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We present the Parareal-CG algorithm for time-dependent differentialequations in this work. The algorithm is a parallel in time iteration algorithmutilizes Chebyshev-Gauss spectral collocation method for fine propagator F andbackward Euler method for coarse propagator G. As far as we know, this is thefirst time that the spectral method used as the F propagator of the pararealalgorithm. By constructing the stable function of the Chebyshev-Gauss spectralcollocation method for the symmetric positive definite (SPD) problem, we findout that the Parareal-CG algorithm and the Parareal-TR algorithm, whose Fpropagator is chosen to be a trapezoidal ruler, converge similarly, i.e., theParareal-CG algorithm converge as fast as Parareal-Euler algorithm withsufficient Chebyhsev-Gauss points in every coarse grid. Numerical examplesincluding ordinary differential equations and time-dependent partialdifferential equations are given to illustrate the high efficiency and accuracyof the proposed algorithm.

Parareal algorithm via Chebyshev-Gauss spectral collocation method