Analysis of a New Krylov subspace enhanced parareal algorithm for time-periodic problems

The classical parareal algorithm for time-periodic problems, solving a periodic-like coarse problem, called the periodic parareal algorithm with periodic coarse problem (PP-PC), usually converges slowly. In this paper, we present a new parallel-in-time algorithm for time-periodic problems based on the classical PP-PC algorithm and the Krylov subspace method. In this new algorithm, a new propagator derived by the Krylov subspace is chosen as the coarse propagator instead of the classical coarse propagator in the PP-PC algorithm. And because of the special characteristic of time-periodic problems, the Krylov subspace enhanced PP-PC algorithm needs to solve a periodic coarse problem on the coarse time grid on each iteration. We provide two different kinds of theoretical bounds under different assumptions for the proposed algorithm. Numerical results illustrate our analysis with two effective theoretical bounds for the heat equation, the wave equation, and the viscous Burgers equation, where we could also find that the new proposed algorithm converges faster than the classical PP-PC algorithm.