  For an integrator when applied to a highly oscillatory system, the near
conservation of the oscillatory energy over long times is an important aspect.
In this paper, we study the long-time near conservation of oscillatory energy
for the adopted average vector field (AAVF) method when applied to highly
oscillatory Hamiltonian systems. This AAVF method is an extension of the
average vector field method and preserves the total energy of highly
oscillatory Hamiltonian systems exactly. This paper is devoted to
analysinganother important property of AAVF method, i.e., the near conservation
of its oscillatory energy in a long term. The long-time oscillatory energy
conservation is obtained via constructing a modulated Fourier expansion of the
AAVF method and deriving an almost invariant of the expansion. A similar result
of the method in the multi-frequency case is also presented in this paper.
