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Local structure-preserving algorithms for the nonlinear Schrödinger equation with power law nonlinearity

APPLIED MATHEMATICS AND COMPUTATION 2025年 484卷

作者： Luo, Fangwen Tang, Qiong Huang, Yiting Ding, Yanhui Tang, Sijia Hunan Univ Technol Sch Sci Zhuzhou 412007 Hunan Peoples R China

This paper introduces three local structure-preserving algorithms for the one-dimensional nonlinear Schr & ouml;dinger equation with power law nonlinearity, comprising two local energy- conserving algorithms and one local momentum-conserving algorithm. Additionally, we extend these local conservation algorithms to achieve global conservation under periodic boundary conditions. Theoretical analyses confirm the conservation properties of these algorithms. In numerical experiments, we validate the advantages of these algorithms in maintaining longterm energy or momentum conservation by comparing them with a multi-symplectic Preissman algorithm.

关键词： Nonlinear Schr & ouml dinger equation Power law nonlinearity structure-preserving algorithms Local conservation Global conservation

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structure-preserving algorithms for the two-dimensional fractional Klein-Gordon-Schrodinger equation

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APPLIED NUMERICAL MATHEMATICS 2020年 156卷 77-93页

作者： Fu, Yayun Cai, Wenjun Wang, Yushun Nanjing Normal Univ Jiangsu Collaborat Innovat Ctr Biomedial Funct Ma Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210023 Peoples R China

This paper aims to construct structure-preserving numerical schemes for the two-dimensional space fractional Klein-Gordon-Schrodinger equation, which are based on the newly developed partitioned averaged vector field methods. First, we derive an equivalent equation, and reformulate the equation as an infinite dimensional canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we use the Fourier pseudo-spectral method to discrete the equation in space direction and obtain a semi-discrete conservative system, which can be reformulated a finite dimensional canonical Hamiltonian system. Further applying the partitioned averaged vector field methods to the semi-discrete system gives a class of fully-discrete schemes that can preserve the mass and energy exactly. Numerical examples are provided to confirm our theoretical analysis results at last. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： structure-preserving algorithms Fractional Klein-Gordon-Schrodinger equation Hamiltonian system Partitioned averaged vector field methods

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structure-preserving algorithms for Birkhoffian systems

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JOURNAL OF GEOMETRY AND PHYSICS 2012年第5期62卷 1157-1166页

作者： Kong, Xinlei Wu, Huibin Mei, Fengxiang Beijing Inst Technol Sch Math Beijing 100081 Peoples R China Beijing Inst Technol Sch Aerosp Engn Beijing 100081 Peoples R China

We propose a new approach to construct structure-preserving algorithms for Birkhoffian systems. First, the Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Then, taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhoffian systems. Simulation results of the given example indicate that structure-preserving algorithms obtained by this method have great advantage in conserving conserved quantities. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Birkhoffian system Discrete Birkhoffian equations structure-preserving algorithms Variational integrator

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A Class of Meshless structure-preserving algorithms for the Nonlinear Schrödinger Equation

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COMPUTATIONAL METHODS IN APPLIED MATHEMATICS 2024年

作者： Wang, Jialing Zhou, Zhengting Lin, Zhoujin Nanjing Univ Informat Sci & Technol Sch Math & Stat Nanjing 210044 Peoples R China

This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d-dimensional ( d = 1 {d=1} or 2) nonlinear Schr & ouml;dinger equation. Based on the method of lines, we first derive a finite-dimensional Hamiltonian system by using the radial basis function method of the quasi-interpolation and the technique of left-multiplying a diagonal matrix to discretize the space direction. Then suitable geometric numerical integrations can be used to discretize the time direction, which yields a class of meshless structure-preserving algorithms. In addition to the construction, the structure-preserving properties and their proofs are also provided in detail. Besides the uniform and nonuniform grids, the numerical experiments on the random grids are also emphasized to verify the theoretical research well, which is of great significance for scattering points based on the characteristics of actual problems.

关键词： The Nonlinear Schr & ouml dinger Equation Meshless Scheme Geometric Numerical Integrations structure-preserving algorithms Random Grids

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Explicit high-order structure-preserving algorithms for the two-dimensional fractional nonlinear Schrodinger equation

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2022年第5期99卷 877-894页

作者： Fu, Yayun Shi, Yanhua Zhao, Yanmin Xuchang Univ Sch Sci Xuchang 461000 Peoples R China

The paper aims to construct a class of high-order explicit conservative schemes for the space fractional nonlinear Schrodinger equation by combing the invariant energy quadratization method and Runge-Kutta method. We first derive the Hamiltonian formulation of the equation, and obtain a new equivalent system via introducing a scalar variable. Then, we propose a semi-discrete conservative system by using the Fourier pseudo-spectral method to approximate the equivalent system in space. Further applying the fourth-order modified Runge-Kutta method to the semi-discrete system gives two classes of schemes for the equation. One scheme preserves the energy while the other scheme conserves the mass. Numerical experiments are provided to demonstrate the conservative properties, convergence orders and long time stability of the proposed schemes.

