The scalar auxiliary variable (SAV) approach for gradient flows

Shen, Jie; Xu, Jie; Yang, Jiang

doi:10.1016/j.jcp.2017.10.021

Cited by 805 publications

(422 citation statements)

References 25 publications

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“…It turns out that they are typical error estimates for all fully discrete numerical methods for various phase field models, which can be abstractly written into the form (62). Although for different models, the norms used to measure errors and the orders of errors (indicated by ℓ 1 and ℓ 2 in (62) ) may be different, the nature of the exponential dependence on 1 ε of the constants C 1 and C 2 remains the same for all these methods and models when the above standard perturbation procedure is used to derive the error estimates, see [169,170,158,32,33,432,209,312,155,193,192,194,425,299,284,385,21,302,140,436,395,382].…”

Section: Coarse Errormentioning

confidence: 99%

The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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Section: Coarse Errormentioning

confidence: 99%

The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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“…For the standard PFC model and its modified version, there have been extensive numerical works [3,4,17,36,50,51,52,57], et cetera, in the existing literature. Of course, because of its generality, the new SAV approach of Shen et al [45] could be applied to the PFC and SPFC problems. For the SPFC equation, few if any simulation results exist, to our knowledge, though a closely related equation is solved in [27].…”

Section: Introductionmentioning

confidence: 99%

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

Cheng¹,

Wang²,

Wise³

2019

CICP

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In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower that that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the ∞ (0, T ; 2 ) ∩ 2 (0, T ; H 3 N ) norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

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“…Apart from the many contributions discussed above, an interesting strategy for formulating energystable schemes for gradient-type dynamical systems (gradient flows) based on certain auxiliary variables has emerged recently [45,38]. The invariant energy quadratization (IEQ) method [45] introduces an auxiliary field function related to the square root of the potential free energy density function together with a dynamic equation for this auxiliary variable, and allows one to reformulate the gradient-flow evolution equation to facilitate schemes for ensuring the energy stability relatively easily.…”

Section: Introductionmentioning

confidence: 99%

An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices

Yang

Dong

2019

Journal of Computational Physics

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We present an energy-stable scheme for numerically approximating the governing equations for incompressible two-phase flows with different densities and dynamic viscosities for the two fluids. The proposed scheme employs a scalar-valued auxiliary energy variable in its formulation, and it satisfies a discrete energy stability property. More importantly, the scheme is computationally efficient. Within each time step, it computes two copies of the flow variables (velocity, pressure, phase field function) by solving individually a linear algebraic system involving a constant and time-independent coefficient matrix for each of these field variables. The coefficient matrices involved in these linear systems only need to be computed once and can be pre-computed. Additionally, within each time step the scheme requires the solution of a nonlinear algebraic equation about a scalar-valued number using the Newton's method. The cost for this nonlinear solver is very low, accounting for only a few percent of the total computation time per time step, because this nonlinear equation is about a scalar number, not a field function. Extensive numerical experiments have been presented for several two-phase flow problems involving large density ratios and large viscosity ratios. Comparisons with theory show that the proposed method produces physically accurate results. Simulations with large time step sizes demonstrate the stability of computations and verify the robustness of the proposed method. An implication of this work is that energy-stable schemes for two-phase problems can also become computationally efficient and competitive, eliminating the need for expensive re-computations of coefficient matrices, even at large density ratios and viscosity ratios.

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The scalar auxiliary variable (SAV) approach for gradient flows

Cited by 805 publications

References 25 publications

The phase field method for geometric moving interfaces and their numerical approximations

The phase field method for geometric moving interfaces and their numerical approximations

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices

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