Parareal algorithm via Chebyshev-Gauss spectral collocation method

Subjects: Mathematics >> Computational Mathematics. submitted time 2023-04-20

Zhou, Quan Liu, Yicheng Wu Shu-Lin

Abstract： 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. 

Peer Review Status:Awaiting Review

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