Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation

Zhang, Chenhui; Ouyang, Jie; Wang, Xiaodong; Chai, Yong; Ma, Mengxia

doi:10.1007/s10915-021-01430-1

Cited by 3 publications

(3 citation statements)

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“…The pearling bifurcation is an important physical phenomenon that has been investigated in [15,39,50]. Several numerical simulations have been implemented in recent works [65][66][67], always with a regular potential. In our simulation, the following initial condition will be used:…”

Section: The Pearling Bifurcation and Sensitivity Of The Initial Valuementioning

confidence: 99%

“…The meandering instability is another interesting phenomenon investigated in the literature, see e.g., [15]. Some numerical tests can be found in [25,66] with a regular potential. To proceed with our simulation, we consider the following initial condition: Besides, we set η = 10, ǫ = 0.01, p = 1, h = 1/256 and λ = ln (19) 0.9 .…”

Section: The Meandering Instabilitymentioning

confidence: 99%

“…Besides, the authors performed the convergence analysis and applied an efficient preconditioned steepest descent algorithm to solve their scheme. Recently, theoretical analysis on the energy stability of a stabilized semi-implicit scheme for the FCH equation was carried out in [66]. We also refer to [64] for an unconditionally energy stable second-order numerical scheme based on the scalar auxiliary variable (SAV) approach, to [67] for a highly accurate, linear and unconditionally energy stable large time-stepping scheme, and to [65] for numerical comparison between those schemes based on the stable SAV approach and the classical backward differentiation formula (BDF) method.…”

Section: Introductionmentioning

confidence: 99%

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

Chen¹,

Jing²,

Wu³

2022

Preprint

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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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