Convergence and Error Analysis for the  Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows

Shen, Jie; Xu, Jie

doi:10.1137/17m1159968

Cited by 276 publications

(103 citation statements)

References 23 publications

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“…Introduction. Recently, the so-called scalar auxiliary variable (SAV) approach is developed in [17,18]. This approach is inspired by the invariant energy quadratization (IEQ) approach [21] but fixes most, if not all, of its shortcomings.…”

mentioning

confidence: 99%

Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model

Cheng¹,

Shen²

2018

SIAM J. Sci. Comput.

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104

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We consider in this paper gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar auxiliary variable (SAV) approach, and we develop the multiple scalar auxiliary variable (MSAV) approach to deal with these cases. We apply the MSAV approach to the phase-field vesicle membrane (PF-VMEM) model which, in addition to some usual nonlinear terms in the free energy, has two additional penalty terms to enforce the volume and surface area. The MSAV approach enjoys the same computational advantages as the SAV approach but can handle free energies with multiple disparate terms such as the volume and surface area constraints in the PF-VMEM model. The MSAV schemes are unconditional energy stable and second-order accurate in time and lead to decoupled elliptic equations with constant coefficients to solve at each time step. Hence, these schemes are easy to implement and extremely efficient when coupled with an adaptive time stepping. Ample numerical results are presented to validate the stability and accuracy of the MSAV schemes.

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mentioning

confidence: 99%

Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model

Cheng¹,

Shen²

2018

SIAM J. Sci. Comput.

Self Cite

104

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“…The generalized auxiliary variable introduced here is inspired by the scalar auxiliary variable (SAV) approach proposed by [38], and to a lesser extent, by the invariant energy quadratization (IEQ) method [44], both of which are devised for gradient flows; see also e.g. [37,20,8,49,27,28,47,45] (among others) for extensions and applications of these techniques. In SAV a scalar-valued auxiliary variable is defined, as the square root of the shifted potential energy integral.…”

Section: Introductionmentioning

confidence: 99%

A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity

Yang

Dong

2020

Journal of Computational Physics

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We present a framework for devising discretely energy-stable schemes for general dissipative systems based on a generalized auxiliary variable. The auxiliary variable, a scalar number, can be defined in terms of the energy functional by a general class of functions, not limited to the square root function adopted in previous approaches. The current method has another remarkable property: the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level, regardless of the time step sizes or the external forces. This property of guaranteed positivity is not available in previous approaches. A unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented. The discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems. The current method, termed gPAV (generalized Positive Auxiliary Variable), requires only the solution of linear algebraic equations within a time step. With appropriate choice of the operator in the algorithm, the resultant linear algebraic systems upon discretization involve only constant and time-independent coefficient matrices, which only need to be computed once and can be precomputed. Several specific dissipative systems are studied in relative detail using the gPAV framework. Ample numerical experiments are presented to demonstrate the performance of the method, and the robustness of the scheme at large time step sizes.The rest of the volume terms are denoted by −V (u), not involving f . B s (f b , u) denotes the boundary terms, which may involve the boundary source term (f b ) through the boundary conditions. Substituting equation (2.6) into equation (2.5), we arrive at the following energy balance equation for the system,We assume that the boundary conditions (2.2) satisfy the following property,The dissipative nature of the system ensures that dEtot dt 0 in the absence of the external forces (i.e. f = 0 and f b = 0). Because the domain Ω can be arbitrary, it follows that V (u) must be non-negative, i.e. V (u) 0.(2.10)

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“…Ones have shown that these strategies are general enough to be useful for developing energy stable numerical approximations to any thermodynamically consistent models, i.e., the models satisfy the second law of thermodynamics or are derived from the Onsager principle [31,32,57]. The convex splitting approach, scalar auxiliary variable, energy quadratization approach and other methods have been applied to the Cahn-Hillard model for crystal growth [15,18,19,24,38,41,43,44,47,50,57].…”

Section: Introductionmentioning

confidence: 99%

Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints

Jing

Wang

2020

Computers & Mathematics with Applications

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Convergence and Error Analysis for the Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows

Cited by 276 publications

References 23 publications

Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model

Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model

A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity

Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints

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