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Decoupled conservative schemes for computing dynamics of the strongly coupled nonlinear Schrodinger system

APPLIED NUMERICAL MATHEMATICS 2020年 157卷 276-290页

作者： He, Zhu Cai, Jiaxiang Shen, Bangyu Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China

Two types of first-order decoupled conservative schemes are firstly proposed for the strongly coupled nonlinear Schrodinger system by using pseudospectral method in space and coordinate increment discrete gradient method in time. And then, in order to improve the solution accuracy in time, the composition methods are employed to construct secondand fourth-order schemes. The proposed schemes are efficient for the system in d >= 2 dimensions and also easy to code because of their decoupled feature. A fast solver is proposed to speed up the computation. Ample numerical examples including the motion of single soliton and interaction of multiple solitary waves are carried out to exhibit the performance of the schemes. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Schrodinger equation structure-preserving algorithm Hamiltonian system Discrete gradient method Energy-preserving algorithm

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An SDG Galerkin structure-preserving scheme for the Klein-Gordon-Schrodinger equation

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2020年第9期43卷 6011-6030页

作者： Wang, Jialing Wang, Yushun Nanjing Univ Informat Sci & Technol Sch Math & Stat Nanjing 210044 Peoples R China Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

In this paper, we use the Galerkin weak form to construct a structure-preserving scheme for Klein-Gordon-Schrodinger equation and analyze its conservative and convergent properties. We first discretize the underlying equation in space direction via a selected finite element method, and the Hamiltonian partial differential equation can be casted into Hamiltonian ordinary differential equations based on the weak form of the system afterwards. Then, the resulted ordinary differential equations are solved by the symmetric discrete gradient method, which yields a charge-preserving and energy-preserving scheme. Moreover, the numerical solution of the proposed scheme is proved to be bounded in the discrete L infinity norm and convergent with the convergence order of O(h2+tau 2) in the discrete L2 norm without any grid ratio restrictions, where h and tau are space and time step, respectively. Numerical experiments conducted last to verify the theoretical analysis.

关键词： finite element method Klein-Gordon-Schrodinger equation structure-preserving algorithm symmetric discrete gradient method

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Quadratic conservative scheme for relativistic Vlasov-Maxwell system

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JOURNAL OF COMPUTATIONAL PHYSICS 2019年 379卷 32-50页

作者： Shiroto, Takashi Ohnishi, Naofumi Sentoku, Yasuhiko Tohoku Univ Dept Aerosp Engn Aoba Ku 6-6-01 Aramaki Aza Aoba Sendai Miyagi 9808579 Japan Osaka Univ Inst Laser Engn 2-6 Yamadaoka Suita Osaka 5650871 Japan

For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solved the first-principle equations of kinetic plasmas, such as the relativistic Vlasov-Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vlasov-Maxwell system. A discrete product rule and summation-by-parts are the key players in the construction of the quadratic conservative scheme. Numerical experiments of the relativistic two-stream instability and relativistic Weibel instability prove the validity of our computational theory, and the proposed strategy will open the doors to the first-principle studies of mesoscopic and macroscopic plasma physics. (C) 2018 The Author(s). Published by Elsevier Inc.

关键词： Computational plasma physics Relativistic Vlasov-Maxwell system structure-preserving algorithm Quadratic conservative scheme

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Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 352卷 341-351页

作者： Bhatt, Ashish Moore, Brian E. Univ Stuttgart Dept Math Stuttgart Germany Univ Cent Florida Dept Math Orlando FL 32816 USA

structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as Korteweg-de Vries, Klein-Gordon, Schrodinger, and Camassa-Holm equations, all with damping/driving terms and time-dependent coefficients. Since key features of the PDEs under consideration are described by local conservation laws, which are independent of the boundary conditions, the proposed (second-order in time) discretizations are developed with the intent of preserving those local conservation laws. The methods are respectively applied to a damped-driven nonlinear Schrodinger equation and a damped Camassa-Holm equation. Numerical experiments illustrate the structure-preserving properties of the methods, as well as favorable results over other competitive schemes. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Conformal symplectic Multi-symplectic structure-preserving algorithm Exponential integrators Damped-driven PDE

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Explicit high-order gauge-independent symplectic algorithms for relativistic charged particle dynamics

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COMPUTER PHYSICS COMMUNICATIONS 2019年第0期241卷 19-27页

作者： Xiao, Jianyuan Qin, Hong Univ Sci & Technol China Dept Engn & Appl Phys Hefei 230026 Anhui Peoples R China Princeton Univ Plasma Phys Lab Princeton NJ 08543 USA

