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Geometric Numerical Integration TU M¨unchen Ernst Hairer January – February 2010 Lecture 5: Long-time behavior of wave equations Table of contents 1 Several high frequencies 1 2 Multi-frequency modulated Fourier expansions 4 3 Long-time energy conservation 6 4 Semi-linear wave equations 8 5 Spectral semi-discretization in space 9 6 Full discretizations – main results 13 ¨ 7 The Stormer–Verlet / leapfrog discretization 16 We first extend the technique of modulated Fourier expansion to problems with several high frequencies, which can be resonant or non-resonant. An extension to infinitely many non-resonant high frequencies permits to obtain interesting results on the long-time behavior of nonlinearly perturbed wave equations. 1 Several high frequencies We consider Hamiltonians of the form ℓ H (x, x˙) = 1 x˙ 2 + ω2 x 2 + U (x), (1) 2 j j j j =0 where x = (x , x , . . ., x ) with x ∈ Rdj , ω = λ /ε with λ = 0, distinct 0 1 ℓ j j j 0 λj 0, and small ε 0. After rescaling ε we may assume λj ≥ 1 for all j . The equations of motion are a system of second-order differential equations 2 x