A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Furihata, Daisuke

doi:10.1007/pl00005429

Cited by 252 publications

(145 citation statements)

References 17 publications

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“…Traditional numerical methodologies for dealing with higher-order operators on very simple geometries include finite differences (see applications to the Cahn-Hilliard equation in [35,67]) and spectral approximations (solutions to the Cahn-Hilliard equation can be found in [58,60,76,77]). In real-world engineering applications, simple geometries are not very relevant, and therefore more geometrically flexible technologies need to be utilized.…”

Section: Spatial Discretizationmentioning

confidence: 99%

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

Gómez¹,

Calo²,

Bazilevs³

et al. 2007

135

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The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions are not common because primal variational formulations of fourthorder operators are well defined and integrable only if the finite element basis functions are piecewise smooth and globally C 1 -continuous. There are a very limited number of two-dimensional finite elements possessing C 1 -continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two-and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of C 1 and higher-order continuity. A NURBSbased variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology. There are a very limited number of two-dimensional finite elements possessing C1-continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two-and three-dimensional geometric exibility, compact support, and, most importantly, the possibility of C1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

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Section: Spatial Discretizationmentioning

confidence: 99%

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

Gómez¹,

Calo²,

Bazilevs³

et al. 2007

135

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“…Up to now, the DVDM has been applied to many conservative or dissipative partial differential equations (PDEs). For example, Furihata and Mori (1996) designed a stable dissipative difference scheme for the Cahn-Hilliard equation. Koide and Furihata (2009) proposed four conservative difference schemes for the regularized long wave equation.…”

Section: Introductionmentioning

confidence: 99%

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Yan

Zheng

2017

International Journal of Applied Mathematics and Computer Science

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The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg-de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg-de Vries equation.

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“…The Cahn-Hilliard equation is a diffuse interface model for describing the spinodal decomposition in binary alloys [5]. Various numerical schemes [2,6,10,11,12,13,14,15,17] have been used to solve the Cahn-Hilliard equation. Among them, an unconditionally stable scheme [11,12] were introduced by the convex-concave splitting of the nonconvex free energy as F (φ) = F c (φ) − F e (φ).…”

Section: Introductionmentioning

confidence: 99%

An Unconditionally Gradient Stable Numerical Method for the Ohta-Kawasaki Model

Kim¹,

Shin²

2017

Bulletin of the Korean Mathematical Society

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Abstract. We present a finite difference method for solving the OhtaKawasaki model, representing a model of mesoscopic phase separation for the block copolymer. The numerical methods for solving the OhtaKawasaki model need to inherit the mass conservation and energy dissipation properties. We prove these characteristic properties and solvability and unconditionally gradient stability of the scheme by using Hessian matrices of a discrete functional. We present numerical results that validate the mass conservation, and energy dissipation, and unconditional stability of the method.

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A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Cited by 252 publications

References 17 publications

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

An Unconditionally Gradient Stable Numerical Method for the Ohta-Kawasaki Model

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