Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows

Han, Daozhi; Brylev, Alex; Yang, Xiaofeng; Tan, Zhijun

doi:10.1007/s10915-016-0279-5

Cited by 91 publications

(69 citation statements)

References 52 publications

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“…Therefore, after simple substitutions using new variables U , U , W , Z defined in , the energy is transformed to an equivalent quadratic form. Therefore, we call it ‘Invariant Energy Quadratization’ approach (see also the authors other work ). We emphasize that the positive constant A and B must be introduced in the new variables to avoid the appearance of singularities in R ( φ ), P ( φ ), and Q ( φ ) because their denominators might equal to zero without them.…”

Section: The Model Systemmentioning

confidence: 99%

“…To this end, we adopt a novel approach, that we term it the Invariant Energy Quadratization (IEQ) approach, where some nonlinear transformations are introduced to enforce the free‐energy density as an invariant, quadratic functionals in terms of new, auxiliary variables. The IEQ method has been successfully applied in the context of other models in the authors' other work , while the application to the PF‐DCG model provides new challenges because of the nonlinearities in various terms, in particular, the anisotropic coefficient in the gradient entropy, the fourth order and fifth order nonlinear polynomial potentials. The key point of IEQ method is that we reformulate the governing system of equations using the new variables to an equivalent system.…”

Section: Introductionmentioning

confidence: 99%

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Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

Zhao

Wang

Yang

2016

Numerical Meth Engineering

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180

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Summary We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free‐energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient‐dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free‐energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi‐discretized scheme in time for the system, in which all nonlinear terms are treated semi‐explicitly. The resulting semi‐discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi‐discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi‐discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: The Model Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

Zhao

Wang

Yang

2016

Numerical Meth Engineering

Self Cite

180

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“…Remark 3.1. When S 1 = S 2 = 0, the above scheme is the IEQ type scheme which had been developed in [2,7,[32][33][34][35][36][38][39][40]. Note even though the IEQ method is formally unconditionally energy stable, the spatial oscillations caused by the anisotropic coefficient κ(∇φ) can still make the scheme blow up for large time steps, which are illustrated in Fig.…”

Section: 2mentioning

confidence: 99%

Efficient linear, stabilized, second-order time marching schemes for an anisotropic phase field dendritic crystal growth model

Yang

2019

Computer Methods in Applied Mechanics and Engineering

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“…Summing up the four equations, (22) and (25)- (27), we obtain the energy dissipation law (21) of the continuous system equations.…”

Section: Reformulation Of the System Equationsmentioning

confidence: 99%

“…Therefore it is highly desirable to design such an energy stable method for the q-NSCH model, which dissipates the energy (preserve thermodynamic consistency) at the discrete level. Many time-discrete or fully discrete level energy stable methods [8,21,23,28,33,35] have been presented for the other types of NSCH models for binary incompressible fluid with the solenoidal velocity field. However for the q-NSCH model presented by Lowengrub and Truskinovsky [36] or the other quasi-incompressible type models with the non-solenoidal velocity, relatively few time-discrete energy stable methods are available [16,43,19].…”

Section: Introductionmentioning

confidence: 99%

Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes

Guo

Lin

Lowengrub

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method -comprised of a standard FAS method for the Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier-Stokes equation -for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10 −8 and energy decreases in time.

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Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows

Cited by 91 publications

References 52 publications

Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach

Efficient linear, stabilized, second-order time marching schemes for an anisotropic phase field dendritic crystal growth model

Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: Primitive variable and projection-type schemes

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