A new Lagrange multiplier method for the mass-conserved Allen–Cahn type square phase-field crystal model

For gradient flows, it is always expected that the two important properties of mass conservation and energy dissipation can be maintained simultaneously. In this paper, we propose a mass-conserved Allen–Cahn type square phase-field crystal (AC-SPFC) model by adding a nonlocal Lagrange multiplier. On the other hand, we construct some linear schemes that dissipate the original energy based on the recently introduced new Lagrange multiplier method (Cheng, Liu and Shen 2020). Noticeably, the new approach does not require a lower bound on the nonlinear part of the energy as the scalar auxiliary variable (Shen, Xu and Yang 2018) method does, and the solving process is pretty straightforward. We exhibit the robustness of the mass-conserved AC-SPFC by comparison with the Cahn–Hilliard type square phase-field crystal equation, and numerically demonstrate the stability and the accuracy of the proposed schemes.