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In this paper, we propose integrating factor (IF)‐based methods for solving the Cahn–Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature, and since then, it has appeared in many fields. Being a nonlinear equation, it is of great importance to develop efficient numerical methods for investigating the solution behavior of the CH equation. We present a few important linear and nonlinear schemes for the CH equation and obtain error estimates for those schemes. The integrators that we present here are of first and second order in time. We show numerical results to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In 1977, Thouless, Anderson, and Palmer (TAP) derived a system of consistent equations in terms of the effective magnetization in order to study the free energy in the Sherrington–Kirkpatrick (SK) spin glass model. The solutions to their equations were predicted to contain vital information about the landscapes in the SK Hamiltonian and the TAP free energy and moreover have direct connections to Parisi's replica ansatz. In this work, we aim to investigate the validity of the TAP equations in the generic mixed p-spin model. By utilizing the ultrametricity of the overlaps, we show that the TAP equations are asymptotically satisfied by the conditional local magnetizations on the asymptotic pure states. [ABSTRACT FROM AUTHOR]

Image inpainting is the process of restoring damaged areas in an image using information available from neighboring regions. In this paper, we present a novel, efficient, and simple local image inpainting algorithm based on the Allen–Cahn (AC) equation with a fidelity term. We utilize the phase separation property of the AC equation and introduce a new phase-dependent fidelity parameter to preserve the original values in the neighboring regions of an inpainting region. The governing partial differential equation is solved using the finite difference method, with the values of the neighboring cells serving as the Dirichlet boundary condition. The proposed algorithm is both local and explicit, making it is fast and easy to implement. We demonstrate the performance of the proposed model through several numerical experiments. Furthermore, comparing this method to other image inpainting methods demonstrates its superiority in image inpainting. [ABSTRACT FROM AUTHOR]

A mathematical analysis is performed for a system consisting of two coupled Cahn-Hilliard equations. These equations incorporate a diffusional mobility that depends on concentration. This modeling approach is often used to describe the process of phase separation in a thin layer of a binary liquid mixture covering a substrate, particularly when one of the components wets the substrate. The analysis establishes the existence of a weak formulation for this problem, which is supported by the use of a Lyapunov functional. Additionally, the analysis provides insights into the regularity properties of the weak formulation. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution. [ABSTRACT FROM AUTHOR]

In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings. [ABSTRACT FROM AUTHOR]

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qϵ). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qϵ). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qϵ). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this study, we propose a restoration algorithm for distorted objects using a curvature-driven flow. First, we capture the convex-hull shaped contour of the distorted object using the mean curvature flow. With the supplemented mass on the captured feature, we evolve the constraint mean curvature flow to a steady state, preserving the non-distorted region. With respect to the mass, we select a restorative shape by considering the square of the curvature derivative. The Allen–Cahn and Cahn–Hilliard equations are applied to the generated restored image from the implicit curvature motions represented by the order parameter. We impose the Dirichlet boundary condition for the order parameter and the Neumann boundary for the chemical potential to fix the feature and to inherit the mass conservation, respectively. We provided examples of the restoration of half-circle and parentheses-shaped objects to reconstruct a circle shape. [ABSTRACT FROM AUTHOR]

In this paper, we propose and present a non-overlapping substructuring-type iterative algorithm for the Cahn–Hilliard (CH) equation, which is a prototype for phase-field models. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of CH equation has. Here we present a formulation for the linear and non-linear Dirichlet–Neumann (DN) methods applied to the CH equation and study the convergence behaviour in one and two spatial dimensions in multiple subdomains. We show numerical experiments to illustrate our theoretical findings and effectiveness of the method. [ABSTRACT FROM AUTHOR]

The stress-strain state that occurs in multilayered structures (for example, beams) significantly depends on the nature of the interaction of their layers. In practice, when calculating such structural elements, an ideal contact of the layers is most often considered, this implies the continuity of displacement vectors and stresses at the contact boundary. Strict elasticity theory solutions in this case inevitably lead to the appearance of infinite tangential stresses on the free surfaces at the angular points of structures at the layer interfaces. In this paper, the problem of the stress-strain state of a multilayered beam is considered in which the interaction of layers is assumed nonideal. For this aim, a layer interaction model with a thin contact layer between the adhesive and adherends is chosen. The equations obtained on the basis of this model allowed us to find non-singular solutions that describe the essentially inhomogeneous stress-strain state in each layer of the beam and at their interfaces. [ABSTRACT FROM AUTHOR]

