Parareal multiscale methods for highly oscillatory dynamical systems

Gil Ariel, Seong Jun Kim, Richard Tsai

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme 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. We propose an alignment algorithm for almost-periodic solutions, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.

Original languageEnglish
Pages (from-to)A3540-A3564
JournalSIAM Journal of Scientific Computing
Issue number6
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.


  • Highly oscillatory problems
  • Multiscale computation
  • Parallel algorithms


Dive into the research topics of 'Parareal multiscale methods for highly oscillatory dynamical systems'. Together they form a unique fingerprint.

Cite this