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

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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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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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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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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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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An indirect convergent Jacobi spectral collocation method for fractional optimal control problems

An indirect convergent Jacobi spectral collocation method fo...

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作者： Yang, Yin Zhang, Jiaqi Liu, Huan O. Vasilev, Aleksandr Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan China Ammosov North-Eastern Federal University Yakutsk Russia

In this paper, we present a novel indirect convergent Jacobi spectral collocation method for fractional optimal control problems governed by a dynamical system including both classical derivative and Caputo fractional derivative. First, we present some necessary optimality conditions. Then we suggest a new Jacobi spectral collocation method to discretize the obtained conditions. By the proposed method, we get a system of algebraic equations by solving of which we can approximate the optimal solution of the main problem. Finally, we present a convergence analysis for our method and solve three numerical examples to show the efficiency and capability of the method. © 201. John Wiley & Sons, Ltd.

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Numerical homogenization of elastoplastic deformations of composite material with small proportion of inclusions 4

Numerical homogenization of elastoplastic deformations of co...

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4th International Conference on Supercomputer Technologies of Mathematical Modelling, SCTeMM 201.

作者： Sivtsev, Petr V. Kolesov, Aleksandr E. Zakharov, Petr E. Yang, Ying Research Laboratory multiscale Matheatical Modeling and Computing Ammosov North-Eastern Federal University Belinskogo 58 Yakutsk677000 Russia Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China

Numerical simulation of the stress-strain state of a composite material may be difficult due to large {1.al complexity associated with a grid resolution of a large number of inclusions. To overcome the problem one may use the homogenization method. But for material with plastic properties, proper modeling of yield stress and hardening may be overcomplicated. In this work we use some simplification associated with a small proportion of inclusions and restriction of stress values by matrix material strength. As model problem, we use deformation of concrete deep beam reinforced with steel or basalt fiber inclusions. For the numerical solution, the finite element method was applied using the FEniCS computing platform. © Published under licence by IOP Publishing Ltd.

关键词： Homogenization 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 {1. 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 (201.) [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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Time-splitting finite difference method with the wavelet-adaptive grids for semiclassical Gross–Pitaevskii equation in supercritical case

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Journal of computational Physics 2014年 267卷 146-161页

作者： Xueyang Li 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

The Gross–Pitaevskii equation is the model equation of the single-particle wave function in a Bose–Einstein condensation. A computation difficulty of the Gross–Pitaevskii equation comes from the semiclassical problem in supercritical case. In this paper, we apply a diffeomorphism to transform the original one-dimensional Gross–Pitaevskii equation into a modified equation. The adaptive grids are constructed through the interpolating wavelet method. Then, we use the time-splitting finite difference method with the wavelet-adaptive grids to solve the modified Gross–Pitaevskii equation, where the approximation to the second-order derivative is given by the Lagrange interpolation method. At last, the numerical results are given. It is shown that the obtained time-splitting finite difference method with the wavelet-adaptive grids is very efficient for solving the one-dimensional semiclassical Gross–Pitaevskii equation in supercritical case and it is suitable to deal with the local high oscillation of the solution.

关键词： Gross–Pitaevskii equation Wavelet-adaptive grids Time-splitting finite difference method Interpolating wavelet Lagrange interpolation method

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Parallelization of a DEM code based on CPU-GPU heterogeneous architecture

Parallelization of a DEM code based on CPU-GPU heterogeneous...

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26th International Conference on Parallel computational Fluid Dynamics, ParCFD 201.

作者： Yue, Xiaoqiang Zhang, Hao Luo, Congshu Shu, Shi Feng, Chunsheng School of Mathematics and Computational Science Xiangtan University 411105 Hunan China Heat and Mass Transfer Technological Center Technical University of Catalonia 08222 Terrassa Spain Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University 411105 Hunan China

ISBN: (纸本)9783642539619

Particulate flows are commonly encountered in both engineering and environmental applications. The discrete element method (DEM) has attracted plentiful attentions since it can predict the whole motion of the particulate flow by monitoring every single particle. However the {1.al capability of the method relies strongly on the numerical scheme as well as the hardware environment. In this study, a parallelization of a DEM based code titled Trubal was implemented. Numerical simulations were carried out to show the benefits of this research. It is shown that the final parallel code gave a substantial acceleration on the Trubal. By simulating 6,000 particles using a NVIDIA Tesla C2050 card together with Intel Core-Dual 2.93 GHz CPU, an average speedup of 4.69 in {1.al time was obtained. © Springer-Verlag Berlin Heidelberg 201..

关键词： Finite difference method

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