Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

Peng, Qiujin; Qiao, Zhonghua; Sun, Shuyu

doi:10.4208/jcm.1611-m2016-0623

Cited by 7 publications

(5 citation statements)

References 34 publications

(44 reference statements)

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“…Remark 4.2. For the CH model with Flory Huggins energy potential, there have been some works to address the energy stability in the existing literature [43,47,48,53,62]. However, the positivity-preserving property has not been theoretically justified for these numerical works, so that the existence of the numerical solutions in these works is not available at a theoretical level.…”

Section: Unconditional Energy Stability and Uniform In Time Hmentioning

confidence: 99%

“…At the level of numerical scheme design, the positivity preserving property is very challenging, due to the particularities of the spatial and temporal discretizations involved. There have been extensive numerical works for the CH model with Flory Huggins energy potential [43,44,47,48,52,53,62], while a theoretical justification to assure the positivity of 1 + φ and 1 − φ has not been available (so that the numerical scheme is unconditionally well-defined). Among the existing literature, it is worth mentioning the numerical analysis to theoretically justify this issue in [22].…”

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confidence: 99%

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

Chen

Wang

et al. 2019

Journal of Computational Physics: X

121

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In this paper we present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments, i.e., the phase variable is always between −1 and 1, at a point-wise level. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of −1 and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. Such an analysis technique could also be applied to a second order numerical scheme, in which the BDF temporal stencil is applied, the expansive term is updated by a second order Adams-Bashforth explicit extrapolation formula, and an artificial Douglas-Dupont regularization term is added to improve the stability property. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes.

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Section: Unconditional Energy Stability and Uniform In Time Hmentioning

confidence: 99%

mentioning

confidence: 99%

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

Chen

Wang

et al. 2019

Journal of Computational Physics: X

121

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“…The mass conservation and the energy stability of such problems have been considered earlier [36] and are used here without any comments.…”

Section: Mathematical Model Of Peng-robinson Equation Of Statementioning

confidence: 99%

“…Besides, numerical solution of the naturally mass conservative fourth order parabolic equation with Peng-Robinson equation of state for single-and multi-component cases was also obtained. In the single-component case, the energy stability, unique solvability and l ∞ convergence of a first order convexsplitting scheme [35] and two second order schemes [36] have been established. In the multi-component case, the relations between different components are decoupled and the solution of the new system of equations was derived by a semi-implicit unconditionally stable scheme combined with a mixed finite element method.…”

Section: Introductionmentioning

confidence: 99%

Energy Stable Linear Schemes for Mass-Conserved Gradient Flows with Peng-Robinson Equation of State

Peng¹

2019

EAJAM

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First and second order numerical schemes for the fourth order parabolic equation with Peng-Robinson equation of state, which are based on recently proposed invariant energy quadratisation method are developed. Both schemes are linear, unconditionally energy stable and uniquely solvable. The reduced linear systems are symmetric and positive definite, so that their solutions can be efficiently found. Numerical results demonstrate the good performance of the schemes, consistent with experimental data.

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“…Several approaches have been carried out to construct energy stable schemes for diffuse interface models with Peng-Robinson EOS. In [6,11,12,19,20,22], energy stable schemes are proposed by using the convex-splitting strategy, which is a popular used approach for solving diffuse interface problems. The modified Newton's method with a relaxation parameter, which is mentioned in [9], is another approach to guarantee the energy decay property while constructing the schemes.…”

Section: Introductionmentioning

confidence: 99%

A New Multi-Component Diffuse Interface Model with Peng-Robinson Equation of State and its Scalar Auxiliary Variable (SAV) Approach

Qiao¹

2019

CiCP

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A new multi-component diffuse interface model with the Peng-Robinson equation of state is developed. Initial values of mixtures are given through the NVT flash calculation. This model is physically consistent with constant diffusion parameters, which allows us to use fast solvers in the numerical simulation. In this paper, we employ the scalar auxiliary variable (SAV) approach to design numerical schemes. It reformulates the proposed model into a decoupled linear system with constant coefficients that can be solved fast by using fast Fourier transform. Energy stability is obtained in the sense that the modified discrete energy is non-increasing in time. The calculated interface tension agrees well with laboratory experimental data.

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Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

Cited by 7 publications

References 34 publications

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

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

Energy Stable Linear Schemes for Mass-Conserved Gradient Flows with Peng-Robinson Equation of State

A New Multi-Component Diffuse Interface Model with Peng-Robinson Equation of State and its Scalar Auxiliary Variable (SAV) Approach

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