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes. [ABSTRACT FROM AUTHOR]

When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson--Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter Λ, which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As Λ = 0, the PB-steric equation becomes the conventional PB equation, but as Λ > 0, the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as Λ goes to infinity. Our theoretical results show that the PB-steric equations (for 0 ≤ Λ ≤ ∞ ) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations. [ABSTRACT FROM AUTHOR]

An efficient and easy-to-implement second-order Strang splitting approach is mainly applied to study the scalar Allen-Cahn (AC) equation in this paper. Base on the idea of dimensional splitting, a new time dependent function (called exponential integrating factor) is introduced for the scalar AC equation. Then we propose the Strang splitting approach which is aim to decompose the original equation into linear part and nonlinear part. In particular, the explicit 2-stage strong stability preserving Runge-Kutta(SSP-RK2) method is employed for the nonlinear part. Furthermore, we rigorously demonstrate the maximum principle, energy stability and convergence of the proposed scheme. Various numerical simulations in 2D and 3D are presented to confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

We prove the existence of the monotone traveling wave for the isothermal fluid equations with viscous and capillary terms by the planar dynamical system method. We obtain that the monotone traveling wave is asymptotically stable under the suitable perturbation. In the process of establishing the uniform a priori estimate, we dispose the capillary term reasonably according to the feature of the equations, and find the appropriate weighted function to overcome the difficulty caused by the non-convex pressure function. [ABSTRACT FROM AUTHOR]

We propose a new class of uniformly accurate splitting methods for the Benjamin–Bona-Mahony equation which converge uniformly in the dispersive parameter ε . The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter ε . The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case P (∂ x) = ∂ x our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in H r for solutions in H r without any loss of derivative. This estimate holds uniformly in ε . In regularizing regimes ε = O (1) we even gain a derivative with our time discretisation at the cost of loosing in terms of 1 ε . Numerical experiments underline our theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we propose and analyze a stabilized linear Crank-Nicolson scheme for the Swift-Hohenberg equation. More precisely, we treat the nonlinear term explicitly while two second-order stabilization terms are added to improve the stability of the scheme. We show that our scheme satisfies the discrete energy dissipation. We prove rigorously that our scheme is second-order accurate in time. Moreover, we adopt a spectral-Galerkin approximation for the spacial variables and establish error estimates for the fully discrete scheme. Numerical experiments are presented to show the accuracy and energy stability of our scheme. [ABSTRACT FROM AUTHOR]

The Enskog–Vlasov equation is a phenomenological kinetic equation that extends the Enskog equation for the dense (non-ideal) hard-sphere fluid by adding an attractive soft potential tail to the purely repulsive hard-sphere contribution. Simplifying assumptions about pair correlations lead to a Vlasov-like self-consistent force field that adds to the Enskog non-local hard-sphere collision integral. Within the limitations imposed by the underlying assumptions, the extension gives the Enskog–Vlasov equation the ability to give a unified description of ideal and non-ideal fluid flows as well as of those fluid states in which liquid and vapour regions coexist, being separated by a resolved interface. Furthermore, the Enskog–Vlasov fluid can be arbitrarily far from equilibrium. Thus the Enskog–Vlasov model equation provides an excellent, although approximate, tool for modelling processes with liquid–vapour interfaces and adjacent Knudsen layers, and allows us to look at slip, jump and evaporation coefficients from a different perspective. Here, a set of 26 moment equations is derived from the Enskog–Vlasov equation by means of the Grad moment method. The equations provide a meaningful approximation to the underlying kinetic equation, and include the description of Knudsen layers. This work focuses on the – rather involved – derivation of the moment equations, with only a few applications shown. [ABSTRACT FROM AUTHOR]

A variable time-step BDF2 scheme combined with the convex splitting strategy is proposed for the extended Fisher–Kolmogorov equation. Under the zero-stability condition r k : = τ k / τ k − 1 ≤ r user , (r user < 4.864) , the suggested BDF2 scheme is shown to be uniquely solvable unconditionally, and preserve a discrete (modified) energy dissipation law. With the help of the recent discrete orthogonal convolution kernels technique, a concise L 2 norm error estimate for the convex splitting BDF2 scheme is established under the time-step ratios restriction 0 < r k ≤ r user. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the robustness and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

