A positivity-preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters

Dong, Lixiu; Wang, Cheng; Wise, Steven M.; Zhang, Zhengru

doi:10.1016/j.jcp.2021.110451

Cited by 50 publications

(10 citation statements)

References 44 publications

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“…For pACOK, we take N x = N y = 2 8 , and for pACON, we take N x = N y = 2 9 . The grid size is h = 2/N x .…”

Section: 4mentioning

confidence: 99%

“…In fact, there have been extensive works on the artificial regularization for the energy stable numerical schemes for various gradient flows, such as artificial regularization for the no-slope-selection epitaxial thin film model [25]. For the ternary gradient flow, there has been a recent work [9] on the energy stable numerical schemes for ternary Cahn-Hilliard system, in which both the energy stability analysis and optimal rate convergence estimate have been established. 1.1.…”

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confidence: 99%

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Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

Choi

Zhao

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.</p>

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“…For pACOK, we take N x = N y = 2 8 , and for pACON, we take N x = N y = 2 9 . The grid size is h = 2/N x .…”

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

Choi

Zhao

2022

DCDS-B

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“…In this paper, we design a uniquely solvable, positivity-preserving, unconditionally energy stable, and first order in time convergent scheme for the binary fluid-surfactant system, based on the convex-splitting idea, combined with the centered difference spatial approximation. For the theoretical analysis of the positivity-preserving property, we make use of the singular nature of the logarithmic function, and prove that such a singular nature prevents the numerical solution approaches the singular limit values, following similar ideas of in the analysis for the Cahn-Hilliard model [5,9,10,11], as well as the one for the Poisson-Nernst-Planck system [34,38], droplet liquid film model [67], etc. In addition, an optimal rate convergence analysis is provided, which is the first such work for the surfactant model.…”

Section: Introductionmentioning

confidence: 99%

A positivity-preserving and convergent numerical scheme for the binary fluid-surfactant system

Qin¹,

Wang²,

Zhang³

2021

Preprint

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In this paper, we develop a first order (in time) numerical scheme for the binary fluid surfactant phase field model. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. The resulting coupled system consists of two Cahn-Hilliard type equations. This system is solved numerically by finite difference spatial approximation, in combination with convex splitting temporal discretization. We prove the proposed scheme is unique solvable, positivitypreserving and unconditionally energy stable. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme, which will be the first such result for the binary fluidsurfactant system. Newton iteration is used to solve the discrete system. Some numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.

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“…Examples include the convex splitting idea [89], invariant energy quadratization (IEQ) [93], and scalar auxiliary variable (SAV) scheme [85]. Some theoretical analysis of positivity-preserving property and energy stability have been reported for these numerical schemes for certain systems with singular energy potential, such as Poisson-Nernst-Planck (PNP) system [84], and the Cahn-Hilliard equation with Flory-Huggins energy [15,24,25,26]. However, for a general reaction-diffusion system (1.1), a variational structure may not exist, so that these structure-preserving ideas may be not directly applicable.…”

Section: Introductionmentioning

confidence: 99%

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

Liu,

Wang,

Wang

2020

Preprint

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In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for certain type reaction-diffusion systems involving the Law of Mass Action with detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories, and both the reaction and the diffusion parts dissipate the same free energy. This subtle fact opens a path of an operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. To our best knowledge, it is the first time to report an energy-dissipation-lawbased operator splitting scheme to a nonlinear PDE with variational structures. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.

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A positivity-preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters

Cited by 50 publications

References 44 publications

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

A positivity-preserving and convergent numerical scheme for the binary fluid-surfactant system

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

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