Parallel-in-Time Solution of Time-Periodic Problems with Unknown Period

Iryna Kulchytska-Ruchka and Sebastian Schöps (Technical University of Darmstadt)

 In this talk we consider a novel parallel-in-time algorithm for time-periodic problems where the period is not given. Inheriting the idea of the periodic Parareal approach PP-PC [1], the proposed method uses discretization on a two-level grid and calculates not only the initial values at each subinterval but also the corresponding period iteratively. This approach extends the multiple shooting method with unknown period [2] by the Parareal-based approximation of the Jacobian. A particular diagonalization of the resulting nonlinear time-periodic coarse-grid system [3] is introduced, thereby permitting further parallelization on the coarse level. Performance of the introduced algorithm is illustrated for a Colpitt oscillator model.

[1] M. J. Gander, Y.-L. Jiang, B. Song, and H. Zhang. Analysis of two parareal algorithms for time-periodic problems. SIAM J. Sci. Comput., 35(5):A2393–A2415, 2013.

[2] P. Deuflhard. Computation of periodic solutions of nonlinear ODEs. BIT, (24):456–466, 1984.

[3] I. Kulchytska-Ruchka and S. Schöps. Efficient parallel-in-time solution of time-periodic problems using a multi-harmonic coarse grid correction, 2019. ArXiv: 1908.05245.

This work is supported by the ‘Excellence Initiative’ of the German Federal and State Governments, the Graduate School of Computational Engineering at Technische Universit\”at Darmstadt, the BMBF grant No. 05M2018RDA (PASIROM).