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

Zhao, Jia; Wang, Qi; Yang, Xiaofeng

doi:10.1002/nme.5372

Cited by 176 publications

(93 citation statements)

References 38 publications

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“…For the temporal discretization, instead of using the method studied in [19] which requires iteratively solving a nonlinear system, we explore the method of Invariant Energy Quadratization (IEQ), which was proposed very recently in [31,33]. This method is a generalization of the method of Lagrange multipliers or of auxiliary variables originally proposed in [1,13].…”

Section: Introductionmentioning

confidence: 99%

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Liu

Yin

2019

J Sci Comput

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The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467-501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.1991 Mathematics Subject Classification. 65N12, 65N30, 35K35.

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Section: Introductionmentioning

confidence: 99%

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Liu

Yin

2019

J Sci Comput

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“…These schemes are linear, unconditionally gradient stable and maximum principle preserving (only for the first-order scheme). This approach has been applied to various phase field models [18,[31][32][33]. The convex splitting technique is a powerful tool to construct unconditionally energy stable schemes for gradient flows.…”

Section: Introductionmentioning

confidence: 99%

“…Besides the above three types of schemes, the invariant energy quadratization (IEQ) method developed recently is also a significant approach to design linear and unconditionally gradient stable schemes. This approach has been applied to various phase field models [18,[31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

Xiao

Feng

2019

Numerical Methods Partial

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In this paper, we present two types of unconditionally maximum principle preserving finite element schemes to the standard and conservative surface Allen-Cahn equations. The surface finite element method is applied to the spatial discretization. For the temporal discretization of the standard Allen-Cahn equation, the stabilized semi-implicit and the convex splitting schemes are modified as lumped mass forms which enable schemes to preserve the discrete maximum principle. Based on the above schemes, an operator splitting approach is utilized to solve the conservative Allen-Cahn equation. The proofs of the unconditionally discrete maximum principle preservations of the proposed schemes are provided both for semi-(in time) and fully discrete cases. Numerical examples including simulations of the phase separations and mean curvature flows on various surfaces are presented to illustrate the validity of the proposed schemes. KEYWORDSconvex splitting scheme, lumped mass finite element method, maximum principle preservation, operator splitting approach, stabilized semi-implicit scheme, surface Allen-Cahn type equation 1 Numer Methods Partial Differential Eq. 2020;36:418-438. wileyonlinelibrary.com/journal/num

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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.…”

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

Cited by 176 publications

References 38 publications

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

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

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