An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

Guan, Zhen; Heinonen, Vili; Lowengrub, John; Wang, Cheng; Wise, Steven M.

doi:10.1016/j.jcp.2016.06.007

Cited by 14 publications

(6 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The unconditional energy stability of the proposed scheme (3.8) follows the idea from the convex-concave decomposition of the energy, an idea popularized in Eyre's work [33]. The method has been applied to the phase field crystal (PFC) equation [41,89]; epitaxial thin film growth models [13,87]; nonlocal gradient model [43]; the Cahn-Hilliard model coupled with fluid flow [14,64], etc. Second-order accurate energy stable schemes have also been reported in recent years, based on either the Crank-Nicolson or BDF2 approach [6,7,22,23,42,44,48,81,90], etc.…”

Section: Numerical Scheme For the Reaction Stagementioning

confidence: 99%

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

Liu,

Wang,

Wang

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for certain type reaction-diffusion systems involving the Law of Mass Action with detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories, and both the reaction and the diffusion parts dissipate the same free energy. This subtle fact opens a path of an operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. To our best knowledge, it is the first time to report an energy-dissipation-lawbased operator splitting scheme to a nonlinear PDE with variational structures. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.

show abstract

Section: Numerical Scheme For the Reaction Stagementioning

confidence: 99%

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

Liu,

Wang,

Wang

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The phase-field crystal equation has been widely studied in the computational physics [51][52][53][54] and condensed matter physics literature [55][56][57]. Recently, the phase-field crystal model has been generalized to account for faster dynamics [58][59][60][61]; see [62] for details on the numerical treatment of the generalized equation.…”

Section: (A) Generalized Stefan Problemmentioning

confidence: 99%

A review on computational modelling of phase-transition problems

Gómez

Burés

Moure

2019

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

Phase-transition problems are ubiquitous in science and engineering. They have been widely studied via theory, experiments and computations. This paper reviews the main challenges associated with computational modelling of phase-transition problems, addressing both model development and numerical discretization of the resulting equations. We focus on classical phase-transition problems, including liquid–solid, gas–liquid and solid–solid transformations. Our review has a strong emphasis on the treatment of interfacial phenomena and the phase-field method. This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

show abstract

“…Commonly, the layer normal is defined as a vector via the gradients , but this is only self-consistent when variations of ϕ are negligible. By explicitly considering microscopic density variation on the scale of individual layers 26 , 31 , 41 , these issues can be avoided with computational cost, but a hydrodynamic-scale theory that both circumvents these issues and considers only mesoscopic variations of the lamellar order has not previously been established.…”

Section: Introductionmentioning

confidence: 99%

“…This last equivalence relation is identical to Ψ → Ψ * . We note that equations such as the Swift-Hohenberg 38 , phase field crystals [39][40][41][42][43] and density functional theory [44][45][46][47] can model striped phases via a continuous field approach that goes beyond the traditional simplistic sinusoidal approximation. However, these all must resolve individual layers, necessarily limiting their applicability to short times and length scales.…”

mentioning

confidence: 99%

Complex-tensor theory of simple smectics

et al. 2023

View full text Add to dashboard Cite

Matter self-assembling into layers generates unique properties, including structures of stacked surfaces, directed transport, and compact area maximization that can be highly functionalized in biology and technology. Smectics represent the paradigm of such lamellar materials — they are a state between fluids and solids, characterized by both orientational and partial positional ordering in one layering direction, making them notoriously difficult to model, particularly in confining geometries. We propose a complex tensor order parameter to describe the local degree of lamellar ordering, layer displacement and orientation of the layers for simple, lamellar smectics. The theory accounts for both dislocations and disclinations, by regularizing singularities within defect cores and so remaining continuous everywhere. The ability to describe disclinations and dislocation allows this theory to simulate arrested configurations and inclusion-induced local ordering. This tensorial theory for simple smectics considerably simplifies numerics, facilitating studies on the mesoscopic structure of topologically complex systems.

show abstract

An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

Cited by 14 publications

References 50 publications

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

A review on computational modelling of phase-transition problems

Complex-tensor theory of simple smectics

Contact Info

Product

Resources

About