Optimally accurate integrators with long time conservation for highly oscillatory second-order differential equations

In this paper, we are concerned with optimally accurate integrators for highly oscillatory second-order differential equations $\ddot{q}(t)+\frac{1}{\varepsilon^2}A q(t) =\frac{1}{\varepsilon^m}F(q(t))$ with large initial data, a scaling parameter $0<\varepsilon\ll 1$ and $m=0,1$. 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 proposed 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 $\mathcal{O}(\varepsilon^{r-rm} h^r)$ 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 multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on the Duffing equation and a nonlinear relativistic Klein--Gordon equation show that the proposed integrators used with large stepsizes have optimal uniformly high accuracy, excellent long time energy conservation and competitive efficiency.