Commun. Comput. Phys., 6 (2009), pp. 883-902.

Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs

Shulin Wu 1, Baochang Shi 1, Chengming Huang 1*

1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.

Received 28 September 2008; Accepted (in revised version) 11 March 2009
Available online 6 April 2009


The parareal algorithm, proposed firstly by Lions et al. [J. L. Lions, Y. Maday, and G. Turinici, A "parareal" in time discretization of PDE's, C.R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 661-668], is an effective algorithm to solve the time-dependent problems parallel in time. This algorithm has received much interest from many researchers in the past years. We present in this paper a new variant of the parareal algorithm, which is derived by combining the original parareal algorithm and the Richardson extrapolation, for the numerical solution of the nonlinear ODEs and PDEs. Several nonlinear problems are tested to show the advantage of the new algorithm. The accuracy of the obtained numerical solution is compared with that of its original version (i.e., the parareal algorithm based on the same numerical method).

AMS subject classifications: 65Y05, 65Y10, 65Y20, 37M05, 65-05

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Key words: Parallel computation, parareal algorithm, Richardson extrapolation, accuracy, nonlinear problems.

*Corresponding author.
Email: (S. L. Wu), (B. C. Shi), (C. M. Huang)

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