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     

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.     

A mathematical model for phase separation: A generalized Cahn–Hilliard equation

In this paper we present a mathematical model to describe the phenomenon of phase separation, which is modelled as space regions where an order parameter changes smoothly. The model proposed, including thermal and mixing effects, is deduced for an incompressible fluid, so the resulting differential system couples a generalized Cahn–Hilliard equation with the Navier–Stokes equation. Its consistency with the second law of thermodynamics in the classical Clausius–Duhem form is finally proved. Copyright © 2011 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.     

Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.     

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     

Energy stable numerical scheme for the viscous Cahn–Hilliard–Navier–Stokes equations with moving contact line

In the present article we study the numerics of the viscous Cahn–Hilliard–Navier–Stokes model, endowed with dynamic boundary conditions which allow us to take into account the interaction between the fluids interface and the moving walls of the physical domain. In what follows, we propose an energy stable temporal scheme for the problem and we prove the stability and the unconditional solvability of the discretization proposed. We also propose a fully discrete scheme for which we prove the stability and the unconditional solvability. Numerical simulations are presented to illustrate the theoretical results.     

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.     

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.     

Numerical analysis for the mixed Navier–Stokes and Darcy Problem with the Beavers–Joseph interface condition

In this article, we consider the coupled Navier–Stokes and Darcy problem with the Beavers–Joseph interface condition. With suitable restrictions of physical parameters α and ν, we prove the existence and local uniqueness of a weak solution. Then we propose a coupled finite element scheme and a decoupled and linearized scheme based on two‐grid finite element. Under suitable further restrictions, their optimal error estimates are obtained. Finally numerical experiments indicate the validity of the theoretical results as well as the efficiency and effectiveness of the decoupled and linearized two‐grid algorithm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1009–1030, 2015     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.     

Efficient energy-stable schemes for the hydrodynamics coupled phase-field model

In this article, several efficient and energy-stable semi–implicit schemes are presented for the Cahn–Hilliard phase-field model of two-phase incompressible flows. A scalar auxiliary variable (SAV) approach is implemented to solve the Cahn–Hilliard equation, while a splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of the phase-field variable, velocity, and pressure are totally decoupled. A finite-difference method on staggered grids is adopted to spatially discretize the proposed time-marching schemes. We rigorously prove the unconditional energy stability for the semi-implicit schemes and the fully discrete scheme. Numerical results in both two and three dimensions are obtained, which demonstrate the accuracy and effectiveness of the proposed schemes. Using our numerical schemes, we compare the SAV, invariant energy quadratization (IEQ), and stabilization approaches. Bubble rising dynamics and coarsening dynamics are also investigated in detail. The results demonstrate that the SAV approach is more accurate than the IEQ approach and that the stabilization approach is the least accurate among the three approaches. The energy stability of the SAV approach appears to be better than that of the other approaches at large time steps.     

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.     

A linear,decoupled fractional time‐stepping method for the nonlinear fluid–fluid interaction

In this paper, a linear decoupled fractional time stepping method is proposed and developed for the nonlinear fluid–fluid interaction governed by the two Navier–Stokes equations. Partitioned time stepping method is applied to two‐physics problems with stiffness of the coupling terms being treated explicitly and is also unconditionally stable. As for each fluid, the velocity and pressure are respectively determined by just solving one vector‐valued quasi‐elliptic equation and the Possion equation with homogeneous Neumann boundary condition per time step. Therefore, the cost of the fluid–fluid interaction is dominant to solve four simple linear equations, which greatly reduces the computational cost of the whole system. The method exploits properties of the fluid–fluid system to establish its stability and convergence with the same results as the standard scheme. Finally, numerical experiments are presented to show the performance of the proposed method.     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

Decoupled modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem

In this paper, a modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem with the Beavers–Joseph–Saffman interface condition is presented. In this method, the Navier–Stokes/Darcy equation is decoupled into two equations, one is the Navier–Stokes equation, the other is the Darcy equation, and the Navier–Stokes equation is solved by the modified characteristics finite element method. The theory analysis shows that this method has a good convergence property. In order to show the effect of our method, some numerical results was presented. The numerical results show that this method is highly efficient. Copyright © 2013 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.     

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.     

Finite difference approximate solutions for the Cahn‐Hilliard equation

In this article, we analyze a Crank‐Nicolson‐type finite difference scheme for the nonlinear evolutionary Cahn‐Hilliard equation. We prove existence, uniqueness and convergence of the difference solution. An iterative algorithm for the difference scheme is given and its convergence is proved. A linearized difference scheme is presented, which is also second‐order convergent. Finally a new difference method possess a Lyapunov function is presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 437–455, 2007     

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.