A numerical scheme for the solution of viscous Cahn–Hilliard equation

In this paper, we present a numerical scheme for the solution of viscous Cahn–Hilliard equation. The scheme is based on Adomian's decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components. Some numerical examples are presented. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A numerical method for the Cahn–Hilliard equation with a variable mobility

We consider a conservative nonlinear multigrid method for the Cahn–Hilliard equation with a variable mobility of a model for phase separation in a binary mixture. The method uses the standard finite difference approximation in spatial discretization and the Crank–Nicholson semi-implicit scheme in temporal discretization. And the resulting discretized equations are solved by an efficient nonlinear multigrid method. The continuous problem has the conservation of mass and the decrease of the total energy. It is proved that these properties hold for the discrete problem. Also, we show the proposed scheme has a second-order convergence in space and time numerically. For numerical experiments, we investigate the effects of a variable mobility.     

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

We give a detailed study of the infinite‐energy solutions of the Cahn–Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well‐posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties. Copyright © 2013 John Wiley & Sons, Ltd.     

Classical solvability of 1‐D Cahn–Hilliard equation coupled with elasticity

In this paper, we prove the classical solvability of a nonlinear 1‐D system of hyperbolic–parabolic type arising as a model of phase separation in deformable binary alloys. The system is governed by the nonstationary elasticity equation coupled with the Cahn–Hilliard equation. The existence proof is based on the application of the Leray–Schauder fixed point theorem and standard energy methods. Copyright © 2006 John Wiley & Sons, Ltd.     

A Cahn–Hilliard model in bounded domains with permeable walls

In this article, we propose a model of phase separation in a binary mixture confined to a bounded region which may be contained within porous walls. The boundary conditions are derived from a mass conservation law and variational methods. Employing classical methods, that is, fixed point theorems and standard energy methods, we obtain the existence and uniqueness of a global solution to our problem. We then also compare our model of phase separation with other previous Cahn–Hilliard equations with homogeneous Neumann and dynamic boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

We consider in this article the Cahn–Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray–Schauder principle, the existence and uniqueness of solutions. We then construct a robust family of exponential attractors. Copyright © 2005 John Wiley & Sons, Ltd.     

Periodic solutions to the Cahn–Hilliard equation with constraint

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.     

Global regular solutions to Cahn–Hilliard system coupled with viscoelasticity

In this paper we prove the existence and uniqueness of a global in time, regular solution to the Cahn–Hilliard system coupled with viscoelasticity. The system arises as a model, regularized by a viscous damping, of phase separation process in a binary deformable alloy quenched below a critical temperature. The key tools in the analysis are estimates of absorbing type with the property of exponentially time‐decreasing contribution of the initial data. Such estimates allow not only to prolong the solution step by step on the infinite time interval but also to conclude the existence of an absorbing set. Copyright © 2009 John Wiley & Sons, Ltd.     

A Cahn–Hilliard type equation with periodic gradient‐dependent potentials and sources

This paper is concerned with the existence, uniqueness and attractability of time periodic solutions of a Cahn–Hilliard type equation with periodic gradient‐dependent potentials and sources. Copyright © 2009 John Wiley & Sons, Ltd.     

3‐D viscous Cahn–Hilliard equation with memory

We deal with the memory relaxation of the viscous Cahn–Hilliard equation in 3‐D, covering the well‐known hyperbolic version of the model. We study the long‐term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors, which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. Copyright © 2008 John Wiley & Sons, Ltd.     

Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transition

A posteriori estimates of errors in quantities of interest are developed for the nonlinear system of evolution equations embodied in the Cahn–Hilliard model of binary phase transition. These involve the analysis of wellposedness of dual backward‐in‐time problems and the calculation of residuals. Mixed finite element approximations are developed and used to deliver numerical solutions of representative problems in one‐ and two‐dimensional domains. Estimated errors are shown to be quite accurate in these numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the approximate solution to an initial boundary valued problem for the Cahn–Hilliard equation

The particular approximate solution of the initial boundary valued problem to the Cahn–Hilliard equation is provided. The Fourier Method is combined with the Adomian’s decomposition method in order to provide an approximate solution that satisfy the initial and the boundary conditions. The approximate solution also satisfies the mass conservation principle.     

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

We propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier‐Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013     

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn–Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori error estimates. Our analysis is valid for both symmetric and nonsymmetric versions of the discontinuous Galerkin formulation.     

Sixth‐order Cahn–Hilliard systems with dynamic boundary conditions

Our aim in this paper is to define dynamic boundary conditions for several sixth‐order Cahn–Hilliard systems. We then study the well‐posedness and the dissipativity of the systems derived. Copyright © 2014 John Wiley & Sons, Ltd.     

Practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth‐order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

The fast scalar auxiliary variable approach with unconditional energy stability for nonlocal Cahn–Hilliard equation

Comparing with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal Cahn–Hilliard equations equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. In this paper, we construct an accurate and efficient scalar auxiliary variable approach for the nonlocal Cahn–Hilliard equation with general nonlinear potential. The first contribution is that we have proved the unconditional energy stability for nonlocal Cahn–Hilliard model and its semi‐discrete schemes carefully and rigorously. Second, what we need to focus on is that the nonlocality of the nonlocal diffusion term will lead the stiffness matrix to be almost full matrix which generates huge computational work and memory requirement. For spatial discretizaion by finite difference method, we find that the discretizaition for nonlocal operator will lead to a block‐Toeplitz–Toeplitz‐block matrix by applying four transformation operators. Based on this special structure, we present a fast procedure to reduce the computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.     

Asymptotic analysis for Cahn–Hilliard type phase‐field systems related to tumor growth in general domains

This article considers a limit system by passing to the limit in the following Cahn–Hilliard type phase‐field system related to tumor growth as β↘0: in a bounded or an unbounded domain with smooth‐bounded boundary. Here, , T > 0, α > 0, β > 0, p ≥ 0, B is a maximal monotone graph, and π is a Lipschitz continuous function. In the case that Ω is a bounded domain, p and ?Δ + 1 are replaced with p(φ_β) and ?Δ, respectively, and p is a Lipschitz continuous function; Colli, Gilardi, Rocca, and Sprekels (Discrete Contin Dyn Syst Ser S 2017; 10:37–54) have proved existence of solutions to the limit problem with this approach by applying the Aubin–Lions lemma for the compact embedding H¹(Ω)?L²(Ω) and the continuous embedding L²(Ω)?(H¹(Ω))^?. However, the Aubin–Lions lemma cannot be applied directly when Ω is an unbounded domain. The present work establishes existence of weak solutions to the limit problem along with uniqueness and error estimates in terms of the parameter β↘0. To this end, we construct an applicable theory by noting that the embedding H¹(Ω)?L²(Ω) is not compact in the case that Ω is an unbounded domain.     

Stabilized semi‐implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations

Stabilized semi‐implicit spectral deferred correction methods are constructed for the time discretization of Allen–Cahn and Cahn–Hilliard equations. These methods are unconditionally stable, lead to simple linear system to solve at each iteration, and can achieve high‐order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effectiveness of the stabilized semi‐implicit spectral deferred correction methods for solving the Allen–Cahn and Cahn–Hilliard equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.