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Long-time energy conservation of numerical methods for oscillatory differential equations

Lubich, Christian
Published in SIAM Journal on Numerical Analysis. 2000, vol. 38, no. 2, p. 414-441
Abstract We consider second-order differential systems where high-frequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determines its coefficients. These invariants are related to the total energy and the oscillatory harmonic energy of the original system. For the numerical solution we study a class of symmetric methods %of order 2 that discretize the linear part without error. We are interested in the case where the product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the near-conservation of the total and the oscillatory energy over very long time intervals.
Keywords Oscillatory differential equationsLong-time energy conservationSecond-order symmetric methodsFrequency expansionBackward error analysisFermi-Pasta-Ulam problem
Stable URL http://archive-ouverte.unige.ch/unige:12329
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Deposited on : 2010-11-05

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