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Unconditionally energy-stable and accurate schemes based on hyperbolic tangent scalar auxiliary variable approach for gradient flows.

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Title: Unconditionally energy-stable and accurate schemes based on hyperbolic tangent scalar auxiliary variable approach for gradient flows.
Authors: Zhou, Haihong1 (AUTHOR) haihongzhou@shu.edu.cn, Zhang, Huan1 (AUTHOR) huanzhang@shu.edu.cn, Pan, Xiaomin1,2,3 (AUTHOR) xmpan@shu.edu.cn
Source: Computers & Mathematics with Applications. Feb2026, Vol. 204, p263-282. 20p.
Subjects: Tangent function, Phase separation, Mathematical models, Derivatives (Mathematics)
Abstract: In this paper, we extend the selection of auxiliary variables by proposing a hyperbolic tangent scalar auxiliary variable (tanh-SAV) approach for solving gradient flows. The proposed tanh-SAV schemes introduce an auxiliary variable based on the hyperbolic tangent function, providing a well-defined formulation that enables the construction of decoupled, linear, and efficient numerical schemes. We demonstrate the construction of first-order, second-order, and higher-order unconditionally energy-stable schemes, utilizing either the Crank–Nicolson method or a k -step backward differentiation formula for time discretization. Only one constant-coefficient equation needs to be solved per time step, and we prove that all resulting tanh-SAV schemes are uniquely solvable at each time level. Furthermore, the theoretical analysis demonstrates the discrete energy stability of the proposed numerical schemes and proves the positivity property of the auxiliary variable. For the tanh-SAV/BDF k schemes (k = 2 , 3 , 4) combined with a Fourier pseudo-spectral spatial discretization, we further establish fully discrete optimal-order error estimates. In addition, we provide numerical simulations of one- and two-dimensional Cahn–Hilliard, Allen–Cahn, and phase-field crystal models. The results demonstrate that, consistent with the theoretical analysis, the proposed schemes preserve the positivity of the auxiliary variable, maintain excellent stability, and achieve the desired temporal accuracy. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:In this paper, we extend the selection of auxiliary variables by proposing a hyperbolic tangent scalar auxiliary variable (tanh-SAV) approach for solving gradient flows. The proposed tanh-SAV schemes introduce an auxiliary variable based on the hyperbolic tangent function, providing a well-defined formulation that enables the construction of decoupled, linear, and efficient numerical schemes. We demonstrate the construction of first-order, second-order, and higher-order unconditionally energy-stable schemes, utilizing either the Crank–Nicolson method or a k -step backward differentiation formula for time discretization. Only one constant-coefficient equation needs to be solved per time step, and we prove that all resulting tanh-SAV schemes are uniquely solvable at each time level. Furthermore, the theoretical analysis demonstrates the discrete energy stability of the proposed numerical schemes and proves the positivity property of the auxiliary variable. For the tanh-SAV/BDF k schemes (k = 2 , 3 , 4) combined with a Fourier pseudo-spectral spatial discretization, we further establish fully discrete optimal-order error estimates. In addition, we provide numerical simulations of one- and two-dimensional Cahn–Hilliard, Allen–Cahn, and phase-field crystal models. The results demonstrate that, consistent with the theoretical analysis, the proposed schemes preserve the positivity of the auxiliary variable, maintain excellent stability, and achieve the desired temporal accuracy. [ABSTRACT FROM AUTHOR]
ISSN:08981221
DOI:10.1016/j.camwa.2025.12.031