A variant of scalar auxiliary variable approaches for gradient flows

Hou, Dianming; Azaïez, Mejdi; Xu, Chuanju

doi:10.1016/j.jcp.2019.05.037

Cited by 67 publications

(29 citation statements)

References 23 publications

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“…Note that near t = 0, the energy dissipation property seems destroyed but the energy boundedness is still satisfied. Similar situation has also been reported in [22].…”

Section: Numerical Testssupporting

confidence: 89%

Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations

Quan,

Wang

2021

Preprint

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In this article, the energy stability of two high-order L2 schemes for time-fractional phasefield equations is established. We propose a reformulation of the L2 operator and also some new properties on it. We prove the energy boundedness (by initial energy) of an L2 scalar auxiliary variable scheme for any phase-field equation and the fractional energy law of an implicit-explicit L2 Adams-Bashforth scheme for the Allen-Cahn equation. The stability analysis is based on a new Cholesky decomposition proposed recently by some of us.

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“…Note that near t = 0, the energy dissipation property seems destroyed but the energy boundedness is still satisfied. Similar situation has also been reported in [22].…”

Section: Numerical Testssupporting

confidence: 89%

Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations

Quan,

Wang

2021

Preprint

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“…Auxiliary variable approach. The proposed schemes are based on a reformulation of the time fractional Allen-Cahn equation by introducing an auxiliary variable -an approach intensively studied recently for gradient flows; see, e.g., [19,35] and the references therein. The key to is to rewrite the original equation (3.1)-(3.2) into the following equivalent form:…”

Section: Numerical Methods -A First Order Schemementioning

confidence: 99%

“…The aim of the present paper is to propose easy-to-implement and unconditionally stable schemes, which preserve a non-local energy dissipation law to be specified. The main idea in constructing the schemes is to use existing efficient approximations to discretize the local part and history part of the fractional derivative respectively, and use auxiliary variable approaches [19,33,34] to deal with the nonlinear potential in the free energy. The contributions of the paper are threefold:…”

Section: Introductionmentioning

confidence: 99%

Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation

Hou¹,

2021

Preprint

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In this paper, we propose and analyze a time-stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equations. The unconditional stability is proved through establishing a discrete nonlocal free energy dispassion law, which is also true for the continuous problem. The main idea used in the analysis is to split the time fractional derivative into two parts: a local part and a history part, which are discretized by the well known L1, L1-CN, and L1 + -CN schemes. Then an extended auxiliary variable approach is used to deal with the nonlinear and history term. The main contributions of the paper are: First, it is found that the time fractional Allen-Chan equation is a dissipative system related to a nonlocal free energy. Second, we construct efficient time stepping schemes satisfying the same dissipation law at the discrete level. In particular, we prove that the proposed schemes are unconditionally stable for quite general meshes. Finally, the efficiency of the proposed method is verified by a series of numerical experiments.

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“…One has to use energy splitting method [14] or stabilized method [22] to deal with the nonlinear part of energy. However, G-SAV approach can be directly applied since there is no requirement of free energy to be bounded from below.…”

Section: Validation and Comparisonmentioning

confidence: 99%

The generalized scalar auxiliary variable approach (G-SAV) for gradient flows

Cheng

2020

Preprint

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We establish a general framework for developing, efficient energy stable numerical schemes for gradient flows and develop three classes of generalized scalar auxiliary variable approaches (G-SAV). Numerical schemes based on the G-SAV approaches are as efficient as the original SAV schemes [30,13] for gradient flows, i.e., only require solving linear equations with constant coefficients at each time step, can be unconditionally energy stable. But G-SAV approaches remove the definition restriction that auxiliary variables can only be square root function. The definition form of auxiliary variable is applicable to any reversible function for G-SAV approaches . Ample numerical results for phase field models are presented to validate the effectiveness and accuracy of the proposed G-SAV numerical schemes.

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A variant of scalar auxiliary variable approaches for gradient flows

Cited by 67 publications

References 23 publications

Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations

Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations

Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation

The generalized scalar auxiliary variable approach (G-SAV) for gradient flows

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