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In this article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption. [ABSTRACT FROM AUTHOR]

A fully discrete semi-convex-splitting finite-element scheme with stabilization for a Cahn–Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFem illustrate the phase segregation and pattern formation. [ABSTRACT FROM AUTHOR]

A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, is approximated in a semi-implicit manner. In the spatial discretization, the marker and cell difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

An unconditionally gradient-stable (dissipative) numerical method is developed for solving a conservative model of the density gradient in the spherically symmetric formulation. The algorithm is constructed using Eyre's method based on a convex splitting of the free energy of the system. The gradient stability of the constructed algorithm in the semidiscrete and fully discrete cases is proved. The theoretical results are confirmed by test computations. The proposed numerical method is used to analyze the influence exerted by the way of specifying the diffusion mobility on the evolution of the phase boundary. [ABSTRACT FROM AUTHOR]

In image inpainting, the identification and inpainting of local detail features and the preservation of global features are crucial. Models based on fractional-order partial differential equations exhibit rich evolutionary behaviors. These behaviors enable them to effectively comprehend image details. Additionally, these models possess a certain sharpening effect in image inpainting. However, they are also prone to issues such as inaccurate identification of large-scale features and over-sharpening. The optimal control model proposed in this paper uses the total variation energy of image global features as the objective function. It also employs the spatial fractional-order vector-valued Cahn–Hilliard equation as the constraint, aiming to achieve a balanced effect between local detail restoration and preservation of global features. The paper aims to optimize the objective function by designing numerical computation schemes for non-convex constraint conditions using L 2 gradient flow, H - 1 gradient flow, and convex splitting. The Split Bregman method is used for further optimization, and a dynamic grayscale adjustment strategy is introduced to maintain grayscale discrimination capability while enhancing computational efficiency. Numerical experiments demonstrate that the new model exhibits certain advantages over other inpainting methods in terms of PSNR values. It also shows strong competitiveness in terms of SSIM values, especially in severely damaged images where it demonstrates greater stability. Compared to traditional fractional-order equation models, the proposed model captures global features and incorporates the dynamic grayscale adjustment strategy, resulting in significantly reduced computation time. [ABSTRACT FROM AUTHOR]

In this paper, we develop a second-order backward differentiation formula (BDF) numerical scheme for the Ericksen-Leslie model of nematic liquid crystal. It combines a convex splitting method to discretize the penalty function. In the meanwhile, a pressure-correction strategy is used to decouple the pressure from that of the velocity. We prove that the proposed scheme is uniquely solvable and unconditionally stable in energy. Some numerical simulations are also performed to demonstrate the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a linearized energy stable numerical scheme is used to solve the modified Cahn-Hilliard-Navier-Stokes model, which is a phase-field model for two-phase incompressible flows. The time discretization is based on the convex splitting of the energy functional, which leads to a linearized system. In order to maintain the energy stability, the definition domain of energy function is extended to infinity. The stability of the scheme is proved and the error estimate is given. Numerical experiments are done to demonstrate the effectiveness for the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L 2 -gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L 2 -gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H − 1 -gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework. [ABSTRACT FROM AUTHOR]

In this paper, a decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system with double well potential is constructed. The proposed scheme is based on the convex splitting of the energy functional in time and uses the mixed finite element discretzation in space. The computation of the velocity u is separated from the computation of the pressure p by using an operator-splitting strategy. A Possion equation is solved to update the pressure at each time step. Unconditional stability and error estimates are analyzed in detail theoretically. Furthermore, the theoretical part is verified by several numerical examples, whose results show that the numerical examples are consistent with the results of the theoretical part.

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]

The conservative Allen–Cahn equation with a nonlocal Lagrange multiplier satisfies mass conservation and energy dissipation property. A challenge to numerically solving the equation is how to treat the nonlinear and nonlocal terms to preserve mass conservation and energy stability without compromising accuracy. To resolve this problem, we first apply the convex splitting idea to not only the term corresponding to the Allen–Cahn equation but also the nonlocal term. A wise implementation of the convex splitting for the nonlocal term ensures numerically exact mass conservation. And we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method. We show analytically that the scheme is uniquely solvable and unconditionally energy stable by using the fact that the scheme guarantees exact mass conservation. Numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme. [ABSTRACT FROM AUTHOR]

We present two graph-based algorithms for multiclass segmentation of high-dimensional data on graphs. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation and graph cuts. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, image labeling, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art in multiclass graph-based segmentation algorithms for high-dimensional data.

