Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

Chen, Wenbin; Wang, Cheng; Wang, Xiaoming; Wise, Steven M.

doi:10.1016/j.jcpx.2019.100031

Cited by 109 publications

(129 citation statements)

References 65 publications

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“…It is worth mentioning two relevant works in this direction. One is the work of Copetti and Elliott [4] who analyzed the implicit Euler scheme and obtained the uniform maximum bound of the numerical solutions; another one is due to Chen et al [2] who studied the first-order and second-order partially implicit scheme and obtained the L ∞ -bound of the numerical solutions. Of course, more interesting and challenging issue is to analyze the energy stability of semi-implicit schemes for the Cahn-Hilliard equation with a logarithmic free energy.…”

Section: Discussionmentioning

confidence: 99%

Revisit of Semi-Implicit Schemes for Phase-Field Equations

sci¹

2020

ATA

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It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semiimplicit schemes for the Allen-Cahn equation with general potential function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang & Yang (J. Comput. Math., 34(5) (2016), pp. 471-481), which studies the semi-implicit scheme for the Allen-Cahn equation with polynomial potentials.

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Section: Discussionmentioning

confidence: 99%

Revisit of Semi-Implicit Schemes for Phase-Field Equations

sci¹

2020

ATA

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“…. In order to avoid this, one can consider to apply a regularization to the logarithmic function [24,25] or develop a positivity-preserving scheme [26]. An extension of our method to the case of logarithmic free energy density using such approaches requires further investigation.…”

Section: Discussionmentioning

confidence: 99%

Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation

Lee

2019

Mathematics

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Strong-stability-preserving (SSP) implicit–explicit (IMEX) Runge–Kutta (RK) methods for the Cahn–Hilliard (CH) equation with a polynomial double-well free energy density were presented in a previous work, specifically H. Song’s “Energy SSP-IMEX Runge–Kutta Methods for the Cahn–Hilliard Equation” (2016). A linear convex splitting of the energy for the CH equation with an extra stabilizing term was used and the IMEX technique was combined with the SSP methods. And unconditional strong energy stability was proved only for the first-order methods. Here, we use a nonlinear convex splitting of the energy to remove the condition for the convexity of split energies and give a stability condition for the coefficients of the second-order method to preserve the discrete energy dissipation law. Along with a rigorous proof, numerical experiments are presented to demonstrate the accuracy and unconditional strong energy stability of the second-order method.

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“…Step 4, update p n+1 and u n+1 using [12]   property of ψ is very challenging due to the particularities of the temporal discretization involved [56]. In this study, we use a regularized logarithmic potential [51] in equation (17)…”

Section: First-order Schemementioning

confidence: 99%

Numerical Approximation of a Phase-Field Surfactant Model with Fluid Flow

et al. 2019

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Modelling interfacial dynamics with soluble surfactants in a multiphase system is a challenging task. Here, we consider the numerical approximation of a phase-field surfactant model with fluid flow. The nonlinearly coupled model consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes equation. With the introduction of two auxiliary variables, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semi-explicitly. By certain subtle explicit-implicit treatments to stress and convective terms, we construct first and second-order time marching schemes, which are extremely efficient and easy-to-implement, for the transformed governing system. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of phasefield variables, velocity and pressure are fully decoupled. We further establish a rigorous proof of unconditional energy stability for the first-order scheme. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow, where the increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence.2 12]. Unlike sharp interface models [1,[13][14][15][16][17][18][19][20], the phase-field method utilizes an appropriate free energy functional [21][22][23][24][25][26][27][28], which determines the thermodynamics of a system, to resolve the interfacial dynamics, thus it has a firm physical basis for multiphase flows. In the pioneering work of Laradji et al., the phase-field model was first used to study the dynamics of phase separation of in a binary systems containing surfactants [29]. Since then, a variety of phase-field surfactant models (free energy functional) have been proposed and well investigated. Komura et al. observed some drawbacks of Laradji et al. model and proposed a different two-order-parameter time dependent Ginzburg-Landau model [30]. Theissen and Gompper slightly modified the local coupling term of Laradji et al. model to deal with the same solubility of surfactants in the bulk phases [31]. Samn and Graaf introduced the logarithmic Floy-Huggins potential to restrict the range of local concentration of surfactants [32,33]. In [34], the authors analyzed the well-posedness of the model proposed by Samn and Graaf, and provided strong evidence that it was mathematically ill-posed for a large set of physically relevant parameters. They made critical changes to the model and substantially increased the domain of validity. This modified model will be used in this study. In [6], the authors introduced a new model that can recover Langmuir and Frumkin adsorption isotherms in equilibrium.The free energy functional only determines the thermodynamics of a system, and hydrodynamics should also be incorporated into a d...

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Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

Cited by 109 publications

References 65 publications

Revisit of Semi-Implicit Schemes for Phase-Field Equations

Revisit of Semi-Implicit Schemes for Phase-Field Equations

Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation

Numerical Approximation of a Phase-Field Surfactant Model with Fluid Flow

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