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

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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A dissipation-preserving scheme for damped oscillatory Hamiltonian systems based on splitting

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APPLIED NUMERICAL MATHEMATICS 2021年 170卷 242-254页

作者： Liu, Kai Fu, Ting Shi, Wei Nanjing Univ Finance & Econ Coll Appl Math Nanjing 210023 Peoples R China Nanjing Tech Univ Coll Math Sci Nanjing 211816 Peoples R China

In this paper, a new dissipation-preserving scheme is established for weakly dissipative perturbations of oscillatory Hamiltonian systems. The system exhibits a nonlinear oscillatory structure. The main oscillation is governed by a matrix M and the damping is governed by a matrix Gamma. The new scheme preserves the oscillatory structure of the systems by incorporating the matrix M in the scheme based on the idea of ERKN methods. Meanwhile, the discrete gradient and splitting are used to construct the scheme such that the numerical solution possesses a nearly correct damping rate of the system. A main feature of the new scheme is that a relatively large stepsize can be chosen since the convergence of the implicit iterations in the scheme is shown to be independent of the matrices M and Gamma. Three numerical experiments of perturbed Hamiltonian systems are conducted to show the effectiveness and the efficiency of the new scheme in comparison with the traditional discrete gradient methods. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Weakly dissipative Hamiltonian systems Oscillatory systems structure-preserving algorithm Discrete gradient Sine-Gordon equation Continuous Fermi-Pasta-Ulam system

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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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A structure-preserving method for the quaternion LU decomposition in quaternionic quantum theory

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COMPUTER PHYSICS COMMUNICATIONS 2013年第9期184卷 2182-2186页

作者： Wang, Minghui Ma, Wenhao Qingdao Univ Sci & Technol Dept Math Qingdao 266061 Peoples R China

In this paper, for the first time, the structure-preserving Gauss transformation is defined. Then by means of its real representation matrix, we present a novel structure-preserving algorithm for the LU decomposition of a quaternion matrix. Numerical experiments show that the structure-preserving algorithm is better than that in the newest quaternion toolbox for matlab (QTFM). (C) 2013 Elsevier B.V. All rights reserved.

关键词： Quaternion matrix LU decomposition structure-preserving algorithm

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A high-order structure-preserving difference scheme for generalized fractional Schrodinger equation with wave operator

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MATHEMATICS AND COMPUTERS IN SIMULATION 2023年第1期210卷 532-546页

作者： Zhang, Xi Ran, Maohua Liu, Yang Zhang, Li Sichuan Normal Univ Sch Math Sci Chengdu 610068 Peoples R China Sichuan Normal Univ VC & VR Key Lab Chengdu 610068 Peoples R China Aba Teachers Univ Sch Math Aba 623002 Peoples R China Sichuan Normal Univ VC & VR Key Lab Sichuan Prov Chengdu 610068 Peoples R China

This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrodinger equation with wave operator. A fourth-order energy-conserving difference scheme is developed for the resulting equivalent system based on scalar auxiliary variable approach. The discrete energy conservation law, boundedness and convergence of difference solutions are proved in detail. Numerical experiments are performed to verify our theoretical analysis results.& COPY;2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Fractional Schrodinger equation with wave operator Scalar auxiliary variable approach structure-preserving algorithm Energy-conserving scheme Boundedness Convergence

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Multi-conformal-symplectic PDEs and discretizations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 323卷 1-15页

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

Past work on integration methods that preserve a conformal symplectic structure focuses on Hamiltonian systems with weak linear damping. In this work, systems of PDEs that have conformal symplectic structure in time and space are considered, meaning conformal symplecticity is fully generalized for PDEs. Using multiple examples, it is shown that PDEs with this particular structure have interesting applications. What it means to preserve a multi-conformal-symplectic conseivation law numerically is explained, along with presentation of two numerical methods that preserve such properties. Then, the advantages of the methods are briefly explored through applications to linear equations, consideration of momentum and energy dissipation, and backward error analysis. Numerical simulations for two PDEs illustrate the properties of the methods, as well as the advantages over other standard methods. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Multi-symplectic PDE structure-preserving algorithm Conformal symplectic

