Invariant Energy Quadratization Research Articles

An accurate and efficient algorithm for the time-fractional molecular beam epitaxy model with slope selection

In this paper, we introduce a time-fractional molecular beam epitaxy (MBE) model with slope selection using the classical Caputo fractional derivative of order α (0<α<1), which is shown to possess an energy dissipation law. Then we develop its efficient and accurate, full discrete, linear numerical approximation. Utilizing the classical L1 numerical treatment for the time-fractional derivative and the invariant energy quadratization strategy, the resulted semi-discrete scheme is shown to preserve the energy dissipation law and the total mass in the time discrete level. The semi-discrete scheme is further discretized in space using the Fourier spectral method, resulting in a fully discrete linear scheme. The fast algorithm for approximating the time-fractional derivative is also introduced to result in an other efficient full discrete linear scheme. Time refinement tests are conducted for both schemes, verifying their first order convergence in time for arbitrary fractional order α∈(0,1]. Several numerical simulations are presented to demonstrate the accuracy and efficiency of the newly proposed schemes. By exploring the fast algorithm calculating the Caputo fractional derivative, our numerical scheme makes it practical for long time simulation of the MBE model while preserving its energy stability, which is essential for MBE model predictions. With the proposed fractional MBE model, we observe that the effective energy decaying scales as O(t−α3) and the roughness increases as O(tα3), during the coarsening dynamics with the random initial condition. That is to say, the coarsening rate of time fractional MBE model could be manipulated by the fractional order α as a power law proportional to α. This is the first time in literature to report/discover such scaling correlation for the MBE model. It provides a potential application field for fractional differential equations to study anomalous coarsening. Besides, the numerical approximation strategy proposed in this paper can be readily applied to study many classes of time-fractional phase field models.

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energy-conserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete $$H_h^1$$ -norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.

Efficient modified techniques of invariant energy quadratization approach for gradient flows

Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

Semi-Implicit Spectral Deferred Correction Method Based on the Invariant Energy Quadratization Approach for Phase Field Problems

Semi-Implicit Spectral Deferred Correction Method Based on the Invariant Energy Quadratization Approach for Phase Field Problems

An efficient energy-preserving algorithm for the Lorentz force system

In this paper, by utilizing the invariant energy quadratization method to transform the original Hamiltonian energy functional into a quadratic form, we propose a novel energy-preserving scheme to solve the Lorentz force system. The method is a linear-implicit scheme for canonical Hamiltonian system, and can efficiently simulate the motion of charged particles in constant magnetic field. With comparison to the well-used Boris method and a similar energy-preserving BDLI method [26], numerical experiments are presented to demonstrate the energy-preserving property and computational efficiency of the method.

A linear, symmetric and energy-conservative scheme for the space-fractional Klein–Gordon–Schrödinger equations

In this paper, we propose an efficient numerical scheme for the space-fractional Klein–Gordon–Schrödinger (SFKGS) equations. Motivated by the “Invariant Energy Quadratization” (IEQ) approach, we introduce two auxiliary variables to transform the SFKGS system into a new equivalent system in which the time derivative is discretized by the Crank–Nicolson method, and the space discretization is based on the Fourier spectral method. Consequently, the numerical scheme shares two good features. The first feature is that the nonlinear terms are treated semi-explicitly and a linear symmetric system is solved at each time step. The second feature is the energy conservation at the discrete level. These two advantages are proved by the theoretical analysis and illustrated by a given numerical example.

Efficient numerical scheme for a dendritic solidification phase field model with melt convection

In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented.

Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation

We consider numerical approximations for the modified phase field crystal equation in this paper. The model is a nonlinear damped wave equation that includes both diffusive dynamics and elastic interactions. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we adopt the “Invariant Energy Quadratization” approach. By using the first-order backward Euler, the second-order Crank–Nicolson, and the second-order BDF2 formulas, we obtain three linear and symmetric positive definite schemes. We rigorously prove their unconditional energy stabilities and implement a number of 2D and 3D numerical experiments to demonstrate the accuracy, stability, and efficiency.

Efficient linear, stabilized, second-order time marching schemes for an anisotropic phase field dendritic crystal growth model

We consider numerical approximations for a phase field dendritic crystal growth model, which is a highly nonlinear system that couples the anisotropic Allen–Cahn type equation and the heat equation together. We propose two efficient, linear, second-order time marching schemes. The first one is based on the linear stabilization approach where all nonlinear terms are treated explicitly and one only needs to solve two linear and decoupled second-order equations. The second one combines the recently developed Invariant Energy Quadratization approach with the linear stabilization technique. Two linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficients numerically, are added to enhance the stability while keeping the required accuracy. We further show the obtained linear system is well-posed and prove its unconditional energy stability rigorously. Various 2D and 3D numerical simulations are implemented to demonstrate the stability and accuracy of the schemes.

Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation

In this paper, we develop four energy-preserving algorithms for the regularized long wave (RLW) equation. On the one hand, we combine the discrete variational derivative method (DVDM) in time and the modified finite volume method (mFVM) in space to derive a fully implicit energy-preserving scheme and a linear-implicit conservative scheme. On the other hand, based on the (invariant) energy quadratization technique, we first reformulate the RLW equation to an equivalent form with a quadratic energy functional. Then we discretize the reformulated system by the mFVM in space and the linear-implicit Crank-Nicolson method and the leap-frog method in time, respectively, to arrive at two new linear structure-preserving schemes. All proposed fully discrete schemes are proved to preserve the corresponding discrete energy conservation law. The proposed linear energy-preserving schemes not only possess excellent nonlinear stability, but also are very cheap because only one linear equation system needs to be solved at each time step. Numerical experiments are presented to show the energy conservative property and efficiency of the proposed methods.

