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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: longtime 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.
As a new analytical tool developed in the past decade, modulated Fourier expansions have been found useful to explain various long-time phenomena in both continuous and discretized oscillatory Hamiltonian systems, ordinary differential equations as well as partial differential equations. In addition, modulated Fourier expansions have turned out useful as a numerical approximation method in oscillatory systems.
In this review paper we first show some long-time phenomena in oscillatory systems, then give theoretical results that explain these phenomena, and finally outline the basics of modulated Fourier expansions with which these results are proved.
The numerical treatment of ordinary differential equations has continued to be a lively area of numerical analysis for more than a century, with interesting applications in various fields and rich theory. There are three main developments in the design of numerical techniques and in the analysis of the algorithms:
Non-stiff differential equations. In the 19th century (Adams, Bashforth, and later Runge, Heun and Kutta), numerical integrators have been designed that are efficient (high order) and easy to apply (explicit) in practical situations.
Stiff differential equations. In the middle of the 20th century one became aware that earlier developed methods are impractical for a certain class of differential equations (stiff problems) due to stability restrictions. New integrators (typically implicit) were needed as well as new theories for a better understanding of the algorithms.
Geometric numerical integration. In long-time simulations of Hamiltonian systems (molecular dynamics, astronomy) neither classical explicit methods nor implicit integrators for stiff problems give satisfactory results. In the last few decades, special numerical methods have been designed that preserve the geometric structure of the exact flow and thus have an improved long-time behaviour.
The basic developments (algorithmic and theoretical) of these epochs are documented in the monographs [HNW93], [HW96], and [HLW06]. Within geometric numerical integration we can also distinguish between non-stiff and stiff situations. Since here the main emphasis is on conservative Hamiltonian systems, the term “stiff” has to be interpreted as “highly oscillatory”.
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