series in Comput. Math., vol. 31, 2006. B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics. Cam-bridge Monographs on Applied and Computational Mathematics, vol. 14, 2004. J.M. Sanz-Serna, M.P. Calvo, Numerical Hamiltonian Problems. Chap-man & Hall, Appl. Math. and Math. Comput., vol. 7, 1994. Section de Math′ematiques, Universit′e de Gen`eve, Switzerland E-mail address: Ernst.Hairer@unige.ch URL: http://www.unige.ch/～hairer/ | 
