Large-stepsize integrators with improved uniform accuracy and long time conservation for highly oscillatory systems with large initial data

In this paper, we are concerned with large-stepsize highly accurate integrators for highly oscillatory second-order differential equations with large initial data and a scaling parameter $0 <\varepsilon\ll 1$. The highly oscillatory property of this model problem corresponds to the parameter $\varepsilon$. We propose and analyze a novel class of highly accurate integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy is improved to be $\mathcal{O}(\varepsilon^2 h^r)$ in the absolute position error for the time stepsize $h$ and the order $r$ of the integrator. The near energy conservation over long times is established for the integrators with large time stepsizes. Numerical results show that the proposed integrators used with large stepsizes have improved uniformly high accuracy and excellent long time energy conservation.