In this paper, two multiscale time integrators (MTIs), motivated from two
types of multiscale decomposition by either frequency or frequency and amplitude, are
proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter 0 < ε ≤ 1. In fact, the solution to this equation
propagates waves with wavelength at O(ε^{2}) when 0<ε≪1, which brings significantly
numerical burdens in practical computation. We rigorously establish two independent
error bounds for the two MTIs at O(τ^{2}/ε^{2}) and O(ε^{2}) for ε ∈ (0,1] with τ > 0 as step
size, which imply that the two MTIs converge uniformly with linear convergence rate
at O(τ) for ε ∈ (0,1] and optimally with quadratic convergence rate at O(τ^{2}) in the
regimes when either ε=O(1) or 0<ε≤τ. Thus the meshing strategy requirement (or
ε-scalability) of the two MTIs is τ =O(1) for 0<ε≪1, which is significantly improved
from τ =O(ε^{3}) and τ =O(ε^{2}) requested by finite difference methods and exponential
wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards
better understanding on the convergence and resolution properties of the two MTIs.
In addition, numerical results support the two error bounds very well.