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We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the "zero-energy-contribution" feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension. [ABSTRACT FROM AUTHOR]

We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier–Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn–Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh–Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically. [ABSTRACT FROM AUTHOR]

In this paper, we consider time marching numerical schemes to solve a two-mode phase field crystal model with face-centered-cubic ordering. We propose two type, linear, and unconditionally energy stable schemes by combining the recently developed IEQ and SAV approaches with the stabilization technique, where several extra stabilizing terms are added to enhance the stability and keep the required accuracy while using large time steps. We further prove the unconditional energy stabilities of the developed schemes rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, semi-implicit, and convex-splitting schemes for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the schemes numerically. • We have developed two type efficient schemes for solving the two-mode PFC-FCC model. • The proposed schemes are linear, second-order accurate, provably unconditionally energy stable, and easy to implement. • The schemes combine the /SAV approaches with the stabilization technique allowing for large time steps in computations. • We prove the unconditional energy stabilities rigorously and implement numerous benchmark 2D and 3D numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations for solving the anisotropic Cahn–Hilliard model. We combine the Scalar Auxiliary Variable (SAV) approach with the stabilization technique to arrive at a novel Stabilized-SAV approach, where three linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient, are added to enhance the stability while keeping the required accuracy. The schemes are very easy-to-implement and fast in the sense that all nonlinear terms are treated in a semi-explicit way, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy. • We propose a novel Stabilized-SAV approach for solving the anisotropic Cahn–Hilliard model. • The scheme is linear, second-order accurate, provably unconditionally energy stable, and non-iterative. • Three crucial linear stabilization terms are added to remove oscillations caused by anisotropy. • One only need to solve three decoupled linear equations at each time step. • We further prove the unconditional energy stabilities rigorously and present numerous simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations for the viscous Cahn–Hilliard equation with hyperbolic relaxation. This type of equations processes energy-dissipative structure. The main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms, allowing large time-marching step, while preserving the energy stability, i.e. the energy dissipative structure at the time-discrete level. We resolve this issue by developing two second-order time-marching schemes using the recently developed “Invariant Energy Quadratization” approach where all nonlinear terms are discretized semi-explicitly. In each time step, one only needs to solve a symmetric positive definite (SPD) linear system. All the proposed schemes are rigorously proven to be unconditionally energy stable, and the second-order convergence in time has been verified by time step refinement tests numerically. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

How to develop efficient numerical schemes while preserving energy stability at the discrete level is challenging for the three-component Cahn-Hilliard phase-field model. In this paper, we develop a set of first- and second-order temporal approximation schemes based on a novel 'Invariant Energy Quadratization' approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to well-posed linear systems with a linear symmetric, positive definite at each time step. We prove that the developed schemes are unconditionally energy stable and present various 2D and 3D numerical simulations to demonstrate the stability and the accuracy of the schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of a binary fluid–surfactant phase field model, in which the free energy contains a nonlinear coupling entropy, a Ginzburg–Landau double well potential, and a logarithmic Flory–Huggins potential. The resulting system consists of two nonlinearly coupled Cahn–Hilliard type equations. We develop a first and a second order time stepping schemes for this system using the “Invariant Energy Quadratization” approach; in particular, the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi-explicitly. Both schemes are linear and lead to symmetric positive definite systems in space at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable in the discrete sense. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes. [ABSTRACT FROM AUTHOR]

We consider the numerical approximations of a two-phase hydrodynamics coupled phase-field model that incorporates the variable densities, viscosities and moving contact line boundary conditions. The model is a nonlinear, coupled system that consists of incompressible Navier–Stokes equations with the generalized Navier boundary condition, and the Cahn–Hilliard equations with moving contact line boundary conditions. By some subtle explicit–implicit treatments to nonlinear terms, we develop two efficient, unconditionally energy stable numerical schemes, in particular, a linear decoupled energy stable scheme for the system with static contact line condition, and a nonlinear energy stable scheme for the system with dynamic contact line condition. An efficient spectral-Galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes. Various numerical results show that the proposed schemes are efficient and accurate. [ABSTRACT FROM AUTHOR]

