Modulated Fourier expansions for continuous and discrete oscillatory systems
E. Hairer and C. Lubich
Abstract. This article reviews some of the phenomena and theoretical results on the long-time energy behaviour of continuous and discretized oscillatory systems that can be explained by modulated Fourier expansions: long-time preservation of total and oscillatory energies in oscillatory Hamiltonian systems and their numerical discretizations, near-conservation of energy and angular momentum of symmetric multistep methods for celestial mechanics, metastable energy strata in nonlinear wave equations. We describe what modulated Fourier expansions are and what they are good for.
Key Words.Modulated Fourier expansion, highly oscillatory differential equations, nonlinear wave equations, numerical integrators, linear multistep methods.