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.

翻译：本文提出用于时间依赖微分方程的Parareal-CG算法。该算法是一种并行时间迭代算法，采用Chebyshev-Gauss谱配置法作为精细传播子F，并采用后向欧拉法作为粗粒度传播子G。据我们所知，这是首次将谱方法作为Parareal算法的F传播子。通过针对对称正定问题构建Chebyshev-Gauss谱配置法的稳定性函数，我们发现Parareal-CG算法与采用梯形法则作为F传播子的Parareal-TR算法具有相似的收敛特性，即在每个粗网格上配置足够多的Chebyshev-Gauss点时，Parareal-CG算法的收敛速度与Parareal-Euler算法相当。文中通过常微分方程和时间依赖偏微分方程的数值算例，验证了所提算法的高效性和精确性。