  We present the Parareal-CG algorithm for time-dependent differential
equations in this work. The algorithm is a parallel in time iteration algorithm
utilizes Chebyshev-Gauss spectral collocation method for fine propagator F and
backward Euler method for coarse propagator G. As far as we know, this is the
first time that the spectral method used as the F propagator of the parareal
algorithm. By constructing the stable function of the Chebyshev-Gauss spectral
collocation method for the symmetric positive definite (SPD) problem, we find
out that the Parareal-CG algorithm and the Parareal-TR algorithm, whose F
propagator is chosen to be a trapezoidal ruler, converge similarly, i.e., the
Parareal-CG algorithm converge as fast as Parareal-Euler algorithm with
sufficient Chebyhsev-Gauss points in every coarse grid. Numerical examples
including ordinary differential equations and time-dependent partial
differential equations are given to illustrate the high efficiency and accuracy
of the proposed algorithm.
