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Waveform relaxation for fractional sub-diffusion equations

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Abstract

We report a new kind of waveform relaxation (WR) method for general semi-linear fractional sub-diffusion equations, and analyze the upper bound for the iteration errors. It indicates that the WR method is convergent superlinearly, and the convergence rate is dependent on the order of the time-fractional derivative and the length of the time interval. In order to accelerate the convergence, we present the windowing WR method. Then, we elaborate the parallelism based on the discrete windowing WR method, and present the corresponding fast evaluation formula. Numerical experiments are carried out to verify the effectiveness of the theoretic work.

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Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments and valuable suggestions to improve the paper significantly.

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 11401589, 11871393, and 11871400), the Fundamental Research Funds for the Central Universities (Nos. 18CX02049A and 17CX02066), and the International Science and Technology Cooperation Program of Shaanxi Research and Development Plan (2019KWZ-08).

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Authors and Affiliations

  1. College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China

    Jun Liu & Yan Wang

  2. Department of Mathematical Sciences, an Jiaotong University, Xi’an, 710049, Shaanxi, China

    Yao-Lin Jiang

  3. School of Science, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    Xiao-Long Wang

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  1. Jun Liu
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  2. Yao-Lin Jiang
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  3. Xiao-Long Wang
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  4. Yan Wang
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Correspondence to Jun Liu.

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Liu, J., Jiang, YL., Wang, XL. et al. Waveform relaxation for fractional sub-diffusion equations. Numer Algor 87, 1445–1478 (2021). https://doi.org/10.1007/s11075-020-01014-4

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