Unconditionally energy stable second-order numerical scheme for the Allen–Cahn equation with a high-order polynomial free energy

Abstract

Abstract In this article, we consider a temporally second-order unconditionally energy stable computational method for the Allen–Cahn (AC) equation with a high-order polynomial free energy potential. By modifying the nonlinear parts in the governing equation, we have a linear convex splitting scheme of the energy for the high-order AC equation. In addition, by combining the linear convex splitting with a strong-stability-preserving implicit–explicit Runge–Kutta (RK) method, the proposed method is linear, temporally second-order accurate, and unconditionally energy stable. Computational tests are performed to demonstrate that the proposed method is accurate, efficient, and energy stable.

Keywords

• Allen–Cahn equation
• Linear convex splitting
• Implicit–explicit RK scheme
• High-order polynomial free energy