Journal of Computational and Applied Mathematics, ISSN 0377-0427, 12/2019, Volume 362, pp. 574 - 595

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–Hilliard equation, with second order accuracy in time and the fourth order...

Energy stability | Optimal rate convergence analysis | Long stencil fourth order finite difference approximation | Second order accuracy in time | Preconditioned steepest descent iteration | Cahn–Hilliard equation | Cahn-Hilliard equation | MATHEMATICS, APPLIED | CONVERGENCE ANALYSIS | ELEMENT-METHOD | 2ND-ORDER | APPROXIMATION | FOURIER | NUMERICAL SCHEME | ALLEN-CAHN | ERROR ANALYSIS | CONVEX SPLITTING SCHEMES | THIN-FILM MODEL

Energy stability | Optimal rate convergence analysis | Long stencil fourth order finite difference approximation | Second order accuracy in time | Preconditioned steepest descent iteration | Cahn–Hilliard equation | Cahn-Hilliard equation | MATHEMATICS, APPLIED | CONVERGENCE ANALYSIS | ELEMENT-METHOD | 2ND-ORDER | APPROXIMATION | FOURIER | NUMERICAL SCHEME | ALLEN-CAHN | ERROR ANALYSIS | CONVEX SPLITTING SCHEMES | THIN-FILM MODEL

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2017, Volume 55, Issue 4, pp. 1689 - 1718

We derive a fractional Cahn-Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space H-alpha, a is an element of [0, 1],...

L∞ boundedness | Error estimates | Stability | Fractional Cahn-Hilliard equation | Mass conservation | Fourier spectral method | SPECTRAL METHOD | MATHEMATICS, APPLIED | mass conservation | MOTION | fractional Cahn Hilliard equation | ALLEN-CAHN | MODEL | L-infinity boundedness | error estimates | stability

L∞ boundedness | Error estimates | Stability | Fractional Cahn-Hilliard equation | Mass conservation | Fourier spectral method | SPECTRAL METHOD | MATHEMATICS, APPLIED | mass conservation | MOTION | fractional Cahn Hilliard equation | ALLEN-CAHN | MODEL | L-infinity boundedness | error estimates | stability

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 02/2017, Volume 42, pp. 462 - 477

•An second-order time-spectral space-scheme for space fractional CH equation.•The proposed scheme satisfies mass conservation and energy degradation.•The...

Periodic and Neumann boundary conditions | Spectral method | Fractional-in-space Cahn–Hilliard equation | Crank–Nicolson scheme | Unconditionally energy stable | EPITAXY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | Crank-Nicolson scheme | ALLEN-CAHN | TIME | Fractional-in-space Cahn-Hilliard equation

Periodic and Neumann boundary conditions | Spectral method | Fractional-in-space Cahn–Hilliard equation | Crank–Nicolson scheme | Unconditionally energy stable | EPITAXY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | Crank-Nicolson scheme | ALLEN-CAHN | TIME | Fractional-in-space Cahn-Hilliard equation

Journal Article

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Efficient local energy dissipation preserving algorithms for the Cahn–Hilliard equation

Journal of Computational Physics, ISSN 0021-9991, 12/2018, Volume 374, pp. 654 - 667

•We derive that Cahn–Hilliard equation possesses a local energy dissipation law (LEDL).•Based on the observation, three LEDL schemes for the CH equation are...

