Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations

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 $0 0$ 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(τ²) in the regimes when either $ε=O(1)$ or $0