Temporal error estimates of the BDF2 numerical scheme with variable time steps for the square phase-field crystal model

In this study, we propose a temporally adaptive semi-discrete computational approach for the square phase-field crystal (SPFC) model, which adopts the variable-time-step BDF2 (VBDF2) temporal discretization. By overcoming the difficulties caused by its high-order non-linear term $\Delta \nabla \cdot ({|\nabla u|}^2\nabla u)$ and complex variable-time-step coefficients, we rigorously prove the unconditional energy stability and convergence results of this scheme. Moreover, we design an adaptive time-stepping algorithm to improve the computational efficiency while guaranteeing the precision. Finally, some numerical simulations validate the previous theoretical analysis.