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A proper orthogonal decomposition-compact difference algorithm for plate vibration models

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Abstract

In this article, we propose a proper orthogonal decomposition-compact difference scheme (POD-CDS) for the displacement-stress form of a simply supported plate vibration model. We prove that the POD-CDS can preserve the same spatial and temporal convergence rates and unconditional stability with the control constant independent of the time levels for the displacement and the stress in discrete \(H^1\)-norm as the compact difference solution does, as well as improve the computing efficiency significantly, which confirms the reliability of the POD-CDS for long time simulation of the vibration models. Furthermore, we conduct the stability and convergence analysis for the corresponding compact difference scheme in discrete \(H^k\)-norm as \(k=0,1,2,\) respectively, which compensates those missing estimates of the references Q. Li and Q. Yang (Adv. Differ. Equ. 328: 1–19, 2019) and Q. Li, Q. Yang, and H. Chen (Numer. Methods Partial Differ. Equ. 36: 1938–1961, 2020). Numerical experiments are provided to verify these theoretical findings and show that the POD-CDS possesses nearly 10–30 times computing efficiency faster than the compact difference scheme does.

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Data Availability

The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to thank the editor and referees for their valuable advices for the improvement of this article.

Funding

Q. Li and H. Chen were supported by the NSF of China under grant Nos. 12171287, 11971276, 11471196, H. Wang was supported by the ARO MURI Grant W911NF-15-1-0562, and by the National Science Foundation under Grant DMS-2012291.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, 250014, Jinan, Shandong, China

    Qing Li & Huanzhen Chen

  2. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Hong Wang

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  1. Qing Li
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  3. Hong Wang
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Correspondence to Huanzhen Chen.

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Li, Q., Chen, H. & Wang, H. A proper orthogonal decomposition-compact difference algorithm for plate vibration models. Numer Algor 94, 1489–1518 (2023). https://doi.org/10.1007/s11075-023-01544-7

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