Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation

Jiang, Maosheng; Zhang, Zengyan; Zhao, Jia

doi:10.1016/j.jcp.2022.110954

Cited by 101 publications

(33 citation statements)

References 46 publications

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“…Up to now, there exist some extensions of the SAV technique, such as the E-SAV [51,52], the G-SAV [37], the relaxed SAV [41], and the relaxed generalized SAV techniques [82] etc. One can combine the GLTDs with those techniques for solving the gradients flows (1.1).…”

Section: Semi-discrete Sav-gl Schemesmentioning

confidence: 99%

A general class of linear unconditionally energy stable schemes for the gradient flows

Tan,

Tang

2022

Preprint

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This paper studies a class of linear unconditionally energy stable schemes for the gradient flows. Such schemes are built on the SAV technique and the general linear time discretization (GLTD) as well as the linearization based on the extrapolation for the nonlinear term, and may be arbitrarily high-order accurate and very general, containing many existing SAV schemes and new SAV schemes. It is shown that the semi-discrete-in-time schemes are unconditionally energy stable when the GLTD is algebraically stable, and are convergent with the order of min{q, ν} under the diagonal stability and some suitable regularity and accurate starting values, where q is the generalized stage order of the GLTD and ν denotes the number of the extrapolation points in time. The energy stability results can be easily extended to the fully discrete schemes, for example, if the Fourier spectral method is employed in space when the periodic boundary conditions are specified. Some numerical experiments on the Allen-Cahn, Cahn-Hilliard, and phase field crystal models are conducted to validate those theories as well as the effectiveness, the energy stability and the accuracy of our schemes.

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Section: Semi-discrete Sav-gl Schemesmentioning

confidence: 99%

A general class of linear unconditionally energy stable schemes for the gradient flows

Tan,

Tang

2022

Preprint

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“…However, the numerical results of R(t) and exp E(φ) are not equal anymore, which means the discrete energies ln R n+1 and E(φ n+1 ) are not equivalent anymore. Inspired by the R-SAV approach in [20], we construct the following E-SAV approach with relaxation (RE-SAV), which not only inherits all the advantages of the new E-SAV approach, but can also significantly improve its accuracy.…”

Section: 2mentioning

confidence: 99%

“…In [19], the authors consider a new SAV approach to construct high-order energy stable schemes. In [20], Jiang et al present a relaxation technique to construct a relaxed SAV (RSAV) approach to improve the accuracy and consistency noticeably.…”

mentioning

confidence: 99%

“…Meanwhile, based on the exponential function, the new E-SAV approach can remove the assumptions of bounded-from-below for the free energy potential or nonlinear term. Besides, to improve its accuracy and consistency noticeably, we apply the relaxation technique which was considered in [20] to propose a relaxed E-SAV (RE-SAV) method for gradient flows. The relaxation technique can improve the accuracy and eliminate the potential failure caused by the exponential growth.…”

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confidence: 99%

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Highly efficient exponential scalar auxiliary variable approaches with relaxation (RE-SAV) for gradient flows

Liu¹,

Li²

2022

Preprint

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For the past few years, scalar auxiliary variable (SAV) and SAV-type approaches became very hot and efficient methods to simulate various gradient flows. Inspired by the new SAV approach in [19], we propose a novel technique to construct a new exponential scalar auxiliary variable (E-SAV) approach to construct high-order numerical energy stable schemes for gradient flows. To improve its accuracy and consistency noticeably, we propose an E-SAV approach with relaxation, which we named the relaxed E-SAV (RE-SAV) method for gradient flows. The RE-SAV approach preserves all the advantages of the traditional SAV approach. In addition, we do not need any the bounded-from-below assumptions for the free energy potential or nonlinear term. Besides, the first-order, second-order and higher-order unconditionally energy stable time-stepping schemes are easy to construct. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

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“…Recently, hybrid analytical-numerical approaches, the finite integral transform method and the generalized integral transform technique (GITT) (Cotta and Mikhailov, 1997;Cotta, 1998;An and Su, 2014), has been developed to solve the structural mechanics problems (Li et al, 2019;Zhang et al, 2019a;Ullah et al, 2019;An et al, 2020;Li et al, 2020;He et al, 2021), and heat and fluid problems (An et al, 2013;Fu et al, 2018;Lisboa et al, 2018Lisboa et al, , 2019Machado dos Santos et al, 2022). The GITT method essentially preserves the properties of partial differential equations, which is different from the physics-preserving schemes of the finite difference method and the finite element method (Zhao et al, 2017;Shen et al, 2018;Jiang et al, 2021Jiang et al, , 2022. Zhang et al (2019a) and Ullah et al (2019) proposed finite integral transform method to investigate the bending of rectangular thin plates with corner supports and buckling behavior of moderately thick clamped rectangular plates, respectively.…”

Section: Introductionmentioning

confidence: 99%

Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

Tuo

Sun

et al. 2022

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PurposeThe purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation.Design/methodology/approachThe bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior.FindingsThe proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates.Originality/valueThe present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

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Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation

Cited by 101 publications

References 46 publications

A general class of linear unconditionally energy stable schemes for the gradient flows

A general class of linear unconditionally energy stable schemes for the gradient flows

Highly efficient exponential scalar auxiliary variable approaches with relaxation (RE-SAV) for gradient flows

Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

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