On Linear and Unconditionally Energy Stable Algorithms for Variable Mobility Cahn-Hilliard Type Equation with Logarithmic Flory-Huggins Potential

Yang, Xiaofeng; Jia, Xin

doi:10.4208/cicp.oa-2017-0259

Cited by 50 publications

(48 citation statements)

References 34 publications

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“…In cases ≥ h 2 , we observe that an order of two is approached, indicating a convergence order of one in and of two in h. The discrete free energy F h , cf. (28), dissipates in all cases monotonically. For all other cases, convergence in terms of ( j , h j ) is not expected.…”

Section: 2mentioning

confidence: 90%

“…Note, although the logarithmic potential is only defined on a finite interval, many authors define an extension over R for convenience in numerical simulations, for instance see [27,28]. Throughout our analysis, we assume the chemical energy density Φ ∈ C 2 (R), that is, Φ is a two times continuously differentiable function with respect to c.…”

Section: Convex-concave Decompositionmentioning

confidence: 99%

“…In all physical scenarios, independent of the chosen discretization and model parameters, mass is conserved and the discrete free energy F h , cf. (28), dissipates in time.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Our goal is to validate the dissipation of discrete free energy F h , cf. (28), and to investigate whether mass is conserved throughout the simulation.…”

Section: Unconditional Energy Stability Validationmentioning

confidence: 99%

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Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

Liu

Frank

Rivière

2019

Numerical Methods Partial

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In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn-Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori error estimates. Our analysis is valid for both symmetric and nonsymmetric versions of the discontinuous Galerkin formulation. KEYWORDSCahn-Hilliard equation, error analysis, interior penalty discontinuous Galerkin methods, stability analysis, unique solvability 1

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Section: 2mentioning

confidence: 90%

Section: Convex-concave Decompositionmentioning

confidence: 99%

“…In all physical scenarios, independent of the chosen discretization and model parameters, mass is conserved and the discrete free energy F h , cf. (28), dissipates in time.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Our goal is to validate the dissipation of discrete free energy F h , cf. (28), and to investigate whether mass is conserved throughout the simulation.…”

Section: Unconditional Energy Stability Validationmentioning

confidence: 99%

See 2 more Smart Citations

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

Liu

Frank

Rivière

2019

Numerical Methods Partial

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“…In recent years, the novel auxiliary variable approaches have been developed and successfully applied to design linear numerical schemes for various diffuse interface models [30,35,45,46,[55][56][57][58][61][62][63]. The first approach is the so-called invariant energy quadratization (IEQ) approach [55][56][57]61] that has been successfully applied to devise efficient, linear schemes for various phase-field models intensively in recent years. The basic idea of IEQ is to define a set of auxiliary variables and then transform the original free energy function into a quadratic form.…”

mentioning

confidence: 99%

Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

2020

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The Allen-Cahn equation is one of fundamental equations of phase-field models, while the logarithmic Flory-Huggins potential is one of the most useful energy potentials in various phasefield models. In this paper, we consider numerical schemes for solving the Allen-Cahn equation with logarithmic Flory-Huggins potential. The main challenge is how to design efficient numerical schemes that preserve the maximum principle and energy dissipation law due to the strong nonlinearity of the energy potential function. We propose a novel energy factorization approach with the stability technique, which is called stabilized energy factorization approach, to deal with the Flory-Huggins potential. One advantage of the proposed approach is that all nonlinear terms can be treated semiimplicitly and the resultant numerical scheme is purely linear and easy to implement. Moreover, the discrete maximum principle and unconditional energy stability of the proposed scheme are rigorously proved using the discrete variational principle. Numerical results are presented to demonstrate the stability and effectiveness of the proposed scheme.

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An efficient and physically consistent numerical method for the Maxwell–Stefan–Darcy model of two‐phase flow in porous media

Kou

Chen

et al. 2022

Numerical Meth Engineering

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Numerical modeling of two-phase flow in porous media has extensive applications in subsurface flow and petroleum industry. A comprehensive Maxwell-Stefan-Darcy (MSD) two-phase flow model has been developedrecently, which takes into consideration the friction between two phases by a thermodynamically consistent way. In this article, we for the first time propose an efficient energy stable numerical method for the MSD model, which can preserve multiple important physical properties of the model. First, the proposed scheme can preserve the original energy dissipation law. This is achieved through a newly-developed energy factorization approach that leads to linear semi-implicit discrete chemical potentials. Second, the scheme preserves the famous Onsager's reciprocal principle and the local mass conservation law for both phases by introducing different upwind strategies for two phase saturations and applying the cell-centered finite volume method to the original formulation of the model. Third, by introducing two auxiliary phase velocities, the scheme has ability to guarantee the positivity of both saturations under proper conditions. Another distinct feature of the scheme is that the resulting discrete system is totally linear, well-posed and unbiased for each phase. Numerical results are also provided to show the excellent performance of the proposed scheme. K E Y W O R D S energy stability, Maxwell-Stefan-Darcy model, Onsager's reciprocal principle, two-phase flow in porous media 546

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On Linear and Unconditionally Energy Stable Algorithms for Variable Mobility Cahn-Hilliard Type Equation with Logarithmic Flory-Huggins Potential

Cited by 50 publications

References 34 publications

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

An efficient and physically consistent numerical method for the Maxwell–Stefan–Darcy model of two‐phase flow in porous media

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