Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

Li, Hongwei; Ju, Lili; Zhang, Chenfei; Peng, Qiujin

doi:10.1007/s10915-017-0576-7

Cited by 43 publications

(29 citation statements)

References 33 publications

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“…Li et. al proved the uniqueness of the weak solution of the EQ scheme with periodic boundary conditions for the nonlocal Allen-Cahn model with a penalizing potential [20]. Here we expand it into both EQ and SAV schemes for the Allen-Cahn model with a penalizing potential subject to a Neumann boundary condition.…”

Section: Solvability Of the Linear Systems Resulting From The Schemesmentioning

confidence: 99%

“…However, in the case of a phase field description, when the phase variable represents the volume fraction of a material component, this model does not warrant the conservation of the volume of that component. In order to preserve the volume, the free energy functional has to be augmented by a volume preserving mechanism with a penalizing potential term or a Lagrange multiplier [20] [24]. This often results in a nonlocal term in the modified Allen-Cahn equation.…”

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confidence: 99%

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Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Jing

Zhao

et al. 2019

J Sci Comput

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We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn-Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end. ]. However, in the case of a phase field description, when the phase variable represents the volume fraction of a material component, this model does not warrant the conservation of the volume of that component. In order to preserve the volume, the free energy functional has to be augmented by a volume preserving mechanism with a penalizing potential term or a Lagrange multiplier [20] [24]. This often results in a nonlocal term in the modified Allen-Cahn equation. In this paper, we call these nonlocal Allen-Cahn equations or Allen-Cahn equations with nonlocal constraints.The Cahn-Hilliard equation is an alternative model for the gradient flow. One feature of the Cahn-Hilliard equation is its volume preserving property. Rubinstein and Sternberg studied the Allen-Cahn model with the volume constraint analytically and compared it with the Cahn-Hilliard model [24]. Their result favors using the Allen-Cahn model with a volume constraint in place of the Cahn-Hilliard model when studying interfacial dynamics of incompressible, immiscible multi-component material systems. For the Allen-Cahn equation as well as the Cahn-Hilliard equation, there have been several popular numerical approaches to construct energy stable schemes for the equations, including the convex splitting approach [11-13, 18, 25, 28, 30] , the stabilizing approach [6, 10, 27], the energy quadratization (EQ) approach [14,15,33,39] and the scalar auxiliary variable approach, which is a special form of EQ method, [9,26,29]. Recently, the energy quadratization (EQ) and its reincarnation in the scalar auxiliary variable (SAV) method have been applied to a host of thermodynamical and hydrodynamic models owing to their simplicity, ease of implementation, computational efficiency, linearity, and most importantly their energy stabi...

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Section: Solvability Of the Linear Systems Resulting From The Schemesmentioning

confidence: 99%

mentioning

confidence: 99%

Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Jing

Zhao

et al. 2019

J Sci Comput

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“…The new variables can be updated with time steps via the semi-implicit linear schemes. A numerical scheme has been developed in [19] applying the IEQ approach to PR-EoS. As a modification of the IEQ approach, the scalar auxiliary variable (SAV) approach has been proposed in [30], which introduces a scalar auxiliary variable instead of the space-dependent new variables in the IEQ approach.…”

mentioning

confidence: 99%

A Novel Energy Factorization Approach for the Diffuse-Interface Model with Peng--Robinson Equation of State

Kou¹,

Sun²,

Wang³

2020

SIAM J. Sci. Comput.

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The Peng-Robinson equation of state (PR-EoS) has become one of the most extensively applied equations of state in chemical engineering and petroleum industry due to its excellent accuracy in predicting the thermodynamic properties of a wide variety of materials, especially hydrocarbons. Although great efforts have been made to construct efficient numerical methods for the diffuse interface models with PR-EoS, there is still not a linear numerical scheme that can be proved to preserve the original energy dissipation law. In order to pursue such a numerical scheme, we propose a novel energy factorization (EF) approach, which first factorizes an energy function into a product of several factors and then treats the factors using their properties to obtain the semiimplicit linear schemes. We apply the EF approach to deal with the Helmholtz free energy density determined by PR-EoS, and then propose a linear semi-implicit numerical scheme that inherits the original energy dissipation law. Moreover, the proposed scheme is proved to satisfy the maximum principle in both the time semi-discrete form and the cell-centered finite difference fully discrete form under certain conditions. Numerical results are presented to demonstrate the stability and efficiency of the proposed scheme.

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“…The volume conservation means V (t) = V (0) for any t > 0. In this modified model, we penalize (V (t) − V (0)) 2 in the energy functional [14]. Specifically, we modify the free energy as follows:…”

Section: Allen-cahn Model With a Penalizing Potentialmentioning

confidence: 99%

Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints

2018

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We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen-Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.

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Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

Cited by 43 publications

References 33 publications

Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

A Novel Energy Factorization Approach for the Diffuse-Interface Model with Peng--Robinson Equation of State

Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints

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