A variable-step, structure-preserving and linear fully discrete scheme for the two-mode phase-field crystal model with face-centered-cubic ordering

Combining the stabilized scalar auxiliary variable approach and the variable-step second-order backward difference formula, an adaptive time-stepping scheme is proposed for the two-mode phase-field crystal model with face-centered-cubic ordering. Specifically, introduce an auxiliary variable to handle the nonlinear term and obtain a new equivalent system, then perform a variable-step second-order approximation on the phase-field variable and a variable-step first-order approximation on the auxiliary variable, that is crucial for proving energy stability. Despite employing a low-order approximation for the auxiliary variable, as long as mild constraints are placed on the constant within this auxiliary variable, the second-order temporal accuracy of the phase-field variable will remain unaffected. By utilizing the boundedness of the $H^4$ norm for the numerical solution of the phase-field variable on nonuniform temporal grids, this paper performs a thorough error analysis of the fully discrete scheme. Some numerical simulations are conducted to verify the temporal accuracy, mass conservation, and energy dissipation. Additionally, to balance the efficiency and accuracy of the numerical experiments, we have selected an appropriate time-adaptive strategy for long-term simulations of phase transition behavior and crystal growth behavior of the phase-field variable.