A model order reduction based adaptive parareal method for time-dependent partial differential equations

In this paper, we propose a model order reduction based adaptive parareal method for time-dependent partial differential equations. By using the data obtained by the fine propagator from neighboring time subintervals and previous neighboring parareal iterations together with some model order reduction technique, we construct some POD subspaces locally in time and use them to construct the coarse propagator for each parareal iteration. We apply this new method to solve some 3D time-dependent advection-diffusion equations with the Kolmogorov flow and the ABC flow. Numerical results show that our method can achieve high accuracy with significant speedup for long-term evolution problems.