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6 - Modulated Fourier Expansions for Continuous and Discrete Oscillatory Systems

Published online by Cambridge University Press:  05 December 2012

By
E. Hairer  and
Ch. Lubich
Edited by
Felipe Cucker ,
Teresa Krick ,
Allan Pinkus  and
Agnes Szanto
E. Hairer
Affiliation:
Université de Genève
Ch. Lubich
Affiliation:
Universität Tübingen
Felipe Cucker
Affiliation:
City University of Hong Kong
Teresa Krick
Affiliation:
Universidad de Buenos Aires, Argentina
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Agnes Szanto
Affiliation:
North Carolina State University
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Summary

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: 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.

Introduction

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.

Type
Chapter
Information
Foundations of Computational Mathematics, Budapest 2011 , pp. 113 - 128
Publisher: Cambridge University Press
Print publication year: 2012

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