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Journal of Computational Mathematics 2020年第3期38卷 452-468页

作者： Jie Chen Zhengkang He Shuyu Sun Shimin Guo Zhangxin Chen School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China Computational Transport Phenomena Laboratory Division of Physical Science and EngineeringKing Abdullah University of Science and TechnologyThuwal 23955-6900Kingdom of Saudi Arabia School of Mathematics and Statistics Xi'an Jiaotong Universityy Xi'an 710049China Department of Chemical&Petroleum Engineering Schulich School of EngineeringUniversity of Calgary2500 University Drive N.W.CalgaryAlberta T2N 1N4Canada

In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel"scalar auxiliary variable"(SAV)approach,a new developed efficient and accurate method for a large class of gradient *** schemes are based on the first order Euler method and the second order backward differential formulas(BDF2)for time discretization,and finite element methods for space *** proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time *** numerical experiments are presented to validate the stability and accuracy of the proposed schemes.

关键词： Phase field models Solid-state dewetting SAV energy stability Surface diffusion Finite element method

Improvement of the energy stability of the Single Ion Microbeam

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Plasma Science and Technology 2008年第2期10卷 250-253页

作者：詹福如齐学红许明亮陈连运余增亮 Institute of Plasma Physics Chinese Academy of Sciences Hefei 230031 China

energy instability strongly affects the state and the beam size of the single ion microbeam. A facility based on the Generating Voltmeter was developed to improve the energy stability of the CAS-LIBB （Chinese Academy of Sciences, key laboratory of ion beam bioengineering） single ion microbeam. This paper presents the analysis of the energy＇ instability of the single ion microbeam. A simplified theoretical model is set up to calculate the relationship between the energy instability and the beam spot size. By using this technique, the energy instability is adjusted to about 1%. Stable run-time is over 6 hours. The radius of the single ion beam is reduced by 10% compared to the previous olin.

关键词： single ion microbeam energy stability electrostatic accelerator

Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

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Science China Mathematics 2022年第10期65卷 2201-2218页

作者： Xiaoli Li Jie Shen School of Mathematics Shandong UniversityJinan 250100China Department of Mathematics Purdue UniversityWest LafayetteIN 47907USA

In this paper,we construct efficient schemes based on the scalar auxiliary variable block-centered finite difference method for the modified phase field crystal equation,which is a sixth-order nonlinear damped wave *** schemes are linear,conserve mass and unconditionally dissipate a pseudo *** prove rigorously second-order error estimates in both time and space for the phase field variable in discrete *** also present some numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy.

关键词： modified phase field crystal scalar auxiliary variable energy stability error estimate numerical experiments

STABLE BOUNDARY CONDITIONS AND DISCRETIZATION FOR P_(N) EQUATIONS

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Journal of Computational Mathematics 2022年第6期40卷 977-1003页

作者： Jonas Bunger Neeraj Sarna Manuel Torrilhon Applied and Computational Mathematics(ACoM) Department of MathematicsRWTH Aachen UniversitySchinkelstr 252062 AachenGermany

A solution to the linear Boltzmann equation satisfies an energy bound,which reflects a natural fact:The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over *** this paper,we present boundary conditions(BCs)for the spherical harmonic(P_(N))approximation,which ensure that this fundamental energy bound is satisfied by the P_(N) *** BCs are compatible with the characteristic waves of P_(N) equations and determine the incoming waves ***,energy bound and compatibility,are shown on abstract formulations of P_(N) equations and BCs to isolate the necessary structures and *** BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step,which is similar to the truncation of the series expansion in the P_(N) *** show that summation by parts(SBP)finite differences on staggered grids in space and the method of simultaneous approximation terms(SAT)allows to maintain the energy bound also on the semi-discrete level.

关键词： Boundary conditions energy stability Spherical harmonic(P_(N))approximation Kinetic theory Moment method Boltzmann Linear transport

Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

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Science China Mathematics 2021年第5期64卷 887-902页

作者： Hong-Lin Liao Xuehua Song Tao Tang Tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211101China Department of Mathematics and International Center for Mathematics Southern University of Science and TechnologyShenzhen 518055China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China NCMIS&LSEC Institute of Computational Mathematics and Scientific Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope *** first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step *** we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0

关键词： molecular beam epitaxial growth variable-step BDF2 scheme discrete orthogonal convolution kernels energy stability convergence analysis

Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 1147-1172页

作者： Zhonghua Qiao Qian Zhang Department of Applied Mathematics The Hong Kong Polytechnic UniversityHung HomHong KongSARChina Research Institute for Smart energy The Hong Kong Polytechnic UniversityHung HomHong KongSARChina School of Science Harbin Institute of TechnologyShenzhenChina

