Invariant Energy Quadratization Research Articles

Linear relaxation method with regularized energy reformulation for phase field models

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models.

Two novel numerical methods for gradient flows: generalizations of the Invariant Energy Quadratization method

Two novel numerical methods for gradient flows: generalizations of the Invariant Energy Quadratization method

Unconditionally energy stable invariant energy quadratization finite element methods for Phase-Field Crystal equation and Swift–Hohenberg equation

In this paper, we design, analyze and numerically validate linearly first- and second-order unconditionally energy stable numerical methods for solving the Phase-Field Crystal equation and Swift–Hohenberg equation which describe a multitude of processes involving spatiotemporal pattern formation. The properties of well-posedness of solution and the decrease of the total energy for the fully discretized schemes are established. Numerical examples are presented to confirm the accuracy, efficiency, and stability of the proposed method.

The subdivision-based IGA-EIEQ numerical scheme for the Navier–Stokes equations coupled with Cahn–Hilliard phase-field model of two-phase incompressible flow on complex curved surfaces

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.

Role of shift constant in Energy Shifted SAV for Hamiltonian Systems

In recent years, considerable work has been devoted to the design of energy-stable numerical methods, a class of geometrical integration technique in which the discrete energy or pseudo-energy remains conserved. This property finds practical applications in systems for which the energy is bounded from below since the growth of the solutions is then itself bounded, yielding a form of stability. Such a property may be further exploited to transform the energy overall into a quadratic form, representing the core of recent numerical techniques such as the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. These methods have been applied to a large class of problems due to their remarkable efficiency. Yet, several aspects of such techniques have seen little investigation. In this work, the role of the “shift” constant in the expression of the potential energy in Hamiltonian systems is investigated. This is a positive gauge constant that may be used to augment the expression of the energy. Since Hamilton’s equations are given in terms of gradients of the Hamiltonian, such a constant has no influence on the resulting dynamics in the continuous system. However, empirical evidence has suggested that the convergence properties of the associated SAV approaches are affected by the magnitude of the shift factor. In this work, the behaviour of the relative global error of SAV schemes in the cases of the harmonic and Duffing oscillators is numerically investigated. The results reveal an optimal shift factor that increases the convergence rate. Using the optimal shift factor, the proposed method displays variable order accuracy ranging from second to twelfth order.

A ternary mixture model with dynamic boundary conditions.

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system

We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier–Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.

Error analysis of second-order IEQ numerical schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation

In this article, we derive error estimates for two second-order numerical schemes for solving the viscous Cahn-Hilliard equation with hyperbolic relaxation, one based on the second-order Crank-Nicolson time marching method and the other on the backward differentiation formula. In both schemes, the nonlinear potential is discretized by the Invariant Energy Quadratization (IEQ) method, which employs an auxiliary variable to produce a linear and unconditional energy-stable structure. After assuming some reasonable boundedness and continuity conditions for the nonlinear function, the optimal error estimate is rigorously derived using mathematical induction. Finally, several numerical experiments are carried out to verify the theoretical predictions of the convergence rate and energy stability of the algorithms.

Error estimates with low-order polynomial dependence for the fully-discrete finite element invariant energy quadratization scheme of the Allen–Cahn equation

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the [Formula: see text]-error bound and the [Formula: see text]-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of [Formula: see text] instead of [Formula: see text] for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.

An efficient linearly implicit and energy‐conservative scheme for two dimensional Klein–Gordon–Schrödinger equations

AbstractThe Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy‐preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the imaginary functions of Klein–Gordon–Schrödinger equations, and transform the original system into its real formulation. Based on the invariant energy quadratization approach, an equivalent system is deduced by introducing a Lagrange multiplier. Then the efficient and unconditionally stable scheme is designed to discretize the deduced equivalent system. A numerical analysis of the proposed scheme is presented to illustrate its uniquely solvability and convergence. Numerical examples are provided to validate accuracy, energy and mass conservation laws, and stability of our proposed method.

