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作者： Huang, Yunqing Li, Jichun Yang, Wei Hunan Key Laboratory for Computation and Simulation in Science Engineering Xiangtan University Hunan 411105 China Department of Mathematical Sciences University of Nevada Las Vegas Las Vegas NV 89154-4020 United States

We investigate the well-posedness of a perfectly matched layer model developed by Cohen and Monk [Comput Methods Appl Mech Eng 169 (1999), 197-217]. A new time-domain finite element method is proposed to solve the model. A novel feature of the scheme is that the number of unknowns is reduced to two from four in the original form. The scheme's effectiveness is demonstrated through numerical experiments. © 2013 Wiley Periodicals, Inc.

关键词： Maxwell equations

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Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations

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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1440-1458页

作者： Xingfa Yang Yin Yang Yanping Chen Jie Liu State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan UniversityChangsha 410082HunanChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105HunanChina School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Center for Research on Leading Technology of Special Equipment School of Mechanical and Electric EngineeringGuangzhou UniversityGuangzhou 510006GuangdongChina

In this paper,we study a class of nonlinear fractional integro-differential equations,the fractional derivative is described in the Caputo *** the properties of the Caputo derivative,we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel,then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra *** provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.

关键词： Spectral method nonlinear fractional derivative Volterra integro-differential equations Caputo derivative

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Two classes of implicit–explicit multistep methods for nonlinear stiff initial-value problems

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Applied Mathematics and computation 2014年 247卷 47-60页

作者： Aiguo Xiao Gengen Zhang Xing Yi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China

The initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications. In order to reduce the computation cost, implicit–explicit (IMEX) methods are often applied to these problems, i.e. the stiff and non-stiff terms are discretized by using implicit and explicit methods, respectively. In this paper, we mainly consider the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, and present two classes of the IMEX multistep methods by combining implicit one-leg methods with explicit linear multistep methods and explicit one-leg methods, respectively. The order conditions and the convergence results of these methods are obtained. Some efficient methods are constructed. Some numerical examples are given to verify the validity of the obtained theoretical results.

关键词： Stiff problems Singular perturbation problems Implicit–explicit multistep methods One-leg methods Linear multistep methods Convergence

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Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion

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Physical Review E 2019年第2期99卷 023302-023302页

作者： Yang Hu Decai Li Licong Jin Xiaodong Niu Shi Shu Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 People's Republic of China

In this work a phase field-lattice Boltzmann model for incompressible two-phase flows is presented. In this model, the interface tracking equation is a linear combination of the local and nonlocal Allen-Cahn equations. We also propose a multiple-relaxation-time lattice Boltzmann model for solving the hybrid Allen-Cahn equation. The second-order convergence rate of the present model in space is validated by simulating the diagonal translation of circular interface. Three other numerical tests, including static bubble immersed in another fluid, bubble rising under gravity, and droplet splashing on a thin liquid film, are simulated to verify the performance of the present model in reducing the numerical dispersion. The numerical results indicate that the order parameter fluctuation can be reduced by one order of magnitude in bulk region.

关键词： Multiphase flows Lattice-Boltzmann methods

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A Posteriori Error Estimates for hp Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

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Advances in Applied Mathematics and Mechanics 2022年第2期14卷 469-493页

作者： Jinling Zhang Yanping Chen Yunqing Huang Fenglin Huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan 411105China School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdong 510631China School of Mathematics and Statistics Xinyang Normal UniversityNo.237 Nanhu RoadShihe DistrictXinyangHenan 464000China

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state *** control problem is approxi-mated by the hp spectral element method with high accuracy and geometricﬂ*** conditions of the continuous and discrete optimal control problems are presented,*** a posteriori error estimates both for the control and state variables are established in *** addition,illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.

关键词： Elliptic equations optimal control control-state constraints a posteriori error esti-mates hp spectral element method.

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Divergence-Free Virtual Element Method for the Stokes Equations with Damping on Polygonal Meshes

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Numerical Mathematics(Theory,Methods and Applications) 2024年第1期17卷 210-242页

作者： Yu Xiong Yanping Chen Jianwei Zhou Qin Liang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtan 411105P.R.China School of Mathematics and Statistics Linyi University Linyi 276005P.R.China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631P.R.China

In this paper,we construct,analyze,and numerically validate a family of divergence-free virtual elements for Stokes equations with nonlinear damping on polygonal *** virtual element method is H1-conforming and exact *** virtue of these properties and the topological degree argument,we rigorously prove the well-posedness of the proposed discrete *** con-vergence analysis is carried out,which imply that the error estimate for the velocity in energy norm does not explicitly depend on the *** experiments on various polygonal meshes validate the accuracy of the theoretical analysis and the asymptotic pressure robustness of the proposed scheme.

关键词： Optimal error estimates divergence-free virtual element nonlinear damping term.

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Parametric symplectic partitioned Runge–Kutta methods with energy-preserving properties for Hamiltonian systems

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Computer Physics Communications 2013年第2期184卷 303-310页

作者： Dongling Wang Aiguo Xiao Xueyang Li School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

Based on W -transformation, some parametric symplectic partitioned Runge–Kutta (PRK) methods depending on a real parameter α are developed. For α = 0 , the corresponding methods become the usual PRK methods, including Radau IA – I A ¯ and Lobatto IIIA – IIIB methods as examples. For any α ≠ 0 , the corresponding methods are symplectic and there exists a value α ∗ such that energy is preserved in the numerical solution at each step. The existence of the parameter and the order of the numerical methods are discussed. Some numerical examples are presented to illustrate these results.

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Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

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Applied Mathematics 2013年第1期4卷 113-118页

作者： Yin Yang College of Civil Engineering and Mechanics Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan China

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

关键词： Legendre Pseudo-Spectral Method Multi-Order Fractional Differential Equations Caputo Derivative

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Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

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Frontiers of Mathematics in China 2012年第3期7卷 397-413页

作者： Yanping CHEN Zuliang LU Ruyi GUO School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China College of Civil Engineering and Mechanics Xiangtan UniversityXiangtan 411105China School of Mathematics and Statistics China Three Gorges UniversityChongqing 404000China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105China

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.

关键词： A priori error estimate quasilinear elliptic equation generalconvex optimal control problem triangular mixed finite element method

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Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

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Communications in Nonlinear science and Numerical simulation 2013年第4期18卷 905-914页

作者： Dongling Wang Aiguo Xiao School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.

关键词： Fractional variational integrators Fractional Euler–Lagrange equations Holonomic constraints

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