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

## TL;DR

This paper introduces linear, energy-stable schemes for a nonlocal Allen-Cahn model that conserves phase mass, compares dynamics with other models, and demonstrates its effectiveness in simulating crystal growth.

## Contribution

It develops and verifies a new class of linear, second-order, energy-stable schemes for a nonlocal Allen-Cahn model with mass conservation, offering an alternative to Cahn-Hilliard for crystal growth simulation.

## Key findings

- Schemes are unconditionally energy stable and linearly solvable.
- Nonlocal Allen-Cahn model shows intermediate dynamics between Allen-Cahn and Cahn-Hilliard.
- Benchmark examples validate the model's predictive capability.

## Abstract

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn model with a nonlocal constraint for crystal growth that conserves the mass of each phase. Solvability conditions are established for the linear systems resulting from the linear schemes. Convergence rates are verified numerically. Dynamics obtained using the nonlocal Allen-Cahn model are compared with the one obtained using the classic Allen-Cahn model as well as the Cahn-Hilliard model, demonstrating slower dynamics than that of the Allen-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 model in simulating crystal growth. Two Benchmark examples are presented to illustrate the prediction made with the nonlocal Allen-Cahn model in comparison to those made with the Allen-Cahn model and the Cahn- Hillard model.