Reformulated Weak Formulation and Efficient Fully Discrete Finite Element Method for a Two-Phase Ferrohydrodynamics Shliomis Model

The two-phase ferrohydrodynamics model consisting of Cahn–Hilliard equations, Navier–Stokes equations, magnetization equations, and magnetostatic equations is a highly nonlinear, coupled, and saddle point structural multiphysics PDE system. While various works exist to develop fully decoupled, linear, second-order in time, and unconditionally energy stable methods for simpler gradient flow models, existing ideas may not be applicable to this complex model or may be only applicable to part of this model. Therefore, significant challenges remain in developing corresponding efficient fully discrete numerical algorithms with the four above-mentioned desired properties, which will be addressed in this paper by dynamically incorporating several key ideas, including a reformulated weak formulation with special test functions for overcoming two major difficulties caused by the magnetostatic equation, the decoupling technique based on the “zero-energy-contribution” property to handle the coupled nonlinear terms, the second-order projection method for the Navier–Stokes equations, and the invariant energy quadratization (IEQ) method for the time marching. Among all these ideas, the reformulated weak formulation serves as a key bridge between the existing techniques and the challenges of the target model, with all of the four desired properties kept in mind. We demonstrate the well-posedness of the proposed scheme and rigorously show that the scheme is unconditionally energy stable. Extensive numerical simulations, including accuracy/stability tests, and several 2D/3D benchmark Rosensweig instability problems for “spiking” phenomena of ferrofluids are performed to verify the effectiveness of the scheme.

Energy stable schemes with second order temporal accuracy and decoupled structure for diffuse interface model of two-phase magnetohydrodynamics

In this paper, two second-order accurate in time, linear, decoupled and unconditionally energy stable schemes for the diffuse interface model of two-phase magnetohydrodynamics (MHD) are presented. The given schemes combine invariant energy quadratization (IEQ) method for the phase-field equations, second order type pressure projection method for the double saddle points MHD system, and some subtle implicit–explicit treatments for nonlinear coupled terms. The strongly coupled, highly nonlinear and double saddle points type two-phase MHD model is spilt into several smaller elliptic type problems, which can be solved efficiently. Rigorous proofs of the unconditional energy stabilities are provided in both temporal discretization and full discretization with finite element method in space approximation. Finally, various numerical results are presented to verify the accuracy and stability of the proposed schemes.

A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation

<abstract><p>In this paper, we present a new method to solve the fractional nonlinear Schrödinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.</p></abstract>

Conservative finite difference methods for the Boussinesq paradigm equation

In this paper, we concentrate on developing and analyzing two types of finite difference schemes with energy conservation properties for solving the Boussinesq paradigm equation. Firstly, by introducing the auxiliary potential function ∂u∂t=Δv and lim|x|→∞v=0, the BPE is reformulated as an equivalent system of coupled equations. Then a class of efficient difference schemes are proposed for solving the resulting system, where one proposed scheme is a two-level nonlinear difference scheme and the other is a three-level linearized difference scheme using the invariant energy quadratization technique. Subsequently, we present the theoretical analysis of the proposed energy conservative finite difference schemes, which involves the discrete energy conservation properties, unique solvability and optimal error estimates. By using the discrete energy method, it is proven that the proposed schemes can achieve the optimal convergence rates of O(Δt2+hx2+hy2) in discrete L2-, H1- and L∞-norms. At last, numerical experiments illustrate the physical behaviors and efficiency of the proposed schemes.

On the convergence of an unconditionally stable numerical scheme for the Q-tensor flow based on the invariant quadratization method

We present a convergence analysis towards a numerical scheme designed for Q-tensor flows of nematic liquid crystals. This scheme is based on the Invariant Energy Quadratization method, which introduces an auxiliary variable to replace the original energy functional. In this work, we show that given an initial value with H2 regularity, we can obtain a uniform H2 estimate on the numerical approximations for the Q-tensor flows and then deduce the convergence to a strong solution of the parabolic-type Q-tensor equation. We also show that the limit of the auxiliary variable is equivalent to the original energy functional term in the strong sense.

Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation

Over the years, energy-dissipation-preserving (EDP) numerical methods (EDP-NMs) for the nonlinear system of energy dissipation have attracted considerable attention because they are good at long-term computations. More recently, the invariant energy quadratization methods (IEQMs) have been proposed by Xiaofeng Yang and his collaborators to construct linearized EDP-NMs for the nonlinear system of energy dissipation, such as gradient flows. However, the system of the linear algebraic equations corresponding to the obtained EDP-NMs involves the variable coefficient matrices in general, thus resulting in complex computations. Besides, most of efficient alternating direction implicit (ADI) methods are not able to preserve the structures of continuous problem, such as, energy dissipation, energy conservation. To overcome these shortcomings, taking two-dimensional (2D) nonlinear Allen-Cahn equation for example, two energy-dissipation-preserving ADI finite difference methods (FDMs) have been developed by using IEQMs. Numerical analyses of them, such as, the discrete energy-dissipation laws, error estimations and stabilities, have been derived in detail. Numerical results support the theoretical analysis, and confirm the efficiency and accuracy of the proposed algorithms.

