Gil Ariel: Parareal methods for highly oscillatory ordinary differential equations

We introduce a multiscale parareal method that efficiently numerically integrates highly oscillatory ordi- nary differential equations. The algorithm computes a low-cost approximation of all slow variables in the system. Then, fast phase-like variables are obtained using the parareal iterative methodology. The method does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. Convergence of the parareal iterations is proven and demonstrated in numerical examples.