In this paper, we develop a series of linear, unconditionally energy stable numerical schemes for solving the classical phase field crystal model. The temporal discretizations are based on the first order Euler method, the second order backward differentiation formulas (BDF2) and the second order Crank–Nicolson method, respectively. The schemes lead to linear elliptic equations to be solved at each time step, and the induced linear systems are symmetric positive definite. We prove that all three schemes are unconditionally energy stable rigorously. Various classical numerical experiments in 2D and 3D are performed to validate the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends. The governing system is derived from the energetic variational approach of a total free energy, that consists of a nonlinear logarithmic Flory–Huggins potential, and a gradient entropy with a concentration-dependent de-Gennes type coefficient. The main challenging issue to solve this kind of models numerically is about the time marching problem, i.e., how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level. We solve this issue in this paper, by developing the first and second order temporal approximation schemes based on the “Invariant Energy Quadratization” method, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step. The unconditional energy stabilities are further proved. Various numerical simulations of 2D and 3D are presented to demonstrate the stability and the accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically. [ABSTRACT FROM AUTHOR]

• The proposed scheme is second order accurate, unconditionally energy stable, efficient and easy to implement. • All nonlinear terms can be treated semi-explicitly, and one only needs to solve a linear system at each time step. • We verify numerically that our schemes are of second order accuracy in time and present ample 2D and 3D numerical tests. In this paper, we consider numerical approximations for solving the hydrodynamically coupled three components Cahn-Hilliard phase-field model. By combining the Invariant Energy Quadratization approach with the stabilization technique, and the projection method for the Navier-Stokes equations, we obtain a linear, second-order, and unconditionally energy stable time marching scheme. We present rigorous proofs for the well-posedness of the obtained linear system and the unconditional energy stability. Various 2D and 3D numerical simulations are implemented to demonstrate the stability and accuracy of the scheme thereafter. [ABSTRACT FROM AUTHOR]

In this paper, we first formulate a hydrodynamically coupled phase-field model for triblock copolymer melts and then develop a time-marching scheme for it. The numerical scheme is based on a combination of the projection method and the IEQ (Invariant Energy Quadratization) method to achieve linear and second-order accuracy in time. To further simplify its implementation, we introduce a new nonlocal variable and its time evolution equation, so we end up with a fully decoupled scheme. Under the premise of satisfying strict unconditional energy stability, the scheme only needs to solve several completely independent elliptic equations with constant coefficients at each time step. We provide the rigorous energy stability proof, the detailed implementation of the proposed scheme, and further perform some numerical simulations of triblock polymer materials under applied shear flow in 2D and 3D to demonstrate the effectiveness of the developed numerical scheme. [ABSTRACT FROM AUTHOR]

For the highly coupled and nonlinear Cahn–Hilliard phase-field model of three-phase incompressible flow, how to establish a fully-decoupled numerical scheme with second-order time accuracy has always been a very difficult and unsolved problem. In this paper, we propose a novel decoupling method, which only needs to solve several decoupling linear elliptic equations with constant coefficients at each time step to obtain a numerical solution with second-order time accuracy. The key idea is to introduce two nonlocal auxiliary variables into the system, one of which is used to linearize the nonlinear potential, and the other is used to introduce an ordinary differential equation to deal with the nonlinear coupling terms with " zero-energy-contribution " characteristics. We strictly prove the solvability and unconditional energy stability of the scheme, and conduct numerical simulations in 2D and 3D to show the accuracy and stability of the scheme numerically. To the best of the author's knowledge, the method developed in this paper is the first second-order fully-decoupled scheme for the hydrodynamics coupled phase-field model. • A novel full decoupling method is proposed for the Navier-Stokes coupled three-phase model. • The key idea is to introduce an ODE for the "zero-energy-contribution" nonlinear terms. • Several decoupled linear elliptic equations are solved at each time step. • The first second-order fully-decoupled scheme for the flow-coupled three-phase model. • Unconditional energy stability and solvability are proved rigorously. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations for the magnetic-coupled phase-field-crystal model for ferromagnetic solid materials. The governing PDE system consists of two coupled and highly nonlinear equations in which one is the Cahn–Hilliard equation for the density of atoms, and the other is the Allen–Cahn equation for the magnetization field. To solve it, we construct an unconditionally energy stable scheme with the second-order accuracy in time based on the recently developed stabilized-SAV approach. The energy stability of the scheme is proved, and the stability and accuracy are then demonstrated numerically by implementing various numerical examples in 2D and 3D, including the crystal growth and phase separations for both of the magnetic-free and magnetic-coupled cases. • The proposed scheme is second-order time-accurate, unconditionally energy stable. • One only needs to solve a linear and decoupled system at each time step. • We present ample numerical results from various 2D and 3D numerical tests. • The simulations are qualitatively consistent with the benchmark results. [ABSTRACT FROM AUTHOR]

