Bound preserving and mass conservative methods for the nonlocal Cahn-Hilliard equation with the logarithmic Flory-Huggins potential
This work addresses a numerical stability issue in simulating phase-field models for materials science, offering an incremental improvement for researchers in computational physics.
The authors tackled the challenge of extending exponential time differencing (ETD) methods to the nonlocal Cahn-Hilliard equation with Flory-Huggins potential, which can produce non-physical solutions, by developing first- and second-order schemes that combine ETD with a projection method to preserve bounds and mass conservation, with error estimates and numerical tests demonstrating performance.
It is well known that the exponential time differencing (ETD) method has been successfully applied to the classic Cahn-Hilliard equation with double well potential. However, this numerical method can not be extended to the Cahn-Hilliard equation with Flory-Huggins potential directly due to the fact that the the numerical solution may go beyond the physical interval which leads the non-physical solution. In this paper, we develop and analyze first- and second-order numerical schemes for the nonlocal Cahn-Hilliard equation with the classic Flory-Huggins energy potential. In more detail, the ETD method is firstly used to obtain the prediction solution, and then this prediction solution is corrected by the projection method to avoid non-physical solution. The proposed method is shown to preserve bound and mass conservation in discrete settings. In addition, error estimates for the numerical solution are rigorously obtained for both schemes. Extensive numerical tests and comparisons are conducted to demonstrate the performance of the proposed schemes.