Numerical energy conservation for multi-frequency oscillatory
differential equations
David Cohen, Ernst Hairer and Christian Lubich
Abstract.
The long-time near-conservation of the total and oscillatory energies of numerical integrators for
Hamiltonian systems with highly oscillatory solutions is studied
in this paper.
The numerical methods considered are second-order
symmetric trigonometric integrators
and the Störmer-Verlet method.
Previously obtained results for systems with a single high frequency
are extended to the multi-frequency
case, and new insight into the long-time behaviour
of numerical solutions is gained for resonant frequencies.
The results are obtained using modulated multi-frequency Fourier expansions
and the Hamiltonian-like structure of the modulation system.
A brief discussion of
conservation properties in the continuous problem is also included.
Key Words.
Gautschi-type numerical methods, Störmer-Verlet method,
Hamiltonian systems, modulated Fourier expansion, energy conservation,
oscillatory solutions.