Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints

Abstract

We present a set of linear, second order, unconditionally energy stable schemes for the Allen–Cahn model with nonlocal constraints for crystal growth that conserves the mass of each phase. Solvability conditions are established for the linear systems resulting from the schemes. Convergence rates are verified numerically. Dynamics obtained using the Allen–Cahn model with nonlocal constraints are compared with the one obtained using the classic Allen–Cahn model as well as the Cahn–Hilliard model, respectively, demonstrating slower dynamics than that of the Allen–Cahn model but faster dynamics than that of the Cahn–Hilliard model. Thus, the Allen–Cahn model with nonlocal constraints can serve as an alternative to the Cahn–Hilliard model in simulating crystal growth while conserving the mass of each phase. Two Benchmark examples are presented to contrast the predictions made with the four models, highlighting the accuracy and effectiveness of the Allen–Cahn model with nonlocal constraints.