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

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

Efficient linear schemes for the nonlocal Cahn–Hilliard equation of phase field models

In this paper, we develop two second-order in time, linear and unconditionally energy stable time marching schemes for solving the nonlocal Cahn–Hilliard phase field model. The main challenge to construct efficient and unconditional energy stable schemes for this model is how to design proper temporal discretizations for the nonlocal term that is induced from a convolutional type potential and the nonlinear cubic term that is induced from the double-well bulk potential. We solve these issues by developing two efficient time-stepping schemes using the Invariant Energy Quadratization approach. Its novelty is that all nonlinear terms can be treated semi-explicitly to produce linear schemes. We further show the well-posedness of the resulting linear system as well as its unconditional energy stability rigorously. Various numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes.

Linear, second order and unconditionally energy stable schemes for the viscous Cahn–Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method

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.

Regularized linear schemes for the molecular beam epitaxy model with slope selection

In this paper, we propose full discrete linear schemes for the molecular beam epitaxy (MBE) model with slope selection, which are shown to be unconditionally energy stable and unique solvable. In details, using the invariant energy quadratization (IEQ) approach, along with a regularized technique, the MBE model is first discretized in time using either Crank–Nicolson or Adam–Bashforth strategies. The semi-discrete schemes are shown to be energy stable and unique solvable. Then we further use Fourier-spectral methods to discretize the space, ending with full discrete schemes that are energy-stable and unique solvable. In particular, the full discrete schemes are linear such that only a linear algebra problem need to be solved at each time step. Through numerical tests, we have shown a proper choice of the regularization parameter provides better stability and accuracy, such that larger time step is feasible. Afterward, we present several numerical simulations to demonstrate the accuracy and efficiency of our newly proposed schemes. The linearizing and regularizing strategy used in this paper could be readily applied to solve a class of phase field models that are derived from energy variation.

Efficient Second Order Unconditionally Stable Schemes for a Phase Field Moving Contact Line Model Using an Invariant Energy Quadratization Approach

We consider the numerical approximations for a phase field model consisting of incompressible Navier--Stokes equations with a generalized Navier boundary condition, and the Cahn--Hilliard equation with a dynamic moving contact line boundary condition. A crucial and challenging issue for solving this model numerically is the time marching problem, due to the high order, nonlinear, and coupled properties of the system. We solve this issue by developing two linear, second order accurate, and energy stable schemes based on the projection method for the Navier--Stokes equations, the invariant energy quadratization for the nonlinear gradient terms in the bulk and boundary, and a subtle implicit-explicit treatment for the stress and convective terms. The well-posedness of the semidiscretized system and the unconditional energy stabilities are proved. Various numerical results based on a spectral-Galerkin spatial discretization are presented to verify the accuracy and efficiency of the proposed schemes.

Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

In this paper, we investigate numerical solution of the diffuse interface model with Peng–Robinson equation of state, that describes real states of hydrocarbon fluids in the petroleum industry. Due to the strong nonlinearity of the source terms in this model, how to design appropriate time discretizations to preserve the energy dissipation law of the system at the discrete level is a major challenge. Based on the “Invariant Energy Quadratization” approach and the penalty formulation, we develop efficient first and second order time stepping schemes for solving the single-component two-phase fluid problem. In both schemes the resulted temporal semi-discretizations lead to linear systems with symmetric positive definite spatial operators at each time step. We rigorously prove their unconditional energy stabilities in the time discrete sense. Various numerical simulations in 2D and 3D spaces are also presented to validate accuracy and stability of the proposed linear schemes and to investigate physical reliability of the target model by comparisons with laboratory data.

Numerical approximations for a three-component Cahn–Hilliard phase-field model based on the invariant energy quadratization method

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.

Numerical Approximations for the Cahn–Hilliard Phase Field Model of the Binary Fluid-Surfactant System

In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn-Hilliard equations. The main challenge in solving the system numerically is how to develop easy-to-implement time stepping schemes while preserving the unconditional energy stability. We solve this issue by developing two linear and decoupled, first order and a second order time-stepping schemes using the so-called "Invariant Energy Quadratization" approach for the double well potentials and a subtle explicit-implicit technique for the nonlinear coupling potential. Moreover, the resulting linear system is well-posed and the linear operator is symmetric positive definite. We rigorously prove the first order scheme is unconditionally energy stable. Various numerical simulations are presented to demonstrate the stability and the accuracy thereafter.

Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model

In this paper, we consider numerical approximations of a hydro-dynamically coupled phase field diblock copolymer model, in which the free energy contains a kinetic potential, a gradient entropy, a Ginzburg–Landau double well potential, and a long range nonlocal type potential. We develop a set of second order time marching schemes for this system using the “Invariant Energy Quadratization” approach for the double well potential, the projection method for the Navier–Stokes equation, and a subtle implicit-explicit treatment for the stress and convective term. The resulting schemes are linear and lead to symmetric positive definite systems at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable. Various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.

Linear and unconditionally energy stable schemes for the binary fluid–surfactant phase field model

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.