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A second-order unconditionally energy stable scheme for the Lifshitz-Petrich model integrated with observational data

Author

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  • Hu, Xiaochuan
  • Kim, Junseok
  • Li, Yibao

Abstract

In this paper, we introduce a modified Lifshitz-Petrich model that incorporates a data assimilation term. This model is used to investigate the nucleation of quasicrystalline structures in polycrystalline materials by leveraging information from observational data. Guided by the principle of feedback control, the data assimilation term drives the solution toward the observational data sampled from the reference process. Using the second-order backward differentiation formula and the scalar auxiliary variable method, we introduce an efficient numerical scheme for the modified Lifshitz-Petrich model. We employ the Fourier spectral method to achieve second-order accuracy and high computational efficiency. And we prove the numerical discrete energy is unconditionally stable. A series of numerical experiments are conducted to evaluate the efficiency and robustness of the proposed method.

Suggested Citation

  • Hu, Xiaochuan & Kim, Junseok & Li, Yibao, 2026. "A second-order unconditionally energy stable scheme for the Lifshitz-Petrich model integrated with observational data," Applied Mathematics and Computation, Elsevier, vol. 518(C).
    • Handle: RePEc:eee:apmaco:v:518:y:2026:i:c:s0096300325006393
    DOI: 10.1016/j.amc.2025.129914
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    References listed on IDEAS

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    1. Li, Dongfang & Li, Xiaoxi & She, Mianfu & Sun, Hai-wei, 2026. "High-order, linearly implicit, and energy-stable methods for Cahn–Hilliard models with degenerate mobility," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 177-190.
    2. Li, Dongfang & Li, Xiaoxi & Mei, Ming & Yuan, Wanqiu, 2023. "A structure-preserving and variable-step BDF2 Fourier pseudo-spectral method for the two-mode phase field crystal model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 483-506.
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