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Implicit-explicit methods have been successfully used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work, we introduce a robust theoretical framework to analyze directly the stability and accuracy of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and provide a rigorous stability and convergence analysis for the original semi-discrete scheme under certain time step constraints. These settle several questions which have been open since the work of Chen and Shen [Comput. Phys. Comm. 108 (1998), pp. 147–158]. [ABSTRACT FROM AUTHOR]

In this paper, we derive the time-fractional Cahn–Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn–Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory–Huggins and double-obstacle type. We apply the Faedo–Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization. [ABSTRACT FROM AUTHOR]

The modified phase field crystal model is a sixth order nonlinear generalized damped wave equation. Two linearized difference schemes are presented based on the method of order reduction. One scheme is second order both in time and space, and the other scheme is convergent with the second temporal order and fourth spatial order. A theoretical analysis is carried out by the energy argument and mathematical induction. The uniqueness of the numerical solution and unconditional convergence in discrete L ∞ -norm are proved rigorously. Numerical results demonstrate that the presented schemes for the modified phase field crystal equation can achieve the theoretical convergence order. [ABSTRACT FROM AUTHOR]

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a "parabolic" type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29-30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange Multiplier approach in time and the Fourier-spectral scheme for space, we propose efficient numerical algorithms to solve the phase field crystal equation. The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential. The unconditional energy stability of the three semi-discrete schemes is proven. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes. [ABSTRACT FROM AUTHOR]

Cahn--Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have nonconstant and often degenerate mobilities. However, in the latter case, the spontaneous appearance of points of vanishing mobility and their impact on the solution are not well understood. In this paper we develop a singular perturbation theory to identify a range of degeneracies for which the solution of the Cahn--Hilliard equation forms a singularity in infinite time. This analysis forms the basis for a rigorous sharp interface theory and enables the systematic development of robust numerical methods for this family of model equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose first- and second-order exponential scalar auxiliary variable (ESAV) schemes for solving the phase field crystal equation with the periodic boundary condition. Specifically, the scalar auxiliary variable (SAV) in this work is constructed based on an exponential function, which differs from the square root form commonly used in the traditional SAV method. This feature allows the proposed schemes that only need to solve one linear system with constant coefficients at each time step. To construct the first-order ESAV scheme, we utilize the backward Euler method. Combining the ESAV method with the leapfrog approach, the second-order ESAV scheme is developed. Theoretically, the proposed schemes are proven to be uniquely solvable and unconditionally energy stable. Furthermore, the error estimates of the second-order ESAV scheme is rigorously proved. The numerical results are provided to demonstrate the effectiveness of the proposed schemes. • The scalar auxiliary variable in this study is constructed based on an exponential function. • The proposed schemes only need to solve one linear system with constant coefficients at each time step. • We use the leapfrog approach to develop the second-order scheme. • The error estimates of the second-order scheme is rigorously proved. [ABSTRACT FROM AUTHOR]

A system of phase-field equations with strong-coupling through state and gradient dependent non-diagonal mobility matrices is studied. Existence of weak solutions is established by the Galerkin approximation and a-priori estimates in strong norms. Relative energy estimates are used to derive a general nonlinear stability estimate. As a consequence, a weak–strong uniqueness principle is obtained and stability with respect to model parameters is investigated. [ABSTRACT FROM AUTHOR]

We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω ⊂ R d , d ≤ 3. We first prove the existence of maximal strong solutions in weighted (in time) L p spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix. [ABSTRACT FROM AUTHOR]

The Cahn–Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. Various dynamic boundary conditions have already been introduced in the literature to model interactions of the materials with the boundary more precisely. To take long-range interactions into account, we propose a new model consisting of a nonlocal Cahn–Hilliard equation with a nonlocal dynamic boundary condition comprising an additional boundary penalization term. We rigorously derive our model as the gradient flow of a nonlocal free energy with respect to a suitable inner product of order H − 1 containing both bulk and surface contributions. In the main model, the chemical potentials are coupled by a Robin type boundary condition depending on a specific relaxation parameter. We prove weak and strong well-posedness of this system, and we investigate the singular limits attained when this relaxation parameter tends to zero or infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider some hyperbolic variants of the mass conserving Allen–Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn–Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with N + 1 transition layers is very slow in time and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen–Cahn and Cahn–Hilliard equations and theirs hyperbolic variations is also performed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Tag localization in passive UHF RFID has become increasingly important, since the RFID system was introduced. Primarily the localization techniques in RFID are dependent on precise reckoning. Multiple researches have been made to solve the challenge of localization of RFID tags. These finding differs on the basis of cost and reliability. UHF RFID system using passive tags is a suitable option because of low cost, high reliability and global acceptance. In this paper a two-dimensional tag matrix-based localization algorithm is designed localize the passive tags using mobility of RFID reader. The algorithm is based on estimation equations for localization of tag. The data is collected from the scenario by the mobility of reader and location is estimated by the using estimation equations. Results revealed that the proposed design gives the desired result with greater level of accuracy as compared to other localization schemes. And it can be easily applied to various RFID applications. [ABSTRACT FROM AUTHOR]

