Analysis of the parareal algorithm for linear parametric differential equations

This paper presents a parareal algorithm with parameterized propagators for linear parametric differential equations over a wide range of parameters. Through transforming the initial value problem into nonparametric ODEs based on Taylor series, we construct the general parameterized fine and coarse propagators for the parareal algorithm in each time subinterval. Furthermore, to accelerate the convergence of the algorithm, the coarse propagator based on the waveform relaxation method is proposed. By analysing the computational complexity of the WR propagator and the general coarse propagator, we find these two propagators are appropriate for the different situations. Finally, the convergence analysis of the parareal algorithm with these propagators is presented and our analysis is illustrated with two numerical experiments.