Local structure-preserving algorithm | Total energy dissipation law | Mass conservation | Cahn–Hilliard equation | Local energy dissipation law | Cahn-Hilliard equation | ACCURATE | INCOMPRESSIBLE FLUIDS | 2ND-ORDER | DIFFERENCE SCHEME | 2-PHASE | PHYSICS, MATHEMATICAL | NONUNIFORM SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | FOURIER-SPECTRAL METHOD | PHASE-FIELD MODELS | FLOWS | FINITE-ELEMENT-METHOD | Algorithms

Local structure-preserving algorithm | Total energy dissipation law | Mass conservation | Cahn–Hilliard equation | Local energy dissipation law | Cahn-Hilliard equation | ACCURATE | INCOMPRESSIBLE FLUIDS | 2ND-ORDER | DIFFERENCE SCHEME | 2-PHASE | PHYSICS, MATHEMATICAL | NONUNIFORM SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | FOURIER-SPECTRAL METHOD | PHASE-FIELD MODELS | FLOWS | FINITE-ELEMENT-METHOD | Algorithms

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 12/2018, Volume 343, pp. 80 - 97

In this paper, we consider numerical approximations for the viscous Cahn–Hilliard equation with hyperbolic relaxation. This type of equations processes...

Phase-field | Variable mobility | Stability | Cahn–Hilliard | Linear | Flory–Huggins | SYSTEM | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | MAGNETIC DIFFUSION | Flory-Huggins | NUMERICAL APPROXIMATIONS | ALLEN-CAHN | LIQUID-CRYSTALS | DYNAMICS | EFFICIENT | Cahn-Hilliard | PHASE-FIELD MODEL | 2-PHASE INCOMPRESSIBLE FLOWS | Usage | Methods | Numerical analysis

Phase-field | Variable mobility | Stability | Cahn–Hilliard | Linear | Flory–Huggins | SYSTEM | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | MAGNETIC DIFFUSION | Flory-Huggins | NUMERICAL APPROXIMATIONS | ALLEN-CAHN | LIQUID-CRYSTALS | DYNAMICS | EFFICIENT | Cahn-Hilliard | PHASE-FIELD MODEL | 2-PHASE INCOMPRESSIBLE FLOWS | Usage | Methods | Numerical analysis

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 07/2019, Volume 73, pp. 217 - 228

•An efficient numerical method for the Cahn–Hilliard equation in complex domains.•Arbitrary domains can be dealt with a straightforward manner.•Most...

Multigrid method | Ternary Cahn–Hilliard system | Complex domain | Phase separation | Cahn–Hilliard equation | Cahn-Hilliard equation | MATHEMATICS, APPLIED | 2-PHASE FLOW | Ternary Cahn-Hilliard system | BOUNDARY-CONDITIONS | PHYSICS, FLUIDS & PLASMAS | ISOGEOMETRIC ANALYSIS | FOURIER-SPECTRAL METHODS | PHYSICS, MATHEMATICAL | SCHEME | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | MODELS | CONVERGENCE | SYSTEMS | FINITE-ELEMENT-METHOD | Analysis | Methods | Algorithms

Multigrid method | Ternary Cahn–Hilliard system | Complex domain | Phase separation | Cahn–Hilliard equation | Cahn-Hilliard equation | MATHEMATICS, APPLIED | 2-PHASE FLOW | Ternary Cahn-Hilliard system | BOUNDARY-CONDITIONS | PHYSICS, FLUIDS & PLASMAS | ISOGEOMETRIC ANALYSIS | FOURIER-SPECTRAL METHODS | PHYSICS, MATHEMATICAL | SCHEME | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | MODELS | CONVERGENCE | SYSTEMS | FINITE-ELEMENT-METHOD | Analysis | Methods | Algorithms

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 04/2017, Volume 73, Issue 8, pp. 1855 - 1864

In this work, we propose a fast and efficient adaptive time step procedure for the Cahn–Hilliard equation. The temporal evolution of the Cahn–Hilliard equation...

Unconditionally stable scheme | Cahn–Hilliard equation | Adaptive time-stepping method | Cahn-Hilliard equation | MATHEMATICS, APPLIED | VARIABLE-MOBILITY | NUMERICAL-METHOD | GROWTH | MODEL | PHASE-FIELD | Analysis | Methods | Algorithms

Unconditionally stable scheme | Cahn–Hilliard equation | Adaptive time-stepping method | Cahn-Hilliard equation | MATHEMATICS, APPLIED | VARIABLE-MOBILITY | NUMERICAL-METHOD | GROWTH | MODEL | PHASE-FIELD | Analysis | Methods | Algorithms

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 01/2016, Volume 305, pp. 360 - 371

Herein, we present a numerical convergence study of the Cahn–Hilliard phase-field model within an isogeometric finite element analysis framework. Using a...

