Xiaobo Jing

Shouwen Sun, Xiaobo Jing, Qi Wang

We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen-Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.

Xiaobo Jing, Qi Wang

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.

Xiaobo Jing, Jun Li, Xueping Zhao et al.

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn-Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end.