We present two graph-based algorithms for multiclass segmentation of high-dimensional data, motivated by the binary diffuse interface model. One algorithm generalizes Ginzburg-Landau (GL) functional minimization on graphs to the Gibbs simplex. The other algorithm uses a reduction of GL minimization, based on the Merriman-Bence-Osher scheme for motion by mean curvature. These yield accurate and efficient algorithms for semi-supervised learning. Our algorithms outperform existing methods, including supervised learning approaches, on the benchmark datasets that we used. We refer to Garcia-Cardona (2014) for a more detailed illustration of the methods, as well as different experimental examples. © 2014 Elsevier Ltd. All rights reserved.

In this paper, we develop a first order (in time) numerical scheme for the binary uid surfactant phase field model. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. The resulting coupled system consists of two Cahn-Hilliard type equations. This system is solved numerically by finite difference spatial approximation, in combination with convex splitting temporal discretization. We prove the proposed scheme is unique solvable, positivity-preserving and unconditionally energy stable. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme, which will be the first such result for the binary uid-surfactant system. Newton iteration is used to solve the discrete system. Some numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn–Hilliard equation is introduced. The proposed numerical scheme employs the L 1 + formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized for spatial discretization. As a result, the fully discrete scheme has several advantages: second-order accurate in time, spectrally accurate in space, uniquely solvable, mass preserving, and unconditionally energy stable. Rigorous proofs are given, along with several numerical results to verify the theoretical results, and to show the accuracy and effectiveness of the proposed scheme. Also, some interesting phase separation dynamics of the time-fractional Cahn–Hilliard equation has been investigated. [ABSTRACT FROM AUTHOR]

In this paper, we propose a high order numerical scheme to simulate the binary uid{surfactant system by combining the semi-implicit spectral deferred correction (SDC) method and the energy stable linear scheme, in the framework of discontinuous Galerkin (DG) methods. The linear scheme we develop in this paper is decoupled and unconditionally energy stable, which is based on the combination of the convex-concave splitting principle and the invariant energy quadratization approach. However, the scheme is only first order accurate with respect to time, and the SDC method can be employed to iteratively improve the temporal accuracy. Specially, the SDC scheme can be extremely accurate when coupled with an adaptive time stepping strategy. Our numerical scheme leads to a set of decoupled and linear algebraic equations; at each time step, we apply a multigrid solver to solve the equations efficiently. In particular, due to the local property of the DG methods, the resulting algebraic equations can be solved in an explicit way when coupled with the multigrid solver, which is an attractive advantage of the DG method. Various numerical experiments are performed to illustrate the high order accuracy, capability, and efficiency of the proposed methods when solving the binary uid{surfactant system. [ABSTRACT FROM AUTHOR]

In this paper, we propose stable second-order numerical schemes for the fractional Cahn–Hilliard and Allen–Cahn equations, which are based on the convex splitting in time and the Fourier spectral method in space. It is shown that the scheme for the fractional Cahn–Hilliard equation preserves mass. Meanwhile, the unique solvability and energy stability of the numerical schemes for the fractional Cahn–Hilliard and Allen–Cahn equations are proved. Finally, we present some numerical experiments to confirm the accuracy and the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

We analyze a first order in time Fourier pseudospectral scheme for Swift-Hohenberg equation. One major challenge for the higher order diffusion non-linear systems is how to ensure the unconditional energy stability and we propose an efficient scheme for the equation based on the convex splitting of the energy. Theoretically, the energy stability of the scheme is proved. Moreover, following the derived aliasing error estimate, the convergence analysis in the discrete l2-norm for the proposed scheme is given. [ABSTRACT FROM AUTHOR]

