In this work,we construct an efficient invariant energy quadratization(IEQ)method of unconditional energy stability to solve the Cahn-Hilliard equation.The constructed numerical scheme is linear,second-order accuracy ...In this work,we construct an efficient invariant energy quadratization(IEQ)method of unconditional energy stability to solve the Cahn-Hilliard equation.The constructed numerical scheme is linear,second-order accuracy in time and unconditional energy stability.We carefully analyze the unique solvability,stability and error estimate of the numerical scheme.The results show that the constructed scheme satisfies unique solvability,unconditional energy stability and the second-order convergence in time direction.Through a large number of 2D and 3D numerical experiments,we further verify the convergence order,unconditional energy stability and effectiveness of the scheme.展开更多
The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial ...The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models.Under the stepratio condition 0<τ_(n)/τ_(n-1)<4.864,the modified energy dissipation law is proven at the discrete levels with regardless of time step size.Nu‐merical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.展开更多
To enhance the computational efficiency of spatio-temporally discretized phase-field models,we present a high-speed solver specifically designed for the Poisson equations,a component frequently used in the numerical c...To enhance the computational efficiency of spatio-temporally discretized phase-field models,we present a high-speed solver specifically designed for the Poisson equations,a component frequently used in the numerical computation of such models.This efficient solver employs algorithms based on discrete cosine transformations(DCT)or discrete sine transformations(DST)and is not restricted by any spatio-temporal schemes.Our proposed methodology is appropriate for a variety of phase-field models and is especially efficient when combined with flow field systems.Meanwhile,this study has conducted an extensive numerical comparison and found that employing DCT and DST techniques not only yields results comparable to those obtained via the Multigrid(MG)method,a conventional approach used in the resolution of the Poisson equations,but also enhances computational efficiency by over 90%.展开更多
In this paper,we construct a new class of efficient and high-order schemes for the Cahn-Hilliard-Navier-Stokes equations with periodic boundary conditions.These schemes are based on two types of scalar auxiliary varia...In this paper,we construct a new class of efficient and high-order schemes for the Cahn-Hilliard-Navier-Stokes equations with periodic boundary conditions.These schemes are based on two types of scalar auxiliary variable approaches.By using a new pressure correction method,the accuracy of the pressure has been greatly improved.Furthermore,one only needs to solve a series of fully decoupled linear equations with constant coefficients at each time step.In addition,we prove the unconditional energy stability of the schemes,rigorously.Finally,plenty of numerical simulations are carried out to verify the convergence rates,stability,and effectiveness of the proposed schemes numerically.展开更多
In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt ...In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.展开更多
In this work,we aim to develop an effective fully discrete Spectral-Galerkin numerical scheme for the multi-vesicular phase-field model of lipid vesicles with adhesion potential.The essence of the scheme is to introdu...In this work,we aim to develop an effective fully discrete Spectral-Galerkin numerical scheme for the multi-vesicular phase-field model of lipid vesicles with adhesion potential.The essence of the scheme is to introduce several additional auxiliary variables and design some corresponding auxiliary ODEs to reformulate the system into an equivalent form so that the explicit discretization for the nonlinear terms can also achieve unconditional energy stability.Moreover,the scheme has a full decoupling structure and can avoid calculating variable-coefficient systems.The advantage of this scheme is its high efficiency and ease of implementation,that is,only by solving two independent linear biharmonic equations with constant coefficients for each phase-field variable,the scheme can achieve the second-order accuracy in time,spectral accuracy in space,and unconditional energy stability.We strictly prove that the fully discrete energy stability that the scheme holds and give a detailed step-by-step implementation process.Further,numerical experiments are carried out in 2D and 3D to verify the convergence rate,energy stability,and effectiveness of the developed algorithm.展开更多
Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a nov...Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.展开更多
In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure syste...In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure system.To handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization.Theoretically,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments.More importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization operations.This implies the potential of our method in large-scale simulations in density functional theory.展开更多
We develop a new conservative Allen-Cahn phase-field model for diblock copolymers in this paper by using the Allen-Cahn type gradient flow approach for the classical Ohta-Kawaski free energy.The change in volume fract...We develop a new conservative Allen-Cahn phase-field model for diblock copolymers in this paper by using the Allen-Cahn type gradient flow approach for the classical Ohta-Kawaski free energy.The change in volume fraction of two composing monomers is eliminated by using a nonlocal Lagrange multiplier.Based on the recently developed stabilized Scalar Auxiliary Variable method,we have further developed an effective numerical scheme to solve the model.The scheme is highly efficient and only two linear and decoupled equations are needed to solve at every time step.We then prove that the numerical method is unconditionally energy stable,the stability and accuracy of the new scheme are demonstrated by numerous numerical examples conducted.By qualitatively comparing the equilibrium solution obtained by the new model and the classic Cahn-Hilliard model,we illustrate the effectiveness of the new model.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.12261017,62062018)the Science and Technology Program of Guizhou Province(Nos.ZK[2022]006,ZK[2022]031,QKHZC[2023]372)+3 种基金Shanxi Province Natural Science Research(Grant No.202203021212249)the Scientific Research Foundation of Guizhou University of Finance and Economics(Grant No.2022KYYB08)the Innovation Exploration and Academic Emerging Project of Guizhou University of Finance and Economics(Grant No.2022XSXMB11)the Special/Youth Foundation of Taiyuan University of Technology(Grant No.2022QN101)。
文摘In this work,we construct an efficient invariant energy quadratization(IEQ)method of unconditional energy stability to solve the Cahn-Hilliard equation.The constructed numerical scheme is linear,second-order accuracy in time and unconditional energy stability.We carefully analyze the unique solvability,stability and error estimate of the numerical scheme.The results show that the constructed scheme satisfies unique solvability,unconditional energy stability and the second-order convergence in time direction.Through a large number of 2D and 3D numerical experiments,we further verify the convergence order,unconditional energy stability and effectiveness of the scheme.
