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energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

Communications on Applied Mathematics and Computation 2024年第1期6卷 30-63页

作者： Maohui Lyu Vrushali A.Bokil Yingda Cheng Fengyan Li State Key Laboratory of Scientific and Engineering Computing(LSEC) Academy of Mathematics and System SciencesChinese Academy of SciencesBeijing100190China Department of Mathematics College of ScienceOregon State UniversityCorvallisOR97331USA Department of Mathematics Michigan State UniversityEast LansingMI48824USA Department of Computational Mathematics Science and EngineeringMichigan State UniversityEast LansingMI48824USA Department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyNY12180USA

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

关键词： Maxwell’s equations Kerr and Raman Discontinuous Galerkin method energy stability

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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 equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The 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

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A SCALAR AUXILIARY VARIABLE (SAV) FINITE ELEMENT NUMERICAL SCHEME FOR THE CAHN-HILLIARD-HELE-SHAW SYSTEM WITH DYNAMIC BOUNDARY CONDITIONS

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Journal of Computational Mathematics 2024年第2期42卷 544-569页

作者： Changhui Yao Fengdan Zhang Cheng Wang School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China Department of Mathematics University of MassachusettsNorth DartmouthMA 02747USA

In this paper,we consider the Cahn-Hilliard-Hele-Shaw(CHHS)system with the dynamic boundary conditions,in which both the bulk and surface energy parts play important roles.The scalar auxiliary variable approach is introduced for the physical system;the mass conservation and energy dissipation is proved for the CHHS system.Subsequently,a fully discrete SAV finite element scheme is proposed,with the mass conservation and energy dissipation laws established at a theoretical level.In addition,the convergence analysis and error estimate is provided for the proposed SAV numerical scheme.

关键词： Cahn-Hilliard-Hele-Shaw system Dynamic boundary conditions Bulk energy and surface energy Scalar auxiliary variable formulation energy stability Convergence analysis

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EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL energy stability FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS

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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 flows.The 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 discretization.The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps.Various 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

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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

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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

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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 time.In 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) approximation.Our BCs are compatible with the characteristic waves of P_(N) equations and determine the incoming waves uniquely.Both,energy bound and compatibility,are shown on abstract formulations of P_(N) equations and BCs to isolate the necessary structures and properties.The 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) method.We 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

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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 paper.This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection.The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image.Efficient exponential time differencing(ETD)solvers are employed in the time integration,and finite difference method is adopted in space discretization.The maximum bound principle and energy stability of the proposed numerical schemes are proved.The 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

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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 equation.The schemes are linear,conserve mass and unconditionally dissipate a pseudo energy.We prove rigorously second-order error estimates in both time and space for the phase field variable in discrete norms.We 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

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A Third Order Accurate in Time,BDF-Type energy Stable Scheme for the Cahn-Hilliard Equation

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Numerical Mathematics(Theory,Methods and Applications) 2022年第2期15卷 279-303页

作者： Kelong Cheng Cheng Wang Steven M.Wise Yanmei Wu School of Science Southwest University of Science and TechnologyMianyangSichuan 621010P.R.China Department of Mathematics The University of MassachusettsNorth DartmouthMA 02747USA Department of Mathematics The University of TennesseeKnoxvilleTN 37996USA

In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.

关键词： Cahn-Hilliard equation third order backward differentiation formula unique solvability energy stability discrete L6 N estimate optimal rate convergence analysis

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