A second-order unconditionally stable method for the anisotropic dendritic crystal growth model with an orientation-field

In this article, we develop a linear, unconditionally energy stable computational scheme for solving the dendritic crystal growth model with the orientational field. We apply the phase field model to describe the evolution of crystal with rotation. The model, which couples the heat equation and anisotropic Allen–Cahn type equation, is a complicated nonlinear system. The time integration is based on the second-order Crank–Nicolson method. The anisotropic coefficient is treated by using the invariant energy quadratization. We mathematically prove that the proposed method is unconditionally energy stable. The second-order spatial and temporal accuracy will be preserved for the numerical approximation. Various computational tests are performed to show the accuracy, stability, and efficiency of the proposed scheme.

Fully discrete Spectral-Galerkin scheme for a ternary Allen–Cahn type mass-conserved Nakazawa–Ohta phase-field model for triblock copolymers

In this paper, we first use the Allen–Cahn relaxation dynamics to simulate the phase-field model of triblock copolymers, where the mass-conserving property is achieved by adding two additional nonlocal Lagrange multipliers. The numerical approximations of the obtained model are further considered, and an efficient fully-discrete numerical scheme based on the Spectral-Galerkin approach for spatial discretization and the so-called explicit-IEQ (invariant energy quadratization) method for time marching is developed. Two auxiliary variables are introduced to reformulate the governing system into an equivalent form, which facilitates the design of numerical algorithms by simply discretizing nonlinear terms explicitly. The resulting scheme is not only second-order accurate in time and spectrally accurate in space, but also easy to implement, i.e., the scheme can be carried out by only solving multiple independently decoupled, linear, and constant-coefficient elliptic equations at each time step. We also prove the unconditional energy stability of the developed scheme and demonstrate its effectiveness by implementing several benchmarking numerical examples in 2D and 3D.

Efficient linear, fully-decoupled and energy stable numerical scheme for a variable density and viscosity, volume-conserved, hydrodynamically coupled phase-field elastic bending energy model of lipid vesicles

We first establish a variable density and viscosity, volume-conserved, hydrodynamically coupled phase-field variable elastic bending energy model for lipid vesicles, and then construct an efficient time-discrete scheme for solving it. The numerical scheme combines the penalty method for solving the Navier–Stokes equation, the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, and the operator-splitting method. Hence it is not only fully decoupled but also owns some desired properties of linearity and unconditional energy stability. The feature of full decoupling is achieved by introducing some auxiliary variables and designing additional ordinary differential equations which are used for discretizing the coupled and nonlinear terms. The solvability and the unconditional energy stability of the numerical scheme have been further rigorously and numerically proven. Several numerical examples are carried out on the sedimentation process of the vesicle cell to show the effectiveness of the model and scheme.

Numerical approximations of flow coupled binary phase field crystal system: Fully discrete finite element scheme with second-order temporal accuracy and decoupling structure

In this article, we first establish a new flow-coupled binary phase-field crystal model and prove its energy law. Then by using some newly introduced variables, we reformulate this three-phase model into an equivalent form, which makes it possible to construct a fully discrete linearized decoupling scheme with unconditional energy stability and second-order time accuracy to solve this model for the first time. The energy law of the reformulated model is also proved. Then we incorporate the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, the projection method for the Navier-Stokes equations, the Crank-Nicolson method for time marching, and the finite element method for spatial discretization together to develop the fully discrete scheme for the reformulated and equivalent system. By using the nonlocal splitting technique, at each time step, only a few decoupled constant-coefficient elliptic equations are required to be solved, even though the original and reformulated models are much more complicated in the form. The developed algorithm is further proved to be unconditionally energy stable, and a detailed implementation process is also provided. Various numerical experiments in 2D and 3D are carried out to verify the effectiveness of the developed scheme, including the binary crystal growth under the action of shear flow and the sedimentation process of many binary particles.