In this study, we apply a finite difference scheme to solve the Cahn–Hilliard equation with generalized mobilities in complex geometries. This method is conservative and unconditionally gradient stable for all positive variable mobility functions and complex geometries. Herein, we present some numerical experiments to demonstrate the performance of this method. In particular, using the fact that variable mobility changes the growth rate of the phases, we employ space-dependent mobility to design a cylindrical biomedical scaffold with controlled porosity and pore size. [ABSTRACT FROM AUTHOR]

We construct and analyze a class of extrapolated and linearized Runge{Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen{Cahn and Cahn{Hilliard phase field equations, based on the scalar auxiliary variable (SAV) formulation. We prove that the proposed q-stage RK{SAV methods have qth-order convergence in time and satisfy a discrete version of the energy decay property. Numerical examples are provided to illustrate the discrete energy decay property and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

A "transparent point" is a particular value of a governing parameter in a nontranslationally invariant system that makes the system "almost" translationally invariant. This concept was introduced recently in the context of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity — it was discovered that a tuning of the lattice spacing parameter h in this model affects the soliton mobility. In this paper, we study the DNLS equation with competing cubic–quintic nonlinearity that also admits the transparent points with respect to the lattice spacing parameter h. We give a geometrical interpretation of the transparent points in terms of dynamical system theory and present a simple asymptotical formula for them at h → 0. Although the derivation of this formula is heuristic and nonrigorous, it gives the values of transparent points with remarkable accuracy even for quite large values of h. [ABSTRACT FROM AUTHOR]

Abstract This article is devoted to designing efficient linear finite element algorithm for the fractional Cahn–Hilliard equation, an important and newly proposed phase field model. Combining the advantages of the classic BiCG algorithm and the Toeplitz-like structure of the coefficient matrix, we develop a fast BiCG(FBiCG) algorithm for the linearized scheme to compute numerically the fractional Cahn–Hilliard equation. Our theoretical analysis and numerical experiments demonstrate that the proposed FBiCG reduces the computation cost and the storage to O (M log M) and O (M) , possesses the same convergence rates as Newton's algorithm does in space and time, and preserves the energy dissipation and equality laws. The numerical experiments also demonstrate that the FBiCG is almost mass conserved, recognizes accurately the phase separation by a very clear coarse graining process and the influences of different indices r and s of fractional Laplacian and different coefficients K and a on the width of the interfaces. [ABSTRACT FROM AUTHOR]

We consider a system which consists of a Cahn–Hilliard equation coupled with a Cahn–Hilliard–Oono type equation in a bounded domain of R d , d = 2 , 3. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters u and v. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for u , v and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for u and v , namely, they both stay uniformly away from the pure phases ± 1 in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state. [ABSTRACT FROM AUTHOR]

We prove an additional result on the linearized Cahn-Hilliard-Cook equation to fill a gap in the main argument in our paper that was published in SIAM J. Numer. Anal., 49 (2011), pp. 2407-2429. The result is a pathwise error estimate, which is proved by an application of the factorization argument for stochastic convolutions. [ABSTRACT FROM AUTHOR]

