The phase-field crystal equation derived from the Swift–Hohenberg energy functional is a sixth order nonlinear equation. We propose numerical methods based on a new convex splitting for the phase-field crystal equation. The first order convex splitting method based on the proposed splitting is unconditionally gradient stable, which means that the discrete energy is non-increasing for any time step. The second order scheme is unconditionally weakly energy stable, which means that the discrete energy is bounded by its initial value for any time step. We prove mass conservation, unique solvability, energy stability, and the order of truncation error for the proposed methods. Numerical experiments are presented to show the accuracy and stability of the proposed splitting methods compared to the existing other splitting methods. Numerical tests indicate that the proposed convex splitting is a good choice for numerical methods of the phase-field crystal equation.
Bibliographical noteFunding Information:
We thank the anonymous reviewers for their through review and valuable comments to improve the quality of the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MOE (2009-0093827) and MSIP (2015-003037).
© 2016 Elsevier Inc.
- Convex splitting method
- Energy stability
- Gradient stability
- Phase-field crystal equation
- Swift–Hohenberg functional