Isogeometric analysis | Manufactured solutions | Bézier extraction | Cahn–Hilliard equation | Cahn-Hilliard equation | Bezier extraction | SPINODAL DECOMPOSITION | FINITE-ELEMENT APPROXIMATION | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DISCONTINUOUS GALERKIN | PHASE-FIELD MODELS | TUMOR-GROWTH | EFFICIENT | STRAIN | Finite element method | Approximation | Computation | Splines | Mathematical analysis | Mathematical models | Estimates | Convergence

Isogeometric analysis | Manufactured solutions | Bézier extraction | Cahn–Hilliard equation | Cahn-Hilliard equation | Bezier extraction | SPINODAL DECOMPOSITION | FINITE-ELEMENT APPROXIMATION | PHYSICS, MATHEMATICAL | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DISCONTINUOUS GALERKIN | PHASE-FIELD MODELS | TUMOR-GROWTH | EFFICIENT | STRAIN | Finite element method | Approximation | Computation | Splines | Mathematical analysis | Mathematical models | Estimates | Convergence

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 04/2016, Volume 310, pp. 85 - 108

We study computationally coarsening rates of the Cahn–Hilliard equation with a smooth double-well potential, and with phase-dependent diffusion mobilities. The...

Degenerate diffusion mobility | Power law | Cahn–Hilliard equation | Coarsening | Cahn-Hilliard equation | BEHAVIOR | LSW-THEORY | MODEL | PHYSICS, MATHEMATICAL | SURFACE MOTION | UPPER-BOUNDS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | FOURIER-SPECTRAL METHOD | ALLEN-CAHN | KINETICS | HELE-SHAW PROBLEM | DEGENERATE MOBILITY | Analysis | Thermoelectricity | Numerical analysis | Phases | Computer simulation | Volume fraction | Computation | Mathematical analysis | Mathematical models | Diffusion

Degenerate diffusion mobility | Power law | Cahn–Hilliard equation | Coarsening | Cahn-Hilliard equation | BEHAVIOR | LSW-THEORY | MODEL | PHYSICS, MATHEMATICAL | SURFACE MOTION | UPPER-BOUNDS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | FOURIER-SPECTRAL METHOD | ALLEN-CAHN | KINETICS | HELE-SHAW PROBLEM | DEGENERATE MOBILITY | Analysis | Thermoelectricity | Numerical analysis | Phases | Computer simulation | Volume fraction | Computation | Mathematical analysis | Mathematical models | Diffusion

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Equation and dynamic boundary condition of Cahn–Hilliard type with singular potentials

Nonlinear Analysis, ISSN 0362-546X, 11/2015, Volume 127, pp. 413 - 433

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn–Hilliard type, is discussed. The existence of a...

Cahn–Hilliard system | Dynamic boundary condition | Strong solution | Mass conservation | Well-posedness | Cahn-Hilliard system | MATHEMATICS | MATHEMATICS, APPLIED | Nonlinear dynamics | Mathematical analysis | Conservation | Nonlinearity | Boundary conditions | Boundaries | Dynamical systems | Regularity

Cahn–Hilliard system | Dynamic boundary condition | Strong solution | Mass conservation | Well-posedness | Cahn-Hilliard system | MATHEMATICS | MATHEMATICS, APPLIED | Nonlinear dynamics | Mathematical analysis | Conservation | Nonlinearity | Boundary conditions | Boundaries | Dynamical systems | Regularity

Journal Article

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Energy stable compact scheme for Cahn–Hilliard equation with periodic boundary condition

Computers and Mathematics with Applications, ISSN 0898-1221, 01/2019, Volume 77, Issue 1, pp. 189 - 198

We present a compact scheme to solve the Cahn–Hilliard equation with a periodic boundary condition, which is fourth-order accurate in space. We introduce...