Abstract 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 schemes for two and three dimensions, which are derived from the one-dimensional compact stencil. The energy stability is completely proven for the proposed scheme based on the application of the compact method and well-known convex splitting methods. Detailed proofs of the mass conservation and unique solvability are also established. Numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be effciently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an ℓ2(0; T;H3h ) stability of the numerical scheme. To overcome the diffculty associated with the convection term ▿ · (ϕu), we perform an ℓ∞(0, T, H1h ) error estimate instead of the classical ℓ∞(0, T, ℓ2 ) one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and effciency of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second-order time accurate convex splitting scheme for the phase field crystal model. The temporal discretization is based on the second-order backward differentiation formula (BDF) and a convex splitting of the energy functional. The mass conservation, unconditionally unique solvability, unconditionally energy stability and convergence of the numerical scheme are proved rigorously. Mixed finite element method is employed to obtain the fully discrete scheme due to a sixth-order spatial derivative. Numerical experiments are presented to demonstrate the accuracy, mass conservation, energy stability and effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider mathematical modeling and numerical simulation of non-isothermal compressible multi-component diffuse-interface two-phase flows with realistic equations of state. A general model with the general reference velocity is derived rigorously through thermodynamical laws and Onsager’s reciprocal principle, and it is capable of characterizing compressibility and partial miscibility between multiple fluids. We prove a novel relation between the pressure, temperature and chemical potentials, which results in a new formulation of the momentum conservation equation indicating that the gradients of chemical potentials and temperature become the primary driving force of the fluid motion except for the external forces. A key challenge in numerical simulation is to develop entropy stable numerical schemes preserving the laws of thermodynamics. Based on the convex–concave splitting of Helmholtz free energy density with respect to molar densities and temperature, we propose an entropy stable numerical method, which solves the total energy balance equation directly, and thus, naturally satisfies the first law of thermodynamics. Unconditional entropy stability (the second law of thermodynamics) of the proposed method is proved by estimating the variations of Helmholtz free energy and kinetic energy with time steps. Numerical results validate the proposed method. [ABSTRACT FROM AUTHOR]

We develop a robust solver for a mixed finite element convex splitting scheme for the Cahn-Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild. [ABSTRACT FROM AUTHOR]

The Landau–Brazovskii model is a well-known Landau model for finding the complex phase structures in microphase-separating systems ranging from block copolymers to liquid crystals. It is critical to design efficient numerical schemes for the Landau–Brazovskii model with energy dissipation and mass conservation properties. Here, we propose a mass conservative and energy stable scheme by combining the spectral deferred correction (SDC) method with the convex splitting technique to solve the Landau–Brazovskii model efficiently. An adaptive correction strategy for the SDC method is implemented to reduce the cost time and preserve energy stability. Numerical experiments, including two- and three-dimensional periodic crystals in Landau–Brazovskii model, are presented to show the efficiency of the proposed numerical method. • We proposed a spectral deferred correction method combined with convex splitting. • We proved the mass conservative and energy stable properties. • The spectral deferred correction method can achieve arbitrary order accuracy. • An adaptive correction strategy can reduce time cost and preserve energy stability. [ABSTRACT FROM AUTHOR]

We propose a Convex Splitting Runge–Kutta (CSRK) scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner. The key feature of the scheme is a combination of a convex splitting method and a specially designed multi-stage two-additive Runge–Kutta method. Our methods are high order accurate in time and assure the gradient (energy) stability for any time step size. We provide detailed proof of the unconditional energy stability and present issues on the practical implementations. We demonstrate the accuracy and stability of the proposed methods using numerical experiments of the Cahn–Hilliard equation. [ABSTRACT FROM AUTHOR]

In this paper, we present the Convex Splitting Runge–Kutta (CSRK) methods which provide a simple unified framework to solve phase-field models such as the Allen–Cahn, Cahn–Hilliard, and phase-field crystal equations. The core idea of the CSRK methods is the combination of convex splitting methods and multi-stage implicit–explicit Runge–Kutta methods. Our CSRK methods are high-order accurate in time and we investigate the energy stability numerically. We present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third-order accuracy. [ABSTRACT FROM AUTHOR]

In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time Hm bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp(CT∊-m), where m is a positive integer, and -∊ is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on ∊-1 only in a polynomial order, rather than exponential. [ABSTRACT FROM AUTHOR]

This paper presents a componentwise convex splitting scheme for numerical simulation of multicomponent two-phase uid mixtures in a closed system at constant temperature, which is modeled by a diffuse interface model equipped with the Van der Waals and the Peng-Robinson equations of state (EoS). The Van der Waals EoS has a rigorous foundation in physics, while the Peng-Robinson EoS is more accurate for hydrocarbon mixtures. First, the phase field theory of thermodynamics and variational calculus are applied to a functional minimization problem of the total Helmholtz free energy. Mass conservation constraints are enforced through Lagrange multipliers. A system of chemical equilibrium equations is obtained which is a set of second-order elliptic equations with extremely strong nonlinear source terms. The steady state equations are transformed into a transient system as a numerical strategy on which the scheme is based. The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arising from the second-order derivative of homogeneous contribution of total Helmholtz free energy; it is also very efficient. This scheme is unconditionally componentwise energy stable and naturally results in unconditional stability for the Van der Waals model. For the Peng-Robinson EoS, it is unconditionally stable through introducing a physics-preserving correction term, which is analogous to the attractive term in the Van der Waals EoS. An efficient numerical algorithm is provided to compute the coefficient in the correction term. Finally, some numerical examples are illustrated to verify the theoretical results and efficiency of the established algorithms. The numerical results match well with laboratory data. [ABSTRACT FROM AUTHOR]

