A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials

Liu, Dong

doi:10.3934/dcds.2021198

Cited by 7 publications

(1 citation statement)

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“…Concerning the Cahn-Hilliard equation ∂ t φ = ∇ • [M∇(−ǫ 2 ∆φ + f (φ))] with logarithmic potential (1.4), we refer to [1,14,22,29,30,42,48] for extensive theoretic analysis on well-posedness as well as long-time behavior of global solutions (see also the review articles [7,62]), while for contributions in the numerical studies, we recall [4, 5, 9, 35-37, 43, 44, 63] and the references therein. Under suitable boundary conditions (e.g., the periodic or Neumann type), the FCH equation (1.1) can be viewed as an H −1 gradient flow of the FCH energy (1.2), which not only dissipates the energy but also preserves the mass along time evolution.…”

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confidence: 99%

A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential

Chen¹,

Jing²,

Wu³

2022

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We propose and analyze a first-order finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states ±1, we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values ±1 in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in l ∞ (0, T ; L 2 h ) ∩ l 2 (0, T ; H 3 h ) under a mild CFL condition C1h ≤ ∆t ≤ C2h. To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states ±1, then the numerical solutions can be kept away from ±1 by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented to demonstrate the accuracy and robustness of the proposed numerical scheme.

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