Efficient schemes with unconditionally energy stability for the anisotropic Cahn-Hilliard Equation using the stabilized-Scalar Augmented Variable (S-SAV) approach
This work provides efficient and stable numerical methods for simulating anisotropic phase-field models, which are important for materials science applications.
The authors developed two second-order unconditionally energy stable numerical schemes for the anisotropic Cahn-Hilliard equation by combining the SAV approach with linear stabilization. The schemes treat all nonlinear terms semi-explicitly, requiring only the solution of three decoupled linear equations per time step, and are validated through 2D and 3D simulations.
In this paper, we consider numerical approximations for the anisotropic Cahn-Hilliard equation. The main challenge of constructing numerical schemes with unconditional energy stabilities for this model is how to design proper temporal discretizations for the nonlinear terms with the strong anisotropy. We propose two, second order time marching schemes by combining the recently developed SAV approach with the linear stabilization approach, where three linear stabilization terms are added. These terms are shown to be crucial to remove the oscillations caused by the anisotropic coefficients, numerically. The novelty of the proposed schemes is that all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. We further prove the unconditional energy stabilities rigorously, and present various 2D and 3D numerical simulations to demonstrate the stability and accuracy.