• For polynomial case, we prove unconditional stability of Strang-splitting. • We introduce a novel modified energy dissipation law. • For logarithmic case, we develop a DIRK propagator together with a Newton solver. • We prove a novel maximum principle and energy dissipation law. We consider a class of second-order Strang splitting methods for Allen-Cahn equations with polynomial or logarithmic nonlinearities. For the polynomial case both the linear and the nonlinear propagators are computed explicitly. We show that this type of Strang splitting scheme is unconditionally stable regardless of the time step. Moreover we establish strict energy dissipation for a judiciously modified energy which coincides with the classical energy up to O (τ) where τ is the time step. For the logarithmic potential case, since the continuous-time nonlinear propagator no longer enjoys explicit analytic treatments, we employ a second order in time two-stage implicit Runge–Kutta (RK) nonlinear propagator together with an efficient Newton iterative solver. We prove a maximum principle which ensures phase separation and establish energy dissipation law under mild restrictions on the time step. These appear to be the first rigorous results on the energy dissipation of Strang-type splitting methods for Allen-Cahn equations. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to define proper dynamic boundary conditions for a generalization of the Cahn-Hilliard system proposed by M. Gurtin. Such boundary conditions take into account the interactions with the walls in confined systems. We then study the existence and uniqueness of weak solutions. [ABSTRACT FROM AUTHOR]

An initial-boundary-value problem for a class of sixth order viscous Cahn-Hilliard type equations with a nonlinear diffusion is considered. The study is motivated by phase-field modelling of various spatial structures, for example arising in oil-water-surfactant mixtures and in modelling of crystal growth on atomic length, known as phase field crystal model. For such problem we prove the existence and uniqueness of a global in time regular solution. First the finite-time existence is proved by means of the Leray-Schauder fixed point theorem. Then, due to suitable estimates, the finite-time solution is extended step by step on the infinite time interval. [ABSTRACT FROM AUTHOR]

We consider the Cahn--Hilliard equation in one space dimension with scaling parameter ∈, i.e., ut = ∈ ²uxx)xx, where W is a nonconvex potential. In the limit ∈ ↓ 0, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation. [ABSTRACT FROM AUTHOR]

We study a sixth order Cahn--Hilliard type equation that arises as a model for the faceting of a growing surface. We show existence of global-in-time weak solutions in two space dimensions, assuming periodic boundary conditions. We also establish exponential-in-time a priori estimates on the H³ norm of solutions. These bounds enable us to prove the uniqueness of weak solutions. We also show the regularizing effect of the equation on the data. [ABSTRACT FROM AUTHOR]

In this paper we present PDE and finite element analyses for a system of PDEs consisting of the Darcy equation and the Cahn--Hilliard equation, which arises as a diffuse interface model for the two-phase Hele-Shaw flow. In the model the two sets of equations are coupled through an extra phase induced force term in the Darcy equations and a fluid induced transport term in the Cahn--Hilliard equation. We propose a fully discrete implicit finite element method for approximating the PDE system, which consists of the implicit Euler method combined with a convex splitting energy strategy for the temporal discretization, the standard finite element discretization for the pressure, and a split (or mixed) finite element discretization for the fourth-order Cahn--Hilliard equation. It is shown that the proposed numerical method satisfies a mass conservation law in addition to a discrete energy law that mimics the basic energy law for the Darcy--Cahn--Hilliard phase field model and holds uniformly in the phase field parameter ε. With the help of the discrete energy law, we first prove that the fully discrete finite method is unconditionally energy stable and uniquely solvable at each time step. We then show that, using the compactness method, the finite element solution has an accumulation point that is a weak solution of the PDE system. As a result, the convergence result also provides a constructive proof of the existence of global-in-time weak solutions to the Darcy--Cahn--Hilliard phase field model in both two and three dimensions. Numerical experiments based on the overall solution method of combining the proposed finite element discretization and a nonlinear multigrid solver are presented to validate the theoretical results and to show the effectiveness of the proposed fully discrete finite element method for approximating the Darcy--Cahn--Hilliard phase field model. [ABSTRACT FROM AUTHOR]

The linearized Cahn–Hilliard–Cook equation is discretized in the spatial variables by a standard finite-element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. Backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn–Hilliard equation. The results should be interpreted as results on the approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn–Hilliard–Cook equation. [ABSTRACT FROM PUBLISHER]

iulian.stoleriu@uaic.ro We introduce two non-local versions of the well known Allen-Cahn and Cahn-Hilliard equations, which are proposed as alternative models for phase transitions in solids. Their analysis can be motivated by the fact that the corresponding local analogues fail to be applicable when the wavelength of microstructure is very small. Though the solutions to the local and non-local equations share some common properties, we have also found some differences between them. We concentrate here on the differences in the coarsening process of their solutions, bringing some numerical results in support of our conclusion. [ABSTRACT FROM AUTHOR]