(untitled) (Journal of Mathematical Study, 2014)
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 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