Long-time energy conservation of numerical
methods for oscillatory differential equations
E. Hairer
and
Ch. Lubich
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.
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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.
Key Words.
Oscillatory differential equations, long-time energy conservation,
second-order symmetric methods,
frequency expansion, backward error analysis, Fermi-Pasta-Ulam problem