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Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation

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Abstract

The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are nonoverlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of these space and time parallel strategies, we present and analyze two parareal algorithms based on the Dirichlet-Neumann and the Neumann-Neumann waveform relaxation method for the heat equation by choosing Dirichlet-Neumann/Neumann-Neumann waveform relaxation as two new kinds of fine propagators instead of the classical fine propagator. Both new proposed algorithms could be viewed as a space-time parallel algorithm, which increases the parallelism both in space and in time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments finally.

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Acknowledgments

We would like to thank the anonymous referee for his valuable comments.

Funding

This work was supported by the Natural Science Foundation of China (NSFC) under grants 11801449, 11871393 and 11871400; the key project of the International Science and Technology Cooperation Program of Shaanxi Research & Development Plan under grant 2019KWZ-08; the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2019JQ-617) and the Fundamental Research Funds for the Central Universities under grant G2018KY0306.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

    Bo Song & Xiaolong Wang

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Yao-Lin Jiang

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  1. Bo Song
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  2. Yao-Lin Jiang
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  3. Xiaolong Wang
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Correspondence to Bo Song.

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Song, B., Jiang, YL. & Wang, X. Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation. Numer Algor 86, 1685–1703 (2021). https://doi.org/10.1007/s11075-020-00949-y

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