Formation of square patterns using a model alike Swift-Hohenberg

Hernandez, Jose Antonio Medina; Castañeda, Felipe Gómez; Cadenas, José Antonio Moreno

doi:10.1109/iceee.2014.6978331

Cited by 4 publications

(5 citation statements)

References 28 publications

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“…Remark The idea of the proposed Scheme 2.10, the BDF temporal approximation combined with a second‐order Douglas‐Dupont regularization, may provide a framework for a more general class of gradient flows, specifically, those with p ‐Laplacian nonlinear terms involved. For example, the square phase field crystal (SPFC) equation models crystal dynamics at the atomic scale in space but on diffusive scales in time, while keeping “square” symmetry crystal lattice structures. This model is an

H^{- 1}

gradient flow of an energy functional containing a four‐Laplacian energy density.…”

Section: The Fully Discrete Scheme With Finite Difference Spatial Dismentioning

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

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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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H^{- 1}

gradient flow of an energy functional containing a four‐Laplacian energy density.…”

Section: The Fully Discrete Scheme With Finite Difference Spatial Dismentioning

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

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“…Suppose that Ω ⊂ R d , d = 2, 3 is a rectangular domain. The energy of square phase field crystal (SPFC) model is given by [16,19,20,30]:…”

Section: Square Phase Field Crystal Modelmentioning

confidence: 99%

“…The highest order term models a small amount of surface diffusion, which smooths out the facets somewhat. In the square Swift-Hohenberg (SH) equation ∂ t u = −(1 + ∆) 2 u − βu + ηu 3 − u 5 + α |∇u| 2 ∇u , α > 0, β, η ∈ R, studied in [12,22,20,30], and the square phase field crystal (SPFC) equation ∂ t u = ∆ γ 0 u + γ 1 ∆u + ε 2 ∆ 2 u − ∇ · |∇u| 2 ∇u , γ 0 ∈ R, γ 1 > 0, studied in [16,19,20,30], the 4-Laplacian term gives preference to square-symmetry patterns. In general, such localized structures play important roles in biological, chemical, and physical processes [23].…”

Section: Introductionmentioning

confidence: 99%

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Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Feng

Salgado

Wang

et al. 2017

Journal of Computational Physics

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We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixthorder nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in generic Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general the theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems -including thin film epitaxy with slope selection and the square phase field crystal model -are carried out to verify the efficiency of the scheme.

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“…Consider the thin epitaxial film model with slope selection (Li & Liu, 2004;Shen, Wang, Wang, & Wise, 2012;Wang, Wang, & Wise, 2010;Xu & Tang 2006), and the square phase field crystal equation (Cross & Hohenberg, 1993;Herná ndez, Castañeda, & Cadenas, 2014;Hoyle, 1995;Lloyd, Sandstede, Avitabile, & Champneys, 2008), if the solution decays at infinity.…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of Nontrivial Stationary Solutions with Decay Order for Some Nonlinear Evolution Equations

Lin¹

2017

JMR

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Formation of square patterns using a model alike Swift-Hohenberg

Cited by 4 publications

References 28 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

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Nonexistence of Nontrivial Stationary Solutions with Decay Order for Some Nonlinear Evolution Equations

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