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High-order integration scheme for relativistic charged particle motion in magnetized plasmas with volume preserving properties

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COMPUTER PHYSICS COMMUNICATIONS 2017年 220卷 285-296页

作者： Matsuyama, Akinobu Furukawa, Masaru Natl Inst Quantum & Radiol Sci & Technol Rokkasho Aomori 0393212 Japan Tottori Univ Grad Sch Engn Tottori Tottori 6808552 Japan

A numerical algorithm for solving full gyro orbit of relativistic charged particle motion in magnetized plasmas is presented. The algorithm developed here achieves the following features simultaneously. (1) The time advancement is explicit, and (2) the integration is performed with respect to the observation time in a laboratory frame. (3) It is suitable for accurate long time integration with its volume preserving properties in the phase space. (4) The algorithm can properly treat the E x B drift velocity in electromagnetic fields for large relativistic factors, and (5) can be extended to arbitrary high orders with the aid of symmetric composition methods. Because our algorithm is formulated in the Lorentz covariant form, explicit conservation of the Minkowski norm is not assumed. Nevertheless, no secular growth of the numerical errors in the norm does occur, and its influence can be minimized up to the levels of round-off errors when a high-order scheme is applied. Numerical results are compared with explicit and implicit Runge-Kutta methods. The numerical accuracy and the computational efficiency are discussed for long time integration in toroidal magnetic field configuration. (C) 2017 Elsevier B.V. All rights reserved.

关键词： structure-preserving algorithm Charged particle motion Hamiltonian mechanics Tokamaks

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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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Conformal conservation laws and geometric integration for damped Hamiltonian PDEs

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年第1期232卷 214-233页

作者： Moore, Brian E. Norena, Laura Schober, Constance M. Univ Cent Florida Dept Math Orlando FL 32816 USA

Conformal conservation laws are defined and derived for a class of multi-symplectic equations with added dissipation. In particular, the conservation laws of energy and momentum are considered, along with those that arise from linear symmetries. Numerical methods that preserve these conformal conservation laws are presented in detail, providing a framework for proving a numerical method exactly preserves the dissipative properties considered. The conformal methods are compared analytically and numerically to standard conservative methods, which includes a thorough inspection of numerical solution behavior for linear equations. Damped Klein-Gordon and sine-Gordon equations, and a damped nonlinear Schrodinger equation, are used as examples to demonstrate the results. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Multi-symplectic PDE Linear dissipation structure-preserving algorithm Preissman box scheme Discrete gradient methods

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Second Order Conformal Symplectic Schemes for Damped Hamiltonian Systems

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JOURNAL OF SCIENTIFIC COMPUTING 2016年第3期66卷 1234-1259页

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

Numerical methods for solving linearly damped Hamiltonian systems are constructed using the popular Stormer-Verlet and implicit midpoint methods. Each method is shown to preserve dissipation of symplecticity and dissipation of angular momentum of an N-body system with pairwise distance dependent interactions. Necessary and sufficient conditions for second order accuracy are derived. Analysis for linear equations gives explicit relationships between the damping parameter and the step size to reveal when the methods are most advantageous;essentially, the damping rate of the numerical solution is exactly preserved under these conditions. The methods are applied to several model problems, both ODEs and PDEs. Additional structure preservation is discovered for the discretized PDEs, in one case dissipation in total linear momentum and in another dissipation in mass are preserved by the methods. The numerical results, along with comparisons to standard Runge-Kutta methods and another structure-preserving method, demonstrate the usefulness and strengths of the methods.

关键词： Conformal symplectic Linear damping Birkhoffian method structure-preserving algorithm Dissipation preservation

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