In this paper, we expand the classical phase-field model for diblock copolymers (BCPs) to the magnetic-coupled case, where the free energy of the system is modeled as the combination of the nonlocal Ohta–Kawasaki free energy and the classical Ginzburg–Landau formalism for ferromagnetic ordering. To solve the model, we develop an efficient, second-order time marching scheme by adopting the recently developed stabilized-SAV (Scalar Auxiliary Variable) approach. The energy stability of the scheme is proved and several numerical examples are presented to show that the morphology patterns are strongly influenced by the magnetic field. [ABSTRACT FROM AUTHOR]

In this article, we construct a fully discrete finite element numerical scheme with linearity, decoupling, unconditional energy stability, and second-order time accuracy for the Navier-Stokes coupled phase-field crystal model. The key idea is based on the design of several auxiliary ODEs, combined with the finite element method for spatial discretization, the projection method for the Navier-Stokes equations, and the IEQ type method for the nonlinear potentials. At each time step, by using the nonlocal splitting technique, only a few decoupled elliptic constant-coefficient equations need to be solved. We further prove that the developed scheme is unconditionally energy stable, and the detailed implementation process is given as well. To verify the effectiveness of the developed scheme, various numerical experiments are carried out, including the crystal growth under the action of shear flow and the sedimentation process of a large number of particles. [ABSTRACT FROM AUTHOR]

We consider in this paper numerical approximations for the hydro-dynamically coupled phase-field model for an immiscible binary mixture of passive/active nematic Liquid Crystals (LCs) and viscous fluids. The passive model is a highly coupled nonlinear system that consists of the incompressible Navier-Stokes equations, the Cahn-Hilliard equations, and the constitutive equation for the polarization field. The active model can be viewed as an assembly of a passive part and a phenomenological active part. We then construct an efficient and linear time marching scheme to solve the passive model and further extend it to solve the active model. Various numerical simulations are presented to demonstrate the stability and accuracy of the proposed scheme in simulating some simulations in 2D and 3D, including the drop deformation dynamics and spinodal decompositions. [ABSTRACT FROM AUTHOR]

• We propose a novel stabilized SAV approach for solving the phase field model for diblock copolymers. • The proposed schemes are second-order accurate, provably unconditionally energy stable, non-iterative. • The added linear stabilization term is shown to be crucial enhance the stability while keeping the required accuracy. • One only need to solve three decoupled linear equations at each time step. • We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D simulations. In this paper, we consider numerical approximations to solving the Cahn-Hilliard phase field model for diblock copolymers. We combine the recently developed SAV (scalar auxiliary variable) approach with the stabilization technique to arrive at a novel stabilized-SAV method, where a crucial linear stabilization term is added to enhancing the stability and keeping the required accuracy while using the large time steps. The scheme is very easy-to-implement and fast in the sense that one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, convex-splitting, and non-stabilized SAV scheme for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically. [ABSTRACT FROM AUTHOR]

In this work, we construct a fully-discrete Spectral-Galerkin scheme for the anisotropic Cahn–Hilliard model. The scheme is based on the combination of a novel so-called explicit-Invariant Energy Quadratization method for time discretization and the Spectral-Galerkin approach for spatial discretization. The designed scheme only needs to solve several independent linear equations with constant coefficients at each time step, which demonstrates the high computational efficiency. The introduction of two auxiliary variables and the design of their associated auxiliary ODEs play a vital role in obtaining the linear structure and unconditional energy stability and thus avoiding the computation of a variable-coefficient system. The unconditional energy stability of the scheme is further rigorously proved, and the implementation process is given in detail. Through several 2D and 3D numerical simulations, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm. • We propose a full discrete spectral-Galerkin approach for solving the anisotropic Cahn–Hilliard model. • The proposed schemes are second-order accurate, provably unconditionally energy stable, and easy to implement. • All nonlinear terms are treated semi-explicitly, and only need to solve several fully decoupled linear equations at each time step. [ABSTRACT FROM AUTHOR]

This paper considers the numerical approximation of a two-component phase-field crystal model consisting of two coupled nonlinear Cahn–Hilliard equations of binary alloys. We develop a highly efficient time-marching scheme with second-order accuracy based on the SAV approach, in which two additional stabilization terms are introduced to improve stability, thus allowing large time steps. Unconditional energy stability is then proved strictly. By simulating a large number of numerical simulations in 2D and 3D, including binary crystal growth and phase separation with vacancies, we then verify the stability and accuracy of the scheme. • A first energy stable, second-order scheme for the phase-field model of binary alloys. • One only needs to solve a linear and decoupled system at each time step. • Numerous 2D and 3D numerical results for binary alloys are given. [ABSTRACT FROM AUTHOR]