文摘The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models.Under the stepratio condition 0<τ_(n)/τ_(n-1)<4.864,the modified energy dissipation law is proven at the discrete levels with regardless of time step size.Nu‐merical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.
基金Supported by Shanxi Province Natural Science Research(202203021212249)Special/Youth Foundation of Taiyuan University of Technology(2022QN101)+3 种基金National Natural Science Foundation of China(12301556)Research Project Supported by Shanxi Scholarship Council of China(2021-029)International Cooperation Base and Platform Project of Shanxi Province(202104041101019)Basic Research Plan of Shanxi Province(202203021211129)。
文摘To enhance the computational efficiency of spatio-temporally discretized phase-field models,we present a high-speed solver specifically designed for the Poisson equations,a component frequently used in the numerical computation of such models.This efficient solver employs algorithms based on discrete cosine transformations(DCT)or discrete sine transformations(DST)and is not restricted by any spatio-temporal schemes.Our proposed methodology is appropriate for a variety of phase-field models and is especially efficient when combined with flow field systems.Meanwhile,this study has conducted an extensive numerical comparison and found that employing DCT and DST techniques not only yields results comparable to those obtained via the Multigrid(MG)method,a conventional approach used in the resolution of the Poisson equations,but also enhances computational efficiency by over 90%.
基金Supported by the Research Project Supported of Shanxi Scholarship Council of China(No.2021-029)Shanxi Provincial International Cooperation Base and Platform Project(202104041101019)Shanxi Province Natural Science Research(202203021211129)。
文摘In this paper,we construct a new class of efficient and high-order schemes for the Cahn-Hilliard-Navier-Stokes equations with periodic boundary conditions.These schemes are based on two types of scalar auxiliary variable approaches.By using a new pressure correction method,the accuracy of the pressure has been greatly improved.Furthermore,one only needs to solve a series of fully decoupled linear equations with constant coefficients at each time step.In addition,we prove the unconditional energy stability of the schemes,rigorously.Finally,plenty of numerical simulations are carried out to verify the convergence rates,stability,and effectiveness of the proposed schemes numerically.
文摘In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.
基金supported by National Natural Science Foundation of China(11771375)Shandong Provincial Natural Science Foundation(ZR2021ZD03,ZR2021MA010)supported by National Science Foundation with grant number DMS-2012490.
文摘In this work,we aim to develop an effective fully discrete Spectral-Galerkin numerical scheme for the multi-vesicular phase-field model of lipid vesicles with adhesion potential.The essence of the scheme is to introduce several additional auxiliary variables and design some corresponding auxiliary ODEs to reformulate the system into an equivalent form so that the explicit discretization for the nonlinear terms can also achieve unconditional energy stability.Moreover,the scheme has a full decoupling structure and can avoid calculating variable-coefficient systems.The advantage of this scheme is its high efficiency and ease of implementation,that is,only by solving two independent linear biharmonic equations with constant coefficients for each phase-field variable,the scheme can achieve the second-order accuracy in time,spectral accuracy in space,and unconditional energy stability.We strictly prove that the fully discrete energy stability that the scheme holds and give a detailed step-by-step implementation process.Further,numerical experiments are carried out in 2D and 3D to verify the convergence rate,energy stability,and effectiveness of the developed algorithm.
文摘Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.
基金The first author would like to thank the support from the UM-Funded PhD Assistantship from University of MacaoThe second author was partially supported by Macao Young Scholar Program(AM201919)+5 种基金excellent youth project of Hunan Education Department(19B543)Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department(2020ZYT003)The third author would like to thank financial support from National Natural Science Foundation of China(Grant Nos.11922120,11871489)FDCT of Macao SAR(Grant No.0082/2020/A2)University of Macao(Grant No.MYRG2020-00265-FST)Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications(Grant No.2020B1212030001).
文摘In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure system.To handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization.Theoretically,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments.More importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization operations.This implies the potential of our method in large-scale simulations in density functional theory.
基金supported by National Natural Science Foundation of China(Nos.71901150,71971143)the Natural Science Foundation of Guangdong Province(No.2020A151501749)+4 种基金supported by Kunming E-Commerce and Internet Finance R&D Center(KEIRDC[2020])the Prominent Educator Program(Yunnan[2018]11)Yunnan Province Young Academic and Technical Leader Candidate Program(No.2018HB027)Yunnan Provincial E-Business Entrepreneur Innovation Interactive Space(No.2017DS012)X.Yang was partially supported by National Science Foundation with grant number DMS-1720212.
文摘We develop a new conservative Allen-Cahn phase-field model for diblock copolymers in this paper by using the Allen-Cahn type gradient flow approach for the classical Ohta-Kawaski free energy.The change in volume fraction of two composing monomers is eliminated by using a nonlocal Lagrange multiplier.Based on the recently developed stabilized Scalar Auxiliary Variable method,we have further developed an effective numerical scheme to solve the model.The scheme is highly efficient and only two linear and decoupled equations are needed to solve at every time step.We then prove that the numerical method is unconditionally energy stable,the stability and accuracy of the new scheme are demonstrated by numerous numerical examples conducted.By qualitatively comparing the equilibrium solution obtained by the new model and the classic Cahn-Hilliard model,we illustrate the effectiveness of the new model.