Symplectic schemes are powerful methods for numerically integrating Hamiltonian systems, and their long-term accuracy and fidelity have been proved both theoretically and numerically. However direct applications of standard symplectic schemes to relativistic charged particle dynamics result in implicit and electromagnetic gauge-dependent algorithms. In the present study, we develop explicit high-order gauge-independent noncanonical symplectic algorithms for relativistic charged particle dynamics using a Hamiltonian splitting method in the 8D phase space. It is also shown that the developed algorithms can be derived as variational integrators by appropriately discretizing the action of the dynamics. Numerical examples are presented to verify the excellent long-term behavior of the algorithms. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Relativistic charged particle dynamics structure-preserving algorithm Noncanonical Poisson bracket Gauge symmetry

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Efficient schemes for the coupled Schrodinger-KdV equations: Decoupled and conserving three invariants

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APPLIED MATHEMATICS LETTERS 2018年 86卷 200-207页

作者： Cai, Jiaxiang Bai, Chuanzhi Zhang, Haihui Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China

We give a systematic method for discretizing Hamiltonian partial differential equations. An application of the method to the Hamiltonian form of the coupled Schrodinger-KdV equations yields a temporal first-order conservative scheme. Temporal second- and fourth-order schemes are developed by employing the composition method. All the schemes are decoupled and exactly conserve three invariants simultaneously. Numerical results show good performance of the schemes and verify the theoretical results. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Schrodinger-KdV equation Decoupled scheme Hamiltonian system Discrete gradient method structure-preserving algorithm

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Efficient high-order structure-preserving methods for the generalized Rosenau-type equation with power law nonlinearity

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2018年 59卷 122-131页

作者： Cai, Jiaxiang Liang, Hua Zhang, Chun Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China

Based on the multi-symplectic Hamiltonian formula of the generalized Rosenau-type equation, a multi-symplectic scheme and an energy-preserving scheme are proposed. To improve the accuracy of the solution, we apply the composition technique to the obtained schemes to develop high-order schemes which are also multi-symplectic and energy-preserving respectively. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results verify that all the proposed schemes have satisfactory performance in providing accurate solution and preserving the discrete mass and energy invariants. Numerical results also show that although each basic time step is divided into several composition steps, the computational efficiency of the composition schemes is much higher than that of the non-composite schemes. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Rosenau-type equation Composition method Spectral method Multi-symplectic conservation law structure-preserving algorithm

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Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrodinger equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2018年 374卷 281-299页

作者： Cai, Jiaxiang Bai, Chuanzhi Zhang, Haihui Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China

We develop two local energy-preserving integrators and a global energy-preserving integrator for the general multisymplectic Hamiltonian system. When applied to the 1D and multi-dimensional N-coupled nonlinear Schrodinger equations, the given schemes have the exact preservation of the local/global conservation law and are decoupled in the components psi(n,) n = 1, 2, ... , N, i.e., each of the components can be solved independently. The decoupled feature is significant and helpful for overcoming the computational difficulty of the N-coupled (N >= 3) nonlinear Schrodinger equations, especially of the multi-dimensional case. The composition method is employed to improve the accuracy of the schemes in time and the discrete fast Fourier transform is used to reduce the computational complexity. Several numerical experiments are carried out to exhibit the behaviors of the wave solutions. Numerical results confirm the theoretical results. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Schrodinger equation structure-preserving algorithm Multisymplectic structure Energy-preserving algorithm Discrete gradient method

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Efficient local structure-preserving schemes for the RLW-type equation

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2017年第5期33卷 1678-1691页

作者： Cai, Jiaxiang Hong, Qi Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China China Acad Engn Phys Grad Sch Beijing 100088 Peoples R China

The multisymplectic schemes have been used in numerical simulations for the RLW-type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure-preserving schemes for the RLW-type equation, which preserve the local energy exactly on any time-space region and can produce richer information of the original problem. The schemes will be mass- and energy-preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant-preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.(c) 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678-1691, 2017

关键词： conservation law regularized long-wave equation structure-preserving algorithm

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Local energy- and momentum-preserving schemes for Klein-Gordon-Schrodinger equations and convergence analysis

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2017年第4期33卷 1329-1351页

作者： Cai, Jiaxiang Hong, Jialin Wang, Yushun Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210046 Jiangsu Peoples R China Chinese Acad Sci State Key Lab Sci & Engn Comp Inst Computat Math & Sci Engn Comp AMSS Beijing 100190 Peoples R China

In this article, we obtain local energy and momentum conservation laws for the Klein-Gordon-Schrodinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy- and momentum-preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time-space region. With suitable boundary conditions, the schemes will be charge- and energy-/momentum-preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order O(2+h2). The theoretical properties are verified by numerical experiments. (c) 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329-1351, 2017

关键词： conservation law convergence analysis Klein-Gordon-Schrodinger equations local structure structure-preserving algorithm

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