Energy stability | Compact scheme | Convex splitting | Cahn–Hilliard equation | Cahn-Hilliard equation | MATHEMATICS, APPLIED | ACCURATE | 2ND-ORDER | DIFFERENCE SCHEME | MODEL | PHASE FIELD APPROACH | NUMERICAL-ANALYSIS | EQUILIBRIUM STATES | 1ST | MINIMIZING WAVELENGTHS | 4TH-ORDER COMPACT | Dimensional stability | Boundary conditions

Energy stability | Compact scheme | Convex splitting | Cahn–Hilliard equation | Cahn-Hilliard equation | MATHEMATICS, APPLIED | ACCURATE | 2ND-ORDER | DIFFERENCE SCHEME | MODEL | PHASE FIELD APPROACH | NUMERICAL-ANALYSIS | EQUILIBRIUM STATES | 1ST | MINIMIZING WAVELENGTHS | 4TH-ORDER COMPACT | Dimensional stability | Boundary conditions

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 09/2017, Volume 102, pp. 264 - 273

We study the well-posedness of the Cahn–Hilliard equation in which the gradient term in the free energy is replaced by a fractional derivative. We begin by...

Error estimates | Non-local energy | Fractional Cahn–Hilliard equation | Fourier spectral method | Well-posedness | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHASE SEGREGATION DYNAMICS | PHYSICS, MULTIDISCIPLINARY | PARTICLE-SYSTEMS | Fractional Cahn-Hilliard equation | LONG-RANGE INTERACTIONS | PHYSICS, MATHEMATICAL | NONUNIFORM SYSTEM

Error estimates | Non-local energy | Fractional Cahn–Hilliard equation | Fourier spectral method | Well-posedness | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHASE SEGREGATION DYNAMICS | PHYSICS, MULTIDISCIPLINARY | PARTICLE-SYSTEMS | Fractional Cahn-Hilliard equation | LONG-RANGE INTERACTIONS | PHYSICS, MATHEMATICAL | NONUNIFORM SYSTEM

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 06/2018, Volume 363, Issue C, pp. 39 - 54

Comparing with the well-known classic Cahn–Hilliard equation, the nonlocal Cahn–Hilliard equation is equipped with a nonlocal diffusion operator and can...

Nonlocal Cahn–Hilliard equation | Energy stability | Nonlocal diffusion operator | Gaussian kernel | Fast Fourier transform | Stabilized linear scheme | ENERGY | SLOPE SELECTION | PHYSICS, MATHEMATICAL | NUMERICAL-ANALYSIS | TIME-STEPPING METHODS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MODELS | ALLEN-CAHN | DIFFUSION | Nonlocal Cahn-Hilliard equation | FOURIER SPECTRAL APPROXIMATIONS | DENSITY-FUNCTIONAL THEORY | BOUNDARY-PROBLEM | Physics | Computer Science

Nonlocal Cahn–Hilliard equation | Energy stability | Nonlocal diffusion operator | Gaussian kernel | Fast Fourier transform | Stabilized linear scheme | ENERGY | SLOPE SELECTION | PHYSICS, MATHEMATICAL | NUMERICAL-ANALYSIS | TIME-STEPPING METHODS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MODELS | ALLEN-CAHN | DIFFUSION | Nonlocal Cahn-Hilliard equation | FOURIER SPECTRAL APPROXIMATIONS | DENSITY-FUNCTIONAL THEORY | BOUNDARY-PROBLEM | Physics | Computer Science

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 07/2016, Volume 140, pp. 38 - 60

We consider a stochastic extension of the nonlocal convective Cahn–Hilliard equation containing an additive Wiener process noise. We first introduce a suitable...

