In this paper, we present the Convex Splitting Runge–Kutta (CSRK) methods which provide a simple unified framework to solve phase-field models such as the Allen–Cahn, Cahn–Hilliard, and phase-field crystal equations. The core idea of the CSRK methods is the combination of convex splitting methods and multi-stage implicit–explicit Runge–Kutta methods. Our CSRK methods are high-order accurate in time and we investigate the energy stability numerically. We present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third-order accuracy.
- Allen–Cahn equation
- Cahn–Hilliard equation
- Convex splitting
- Implicit–explicit Runge–Kutta
- Phase-field crystal equation