We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S. M. Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second order convergence in space. Instead of the (discrete) Ls∞ (0, T;Lh²)∩Ls²(0, T;Hh²) error estimate, which would represent the typical approach, we provide a discrete Ls∞ (0, T;Hh¹)∩Ls² (0, T;Hh³) error estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditional in the sense that the time step s is in no way constrained by the mesh spacing h. This is accomplished with the help of an Ls²(0, T;Hh³) bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality. [ABSTRACT FROM AUTHOR]

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term. [ABSTRACT FROM AUTHOR]

Phase field models are widely used to describe the two-phase system. The evolution of the phase field variables is usually driven by the gradient flow of a total free energy functional. The generalization of the approach to an N phase ( N ≥ 3 ) system requires some extra consistency conditions on the free energy functional in order for the model to give physically relevant results. A projection approach is proposed for the derivation of a consistent free energy functional for the three-phase Cahn–Hilliard equations. The system is then coupled with the Navier–Stokes equations to describe the three-phase flow on solid surfaces with moving contact line. An energy stable scheme is developed for the three-phase flow system. The discrete energy law of the numerical scheme is proved which ensures the stability of the scheme. We also show some numerical results for the dynamics of triple junctions and four phase contact lines. [ABSTRACT FROM AUTHOR]

This research addresses a sequential convex splitting method for numerical simulation of multicomponent two-phase fluids mixture in a single-pore at constant temperature, which is modeled by the gradient theory with Peng-Robinson equation of state. The gradient theory of thermodynamics and variational calculus are utilized to obtain a system of chemical equilibrium equations which are transformed into a transient system as a numerical strategy on which the numerical scheme is based. The proposed numerical algorithm avoids computing Hessian matrix arising from the second-order derivative of homogeneous contribution of free energy; it is also quite robust. This scheme is proved to be unconditionally component-wise energy stable. The Raviart-Thomas mixed finite element method is applied to spatial discretization. [ABSTRACT FROM AUTHOR]

In this paper, we present a local discontinuous Galerkin (LDG) method and two unconditionally energy stable schemes for the phase field crystal (PFC) equation. The semidiscrete energy stability of the LDG method is proved first. The PFC equation is a sixth order nonlinear partial differential equation (PDE), which leads to the severe time step restriction (Δt = O(Δx6)) of explicit time discretization methods to maintain stability. Due to this, we introduce semi-implicit first order and second order time discretization methods which are based on the convex splitting principle of a discrete energy and prove the corresponding unconditional energy stabilities. To improve the temporal accuracy, the spectral deferred correction (SDC) method and a high order semi-implicit Runge-Kutta method combining with the first-order convex splitting method are adopted for the PFC equation with constant and degenerate mobility, respectively. The equations at the implicit time level are nonlinear and we employ an efficient nonlinear multigrid solver to solve the equations. In particular, we show the multigrid solver has optimal complexity numerically. Numerical results are also given to illustrate that the combination of the LDG method for spatial approximation, SDC, and high order semi-implicit time marching methods with the multigrid solver provides an efficient and practical approach when solving the PFC equation. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient and unconditionally energy stable fully-discrete local discontinuous Galerkin (LDG) method for approximating the Cahn–Hilliard–Brinkman (CHB) system, which is comprised of a Cahn–Hilliard type equation and a generalized Brinkman equation modeling fluid flow. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction ( Δ t = O ( Δ x 4 ) ) of explicit time discretization methods for stability, we introduce a semi-implicit scheme which consists of the implicit Euler method combined with a convex splitting of the discrete Cahn–Hilliard energy strategy for the temporal discretization. The unconditional energy stability of this fully-discrete convex splitting scheme is also proved. Obviously, the fully-discrete equations at the implicit time level are nonlinear, and to enhance the efficiency of the proposed approach, the nonlinear Full Approximation Scheme (FAS) multigrid method has been employed to solve this system of algebraic equations. We also show the nearly optimal complexity numerically. Numerical experiments based on the overall solution method of combining the proposed LDG method, convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the CHB system. [ABSTRACT FROM AUTHOR]

