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We present a set of linear, second order, unconditionally energy stableschemes for the Allen-Cahn model with a nonlocal constraint for crystal growththat conserves the mass of each phase. Solvability conditions are establishedfor the linear systems resulting from the linear schemes. Convergence rates areverified numerically. Dynamics obtained using the nonlocal Allen-Cahn model arecompared with the one obtained using the classic Allen-Cahn model as well asthe Cahn-Hilliard model, demonstrating slower dynamics than that of theAllen-Cahn model but faster dynamics than that of the Cahn-Hillard model. Thus,the nonlocal Allen-Cahn model can be an alternative to the Cahn-Hilliard modelin simulating crystal growth. Two Benchmark examples are presented toillustrate the prediction made with the nonlocal Allen-Cahn model in comparisonto those made with the Allen-Cahn model and the Cahn- Hillard model.

Linear Second Order Energy Stable Schemes of Phase Field Model with Nonlocal Constraints for Crystal Growth