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Unconditional long-time stability-preserving second-order BDF fully discrete method for fractional Ginzburg-Landau equation

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Abstract

In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. More precisely, we consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in the \(L^2\), \(H^1\), \(H^{1+\alpha }\) and \(H^2\) norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results.

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Funding

This study was funded by the National Natural Science Foundation of China (Grant Nos. 12271367), Shanghai Science and Technology Planning Projects (Grant No. 20JC1414200), Natural Science Foundation of Shanghai (Grant No. 20ZR1441200), and The Scientific Research Fund of Hunan Provincial Education Department (Grant No. 22B0879).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Yi Huang, Wansheng Wang & Yanming Zhang

  2. School of Mathematics and Statistics, Hunan First Normal University, Changsha, 410006, China

    Yanming Zhang

Authors
  1. Yi Huang
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  2. Wansheng Wang
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  3. Yanming Zhang
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Contributions

Y. Huang and W. S. Wang wrote the main manuscript text and Y. M. Zhang prepared the figures. All authors reviewed the manuscript.

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Correspondence to Wansheng Wang.

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Huang, Y., Wang, W. & Zhang, Y. Unconditional long-time stability-preserving second-order BDF fully discrete method for fractional Ginzburg-Landau equation. Numer Algor 97, 167–189 (2024). https://doi.org/10.1007/s11075-023-01699-3

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Mathematics Subject Classification (2010)