Bound preserving and mass conservative methods for the nonlocal Cahn-Hilliard equation with the logarithmic Flory-Huggins potential

Abstract

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.

Figures (7)

• Figure 1: The evolutions of the supremum norms and mass increment of the numerical solution generated by the classic ETD1 scheme and P-ETD1 scheme with $\kappa=2$, $\tau=0.1$, $h=1/512$, $\epsilon=0.0 2$, $\theta_c=1.6$, $\theta=0.8$, $\sigma=30$
• Figure 2: The evolutions of the iteration number (left) and the Lagrange multiplier (right) generated by P-ETD1 scheme with $\kappa=2$, $\tau=0.1$, $h=1/512$, $\epsilon=0.0 2$, $\theta_c=1.6$, $\theta=0.8$, $\sigma=30$
• Figure 3: The evolutions of the supremum norms and mass increment of the numerical solution generated by the classic ETDRK2 scheme and P-ETDRK2 scheme with $\kappa=2$, $\tau=0.1$, $h=1/512$, $\epsilon=0.02$, $\theta_c=1.6$, $\theta=0.8$, $\sigma=30$
• Figure 4: The evolutions of the iteration number (left) and the Lagrange multiplier (right) generated by P-ETDRK2 scheme with $\kappa=2$, $\tau=0.1$, $h=1/512$, $\epsilon=0.0 2$, $\theta_c=1.6$, $\theta=0.8$, $\sigma=30$
• Figure 5: The phase structures at $t=10$, $20$, $50$, $100$, $200$, and $2000$, respectively (left to right and top to bottom) generated by the P-ETDRK2 scheme with $\kappa=2$, $\tau=0.1$, $h=1/512$, $\epsilon=0.02$, $\theta_c=1.6$, $\theta=0.8$, $\sigma=70$

Theorems & Definitions (12)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1: see High2008
• Lemma 4.2
• Lemma 4.3
• proof
• Theorem 4.1
• proof