In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a nonsteady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in L∞ (0, T;L∞) and the discrete chemical potential is bounded in L∞(0, T; L²), for any time and space step sizes, in two and three dimensions, and for any finite final time T. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. [ABSTRACT FROM AUTHOR]

Abstract: We present an unconditionally energy stable finite-difference scheme for the binary fluid-surfactant system. The proposed method is based on the convex splitting of the energy functional with two variables. Here are two distinct features: (i) the convex splitting energy method is applied to energy functional with two variables, and (ii) the stability issue is related to the decay of the corresponding energy. The full discrete scheme leads to a decoupled system including a linear sub-system and a nonlinear sub-system. Algebraic multigrid and Newton-multigrid methods are adopted to solve the linear and nonlinear systems, respectively. Numerical experiments are shown to verify the stability of such a scheme. [Copyright &y& Elsevier]

Zone melting is an efficient purification method that is widely used in the manufacture of semiconductors. Recently, it has been used in the purification of industrial phosphorus [Z. Zhang et al., Separation Purification Tech., 98 (2012), pp. 249-254]. It is a process consisting of both the melting-solidification process and the solute diffusion process. There is a great demand for a quantitative understanding of the underlying mechanism. In this paper, we start from a two-component, two-phase physical system and propose a variational phase-field model to model the zone melting process. The model consisting of the Allen-Cahn equation, the heat equation, and the solute diffusion equation is solved numerically. To overcome the stiffness caused by the nonlinear term in the Allen-Cahn equation, we propose a semi-implicit scheme based on a convex splitting of the free energy. By a proper splitting of the free energy functional, we show that our semi-implicit scheme is unconditionally energy stable. Numerical experiments are carried out to verify the convergence and stability of our numerical scheme. Furthermore, we present our numerical simulations for the zone melting process and numerically show the feasibility of our model by comparisons with experiments. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction ( ) of explicit time discretization methods for stability, we introduce a semi-implicit time integration scheme which is based on a convex splitting of the discrete Cahn–Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver. [Copyright &y& Elsevier]

In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an effcient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver. [ABSTRACT FROM AUTHOR]

In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second order accurate convex splitting scheme for the modified phase field crystal equation, a generalized damped wave equation for which the usual phase field crystal equation is a special degenerate case. The fully discrete, fully second order finite difference scheme in question was derived in a recent work [A. Baskaran et al., J. Comput. Phys., 250 (2013), pp. 270-292]. An introduction of a new variable ψ, corresponding to the temporal derivative of the phase variable ø, could bring an accuracy reduction in the formal consistency estimate, because of the hyperbolic nature of the equation. A higher order consistency analysis by an asymptotic expansion is performed to overcome this difficulty. In turn, second order convergence in both time and space is established in a discrete L∞ (0, T; H³) norm. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a semi-implicit finite element method is presented for the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn–Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier–Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, an efficient numerical scheme is designed for a phase field model for the moving contact line problem, which consists of a coupled system of the Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition . The nonlinear version of the scheme is semi-implicit in time and is based on a convex splitting of the Cahn–Hilliard free energy (including the boundary energy) together with a projection method for the Navier–Stokes equations. We show, under certain conditions, the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Numerical tests are carried out to verify the accuracy and stability of the scheme. The behavior of the solution near the contact line is examined. It is verified that, when the interface intersects with the boundary, the consistent splitting scheme for the Navier Stokes equations has the better accuracy for pressure. [Copyright &y& Elsevier]

We propose a novel multiphase segmentation model built upon the celebrated phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term that complements it. The proposed sine-sinc model outputs a single multiphase distribution from which each individual segment or phase can be easily extracted. Theoretical analysis is developed for the Γ-convergence behavior of the proposed model and the existence of its minimizers. Since the model is not quadratic nor convex, for computation we adopted the convex-concave procedure (CCCP) that has been developed in the literatures of both computational nonlinear PDEs and neural computation. Numerical details and experiments on both synthetic and natural images are presented. [ABSTRACT FROM AUTHOR]

In this paper, a novel finite element method for solving a modified Cahn–Hilliard–Hele–Shaw system is proposed. The time discretization is based on the convex splitting of the energy functional in the modified Cahn–Hilliard equation, i.e., the high-order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly, respectively. Designing in this way leads to solving a linear system at each time step, which is much efficient compared to solving a nonlinear system resulting from most existing schemes. The proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable. Numerical results are presented to support our theoretical analysis. [ABSTRACT FROM AUTHOR]