Based on a nonlocal Laplacian operator,a novel edge detection method of the grayscale image is proposed in this *** operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge *** nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale *** exponential time differencing(ETD)solvers are employed in the time integration,and finite difference method is adopted in space *** maximum bound principle and energy stability of the proposed numerical schemes are *** capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

关键词： Image segmentation Allen-Cahn equation nonlocal edge detection operator maximum principle energy stability

Highly efficient and unconditionally energy stable semi-discrete time-marching numerical scheme for the two-phase incompressible flow phase-field system with variable-density and viscosity

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Science China Mathematics 2022年第12期65卷 2631-2656页

作者： Chuanjun Chen Xiaofeng Yang School of Mathematics and Information Sciences Yantai UniversityYantai 264005China Department of Mathematics University of South CarolinaSC29208USA

For the variable-density/viscosity Cahn-Hilliard phase-field model of the binary-phase incompressible fluid flow system, the development of easy-to-implement numerical schemes has long been known as a challenging problem. We develop a novel fully-decoupled numerical technique in this article which can achieve unconditional energy stability while explicitly discretizing nonlinear coupling items. The idea is invented on the basis of combining the Strang operator splitting method and the novel decoupling method by using the zero-energy-contribution property. The scheme 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 are with constant coefficients. To demonstrate the effectiveness of the scheme, we provide the rigorous proof of the energy stability/solvability, and also perform ample accuracy and stability tests and 2D/3D numerical simulations, including the Rayleigh-Taylor instability and bubble rising dynamics.

关键词： variable-density decoupled time marching Cahn-Hilliard equation Navier-Stokes equations energy stability

An energy Stable Monolithic Eulerian Fluid-Structure Numerical Scheme

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Chinese Annals of Mathematics,Series B 2018年第2期39卷 213-232页

作者： olivier pironneau Dedicated to Philippe G. Ciarlet on the occasion of his 80th birthda

The conservation laws of continuum mechanics, written in an Eulerian frame,do not distinguish fluids and solids, except in the expression of the stress tensors, usually with Newton's hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. By taking the velocities as unknown monolithic methods for fluid structure interactions(FSI for short) are built. In this paper such a formulation is analysed when the solid is compressible and the fluid is incompressible. The idea is not new but the progress of mesh generators and numerical schemes like the Characteristics-Galerkin method render this approach feasible and reasonably robust. In this paper the method and its discretisation are presented, stability is discussed through an energy estimate. A numerical section discusses implementation issues and presents a few simple tests.

关键词： Fluid-Structure interactions Numerical method energy stability Finiteelement method

A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

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Acta Mathematica Scientia 2024年第2期44卷 650-670页

作者：司智勇 School of Mathematics and Information Science Henan Polytechnic UniversityJiaozuo454003China

This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau *** backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau *** this method,the system is decoupled and linearized to avoid solving the non-linear equation at each *** theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.

关键词： time-dependent Ginzburg-Landau equation generalized scalar auxiliary variable algorithm maximum bound principle energy stability

An energy Stable SPH Method for Incompressible Fluid Flow

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Advances in Applied Mathematics and Mechanics 2022年第5期14卷 1201-1224页

作者： Xingyu Zhu Shuyu Sun Jisheng Kou Computational Transport Phenomena Laboratory Division of Physical Science and EngineeringKing Abdullah University of Science and TechnologyThuwal 23955-6900Saudi Arabia School of Civil Engineering Shaoxing UniversityShaoxing 312000ZhejiangChina School of Mathematics and Statistics Hubei Engineering UniversityXiaogan 432000HubeiChina

In this paper,a novel unconditionally energy stable Smoothed Particle Hydrodynamics(SPH)method is proposed and implemented for incompressible fluid *** this method,we apply operator splitting to break the momentum equation into equations involving the non-pressure term and pressure term *** idea behind the splitting is to simplify the calculation while still maintaining energy stability,and the resulted algorithm,a type of improved pressure correction scheme,is both efficient and energy *** show in detail that energy stability is preserved at each full-time step,ensuring unconditionally numerical *** examples are presented and compared to the analytical solutions,suggesting that the proposed method has better accuracy and ***,we observe that if we are interested in steady-state solutions only,our method has good performance as it can achieve the steady-state solutions rapidly and accurately.

关键词： Particle method Smoothed Particle Hydrodynamics(SPH)method incompressible fluid flow energy stability numerical stability