Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation

Hu, Zhaozheng; Wise, Steven M.; Wang, C.; Lowengrub, John

doi:10.1016/j.jcp.2009.04.020

Cited by 282 publications

(228 citation statements)

References 18 publications

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“…The calculations here are meant to motivate those for the fully discrete scheme that we exhibit in later sections. The scheme is based on a convex splitting approach Eyre [6,7,27, 1] that we have used in earlier works [10,21,23,28,29,35]. There are two important properties that convex splitting schemes generally inherit, unconditional energy stability and unconditional unique solvability [7,35].…”

Section: A Convex Splitting Scheme In Timementioning

confidence: 99%

An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

Collins

Shen

Wise

2013

Commun. comput. phys.

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We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the time and space discrete ∞ 0, T ; H 1 h and 2 0, T ; H 2 h norms. We also present an efficient, practical nonlinear multigrid method -comprised of a standard FAS method for the CahnHilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part -for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy-and boundary-driven flows.

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Section: A Convex Splitting Scheme In Timementioning

confidence: 99%

An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

Collins

Shen

Wise

2013

Commun. comput. phys.

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“…Irrespective of the discretization method used, these last two points lead to the need for robust and scalable software to solve these types of problems. Last but not least, phase-field models should possess strong energy stability [30]: free energy must decrease in time. The goal of this paper is to devise a time discretization scheme that applies to both conserved and non-conserved phase-field models with a polynomial potential, within a high-performance framework, with the longterm goal of obtaining quantitative results for industrially relevant applications.…”

Section: Introductionmentioning

confidence: 99%

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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“…In an effort to have simultaneously unconditional stability and solvability, a second-order time-accurate convex-splitting scheme was proposed by (Hu et al, 2009). The method is given by…”

Section: Second-order Convex Splittingmentioning

confidence: 99%

“…However, the method is not unconditionally stable as defined in (190). A less restrictive statement referred to as weak energy stability (Hu et al, 2009) may be proven if W − is a quadratic polynomial, which applies to many potential functions. An alternative is the multistep scheme proposed in (Guillén-González and Tierra, 2013), which is however restricted to the quartic potential.…”

Section: Second-order Convex Splittingmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation

Cited by 282 publications

References 18 publications

An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

An energy-stable time-integrator for phase-field models

Computational Phase‐Field Modeling

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