Cahn–Hilliard | Stochastic partial differential equation | Variational solution | Cahn-Hilliard | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MOTION | PHASE SEGREGATION DYNAMICS | PARTICLE-SYSTEMS | LONG-RANGE INTERACTIONS | BOUNDARY-PROBLEM | Nonlinearity | Additives | Stochasticity | Noise | Mathematical analysis | Uniqueness

Cahn–Hilliard | Stochastic partial differential equation | Variational solution | Cahn-Hilliard | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MOTION | PHASE SEGREGATION DYNAMICS | PARTICLE-SYSTEMS | LONG-RANGE INTERACTIONS | BOUNDARY-PROBLEM | Nonlinearity | Additives | Stochasticity | Noise | Mathematical analysis | Uniqueness

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 06/2013, Volume 242, pp. 321 - 350

We present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective Cahn–Hilliard equation, a stiff,...

Isogeometric analysis | Bézier extraction | Steady state | Shear flow | Cahn–Hilliard equation | Spinodal decomposition | Cahn-Hilliard equation | FLUIDS | GENERALIZED-ALPHA METHOD | APPROXIMATION | Bezier extraction | PHYSICS, MATHEMATICAL | NONUNIFORM SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PHASE-FIELD MODELS | DEGENERATE MOBILITY | 2-PHASE FLOWS | FINITE-ELEMENT-METHOD | NUMERICAL-SIMULATION | FREE ENERGY

Isogeometric analysis | Bézier extraction | Steady state | Shear flow | Cahn–Hilliard equation | Spinodal decomposition | Cahn-Hilliard equation | FLUIDS | GENERALIZED-ALPHA METHOD | APPROXIMATION | Bezier extraction | PHYSICS, MATHEMATICAL | NONUNIFORM SYSTEM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PHASE-FIELD MODELS | DEGENERATE MOBILITY | 2-PHASE FLOWS | FINITE-ELEMENT-METHOD | NUMERICAL-SIMULATION | FREE ENERGY

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 02/2018, Volume 124, pp. 44 - 56

This paper is concerned with the finite element approximation of the stochastic Cahn–Hilliard–Cook equation driven by an infinite dimensional Wiener type...

SPDEs | Cahn–Hilliard equation | Finite element | Cahn-Hilliard equation | MATHEMATICS, APPLIED | APPROXIMATION | Finite element method | Analysis | Methods

SPDEs | Cahn–Hilliard equation | Finite element | Cahn-Hilliard equation | MATHEMATICS, APPLIED | APPROXIMATION | Finite element method | Analysis | Methods

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 07/2017, Volume 340, pp. 177 - 199

The primal variational formulation of the fourth-order Cahn–Hilliard equation requires C1-continuous finite element discretizations, e.g., in the context of...

Isogeometric analysis | Finite cell method | Variational boundary conditions | The Nitsche method | Cahn–Hilliard equation | Composite electrode | Cahn-Hilliard equation | BATTERY ELECTRODE PARTICLES | SPINODAL DECOMPOSITION | GALERKIN METHODS | MODEL | PHYSICS, MATHEMATICAL | NURBS | ELECTROCHEMICAL REACTIONS | ELEMENTS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | LITHIUM-ION BATTERIES | PHASE-FIELD | Thermodynamics | Analysis | Methods

Isogeometric analysis | Finite cell method | Variational boundary conditions | The Nitsche method | Cahn–Hilliard equation | Composite electrode | Cahn-Hilliard equation | BATTERY ELECTRODE PARTICLES | SPINODAL DECOMPOSITION | GALERKIN METHODS | MODEL | PHYSICS, MATHEMATICAL | NURBS | ELECTROCHEMICAL REACTIONS | ELEMENTS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | LITHIUM-ION BATTERIES | PHASE-FIELD | Thermodynamics | Analysis | Methods

Journal Article