An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation

Guo, Jing; Cheng, Wang; Wise, Steven M.; Yue, Xingye

doi:10.4310/cms.2016.v14.n2.a8

Cited by 151 publications

(90 citation statements)

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“…In addition to the

‖ \cdot ‖_{\infty}

bound for the phase variable, a discrete

W^{1, \infty}

estimate could be derived through a more careful analysis. In more details, by taking a discrete inner product with (2.10) by

- Δ_{h}^{3} ϕ^{k + 1}

, performing a nonlinear estimate, we are able to derive an

ℓ^{\infty} (0, T; H_{h}^{3})

estimate for the numerical solution, following similar ideas as in . Subsequently, the discrete

W^{1, \infty}

bound comes from a similar discrete Sobolev inequality:

‖ f ‖_{\infty} + ‖ \nabla_{h} f ‖_{\infty} \leq C ‖ f ‖_{H_{h}^{3}}

.…”

Section: The Fully Discrete Scheme With Finite Difference Spatial Dismentioning

confidence: 99%

“…As a result, a stronger convexity of the nonlinear term in the BDF one (2.10) is expected to improve the numerical efficiency in the nonlinear iteration. Such a numerical comparison has been undertaken for the Cahn‐Hilliard (CH) model in recent works: the analogous CN‐ES and BDF2‐ES numerical schemes for the CH equation, proposed in , respectively, were tested using a similar numerical setup. The numerical experiments have indicated that, since the nonlinear term in the BDF2‐ES approach has a stronger convexity than the one in the CN‐ES scheme, a 20 to 25% improvement of the computational efficiency is generally available for the CH model.…”

Section: Preconditioned Descent Solversmentioning

confidence: 99%

See 1 more Smart Citation

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Feng

Cheng²,

Wise

et al. 2018

Numerical Methods Partial

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In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an ℓ ∞ ( 0 , T ; H h 2 ) stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.

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“…In addition to the

‖ \cdot ‖_{\infty}

bound for the phase variable, a discrete

W^{1, \infty}

estimate could be derived through a more careful analysis. In more details, by taking a discrete inner product with (2.10) by

- Δ_{h}^{3} ϕ^{k + 1}

, performing a nonlinear estimate, we are able to derive an

ℓ^{\infty} (0, T; H_{h}^{3})

estimate for the numerical solution, following similar ideas as in . Subsequently, the discrete

W^{1, \infty}

bound comes from a similar discrete Sobolev inequality:

‖ f ‖_{\infty} + ‖ \nabla_{h} f ‖_{\infty} \leq C ‖ f ‖_{H_{h}^{3}}

.…”

Section: The Fully Discrete Scheme With Finite Difference Spatial Dismentioning

confidence: 99%

Section: Preconditioned Descent Solversmentioning

confidence: 99%

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Feng

Cheng²,

Wise

et al. 2018

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore it is important to develop efficient and accurate numerical schemes for their simulation. There exists an extensive literature on the numerical analysis of gradient flows, see for instance [3,14,7,10,23,9,15] and the references therein.…”

mentioning

confidence: 99%

Energy stability and convergence of SAV block-centered finite difference method for gradient flows

2019

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We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

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“…On the other hand, the Cahn-Hilliard equation represents another type of gradient flows for the phase transition, it has also been numerically studied extensively in the literature [20][21][22][23][24][25][26][27][28]. Here just mention some recent work, Guo et al [23] proposed an unconditionally energy stable finite difference scheme with a modified Crank-Nicolson temporal approximation, a global in time H 2 bound for the numerical solution is derived. Diegel et al [24] proposed a second-order mixed finite element method.…”

Section: Introductionmentioning

confidence: 99%

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

Pan

2018

Numerical Methods Partial

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In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.

show abstract

An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation

Cited by 151 publications

References 0 publications

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

Energy stability and convergence of SAV block-centered finite difference method for gradient flows

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

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