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

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.