Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations (Journal of Mathematical Study, 2014)
In this paper, two multiscale time integrators (MTIs), motivated from twotypes of multiscale decomposition by either frequency or frequency and amplitude, areproposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0 < ε≤ 1.$ In fact, the solution to this equationpropagates waves with wavelength at $O(ε^2)$ when $0 0$ as stepsize, which imply that the two MTIs converge uniformly with linear convergence rateat $O(τ)$ for $ε ∈ (0,1]$ and optimally with quadratic convergence rate at $O(τ^2)$ in theregimes when either $ε=O(1)$ or $0