We develop an accurate and robust numerical scheme for solving the incompressible hydrodynamically coupled Cahn–Hilliard system of the two-phase fluid flow system on complex surfaces. Our algorithm leverages a number of efficient techniques, including the subdivision-based isogeometric analysis (IGA) method for spatial discretization, the explicit Invariant Energy Quadratization (EIEQ) method for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling, and the projection method for the Navier–Stokes equation to facilitate fully decoupled type implementations. The integration of these methodologies results in a fully discrete scheme with desired properties such as linearity, second-order temporal accuracy, full decoupling, and unconditional energy stability. The implementation of the scheme is straightforward, requiring the solution of a few elliptic equations with constant coefficients at each time step. The rigorous stability proof of unconditional energy stability and the implementation procedure are given in detail. Numerous numerical simulations on complex curved surfaces are carried out to verify the effectiveness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces. [ABSTRACT FROM AUTHOR]

We present an efficient fully discrete algorithm for solving the Allen–Cahn and Cahn–Hilliard equations on complex curved surfaces. The spatial discretization employs the recently developed IGA (isogeometric analysis) framework, where we adopt the strategy of Loop subdivision with the superior adaptability of any topological structure, and the basis functions are quartic box-splines used to define the subdivided surface. The time discretization is based on the so-called EIEQ (explicit-Invariant Energy Quadratization) approach, which applies multiple newly defined variables to linearize the nonlinear potential and realize the efficient decoupled type computation. The combination of these two methods can help us to gain a linear, second-order time accurate scheme with the property of unconditional energy stability, whose rigorous proof is given. We also develop a nonlocal splitting technique such that we only need to solve decoupled, constant-coefficient elliptic equations at each time step. Finally, the effectiveness of the developed numerical algorithm is verified by various numerical experiments on the complex benchmark curved surfaces such as bunny, splayed, and head surfaces. [ABSTRACT FROM AUTHOR]

We construct in this paper a fully-decoupled and second-order accurate numerical scheme for solving the Cahn–Hilliard–Navier–Stokes phase-field model of two-phase incompressible flows. A full decoupling method is used by introducing several nonlocal variables and their ordinary differential equation to deal with the nonlinear and coupling terms. By combining with some effective methods to handle the Navier–Stokes equation, we obtain an efficient and easy-to-implement numerical scheme in which one only needs to solve several fully-decoupled linear elliptic equations with constant coefficients at each time step. We further prove the unconditional energy stability and solvability rigorously, and present various numerical simulations in 2D and 3D to demonstrate the efficiency and stability of the proposed scheme, numerically. • The proposed numerical scheme is linear, second-order accurate in time. • The scheme is also decoupled, and unconditionally energy stable. • One only needs to solve several decoupled, linear elliptic equations at each time step. • The unconditional energy stability is rigorously proven. • We present various numerical simulations to demonstrate the efficiency and stabilty. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations of the volume-conserved phase-field elastic bending energy model for lipid vesicles where a nonlocal term is added to the model such that the total volume can be conserved precisely. We further develop two linear and unconditionally energy stable schemes by combining the recently developed IEQ and SAV approaches with the stabilization technique, where several extra stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with some other type numerical schemes for simulating numerous benchmarking numerical examples in 2D and 3D, we demonstrate the robustness of the new nonlocal volume-conserved model, as well as the stability and the accuracy of the developed schemes, numerically. • We develop a new volume-conserved elastic bending energy model for lipid vesicles. • The two proposed schemes are linear, second-order accurate. • The schemes are provably unconditionally energy stable, and easy to implement. • The effectiveness of the model and the schemes are demonstrated numerically. [ABSTRACT FROM AUTHOR]

• We propose a novel stabilized SAV approach for solving the phase field model for tri-block copolymers. • The proposed schemes are second-order accurate, provably unconditionally energy stable, non-iterative. • The added linear stabilization term is shown to be crucial enhance the stability while keeping the required accuracy. • One only need to solve three decoupled linear equations at each time step. • We further prove the unconditional energy stabilities rigorously and present numerious 2D and 3D simulations. We establish a ternary Cahn-Hilliard type Nakazawa-Ohta phase-field model for the triblock copolymer, and study its numerical approximation. The model is a highly coupled nonlinear system, consisting of two Cahn-Hilliard equations and two nonlocal equations. We solve the model by constructing a second-order accurate, time-marching scheme via the Scalar Auxiliary Variable (SAV) approach combined with the stabilization technique. At every time step, the scheme is composed of several decoupled type bi-Laplace equations, which makes it the first linear and fully-decoupled scheme. We further prove the unconditional energy stability rigorously and perform numerous numerical simulations in 2D and 3D to illustrate it numerically. [ABSTRACT FROM AUTHOR]