关键词： Fractional nonlinear Schrodinger equation structure-preserving algorithms invariant energy quadratization Runge-Kutta method explicit conservative schemes

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LOCAL structure-preserving algorithms FOR THE KDV EQUATION

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Journal of Computational Mathematics 2017年第3期35卷 289-318页

作者： Jialing Wang Yushun Wang Jiangsu Key Laboratory of NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing 210023 China

In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplectic algorithms, eight local energy-conserving algo- rithms and eight local momentum-conserving algorithms. Among these algorithms, some have been discussed and widely used while the most are new. The outstanding advantage of these proposed algorithms is that they conserve the local structures in any time-space re- gion exactly. Therefore, the local structure-preserving algorithms overcome the restriction of global structure-preserving algorithms on the boundary conditions. Numerical experiments are conducted to show the performance of the proposed methods. Moreover, the unified framework can be easily applied to many other equations.

关键词： Korteweg-de Vries （KdV） equation structure-preserving algorithms Concate-nating method Multi-symplectic conservation law.

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Three sixth-order explicit symplectic Runge-Kutta-Nystrom methods with exact parameters

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RESULTS IN APPLIED MATHEMATICS 2025年 26卷

作者： Pan, Mengjiao Zhang, Jingjing Zhang, Shangyou East China Jiaotong Univ Sch Sci Nanchang 330013 Peoples R China Univ Delaware Dept Math Sci Newark DE 19716 USA

For separable Hamiltonian systems, we construct first ever three symplectic explicit Runge-Kutta-Nystr & ouml;m methods of six orders with exact parameters. We numerically and theoretically compare these three new exact methods with the only existing exact 6-th order symplectic explicit partitioned Runge-Kutta method and two approximate 6-th order symplectic explicit Runge-Kutta-Nystr & ouml;m methods. These new methods are more accurate and stable.

关键词： Symplectic schemes structure-preserving algorithms Explicit runge-kutta-nystr & ouml m method Geometric numerical integration

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Novel improved multidimensional Stormer-Verlet formulas with applications to four aspects in scientific computation

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MATHEMATICAL AND COMPUTER MODELLING 2013年第3-4期57卷 857-872页

作者： Wang, Bin Wu, Xinyuan Zhao, Hua Nanjing Univ Dept Math State Key Lab Novel Software Technol Nanjing 210093 Jiangsu Peoples R China Univ Cambridge Dept Appl Math & Theoret Phys Cambridge CB3 0WA England Beijing Inst Tracking & Tele Commun Technol Beijing 100094 Peoples R China

This paper presents two novel improved multidimensional Stormer-Verlet formulas with four applications to time-independent Schrodinger equations, wave equations, orbital problems and the problem of Fermi, Pasta & Ulam. For solving the system of second-order ordinary differential equations y '' + My = f (t, y) with M is an element of R-mxm, the multidimensional ARKN methods (adapted Runge-Kutta-Nystrom methods) were formulated by Wu et al. (2009) [1]. Very recently, the multidimensional ERKN methods (extended Runge-Kutta-Nystrom methods) were proposed by Wu et al. (2010) [26]. Both the ARKN methods and the ERKN methods perform numerically much better than the classical Runge-Kutta-Nystrom methods due to the use of the special structure of the equation brought by the linear term My. Based on the two kinds of multidimensional schemes, we derive two novel improved multidimensional Stormer-Verlet formulas, which are shown to be symplectic and of order two. Each new formula is a blend of existing trigonometric integrators and symplectic integrators. Meantime, the symplecticity conditions for the one-stage explicit multidimensional ARKN methods are presented. Stability and phase properties of the two improved formulas are analyzed. Numerical experiments demonstrate that the two improved multidimensional Stormer-Verlet formulas are more efficient than the classical Stormer-Verlet formula and the two other improved Stormer-Verlet methods appeared in the literature. In particular, when applied to a Hamiltonian system, the two symplectic improved multidimensional Stormer-Verlet formulas preserve well the Hamiltonian in the sense of numerical approximation, and have better accuracy than the classical Stormer-Verlet formula and the two other improved Stormer-Verlet methods with the same computational cost. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Schrodinger equations Wave equations Orbital problems Fermi-Pasta-Ulam problem structure-preserving algorithms Improved multidimensional Stormer-Verlet formulas

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Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

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CALCOLO 2017年第1期54卷 117-140页

作者： Wang, Bin Yang, Hongli Meng, Fanwei Qufu Normal Univ Sch Math Sci Qufu 273165 Shandong Peoples R China Nanjing Inst Technol Dept Math & Phys Nanjing 211167 Jiangsu Peoples R China

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge-Kutta-Nystrom (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge-Kutta-Nystrom methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.

关键词： Symplectic and symmetric explicit schemes Sixth-order extended Runge-Kutta-Nystrom schemes Multi-frequency oscillatory nonlinear Hamiltonian equations structure-preserving algorithms

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The accurate particle tracer code

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COMPUTER PHYSICS COMMUNICATIONS 2017年 220卷 212-229页

作者： Wang, Yulei Liu, Jian Qin, Hong Yu, Zhi Yao, Yicun Univ Sci & Technol China Sch Nucl Sci & Technol Hefei 230026 Anhui Peoples R China Univ Sci & Technol China Dept Modern Phys Hefei 230026 Anhui Peoples R China Princeton Univ Plasma Phys Lab POB 451 Princeton NJ 08543 USA Chinese Acad Sci Inst Plasma Phys Theory & Simulat Div Hefei 230031 Anhui Peoples R China

The Accurate Particle Tracer (APT) code is designed for systematic large-scale applications of geometric algorithms for particle dynamical simulations. Based on a large variety of advanced geometric algorithms, APT possesses long-term numerical accuracy and stability, which are critical for solving multi-scale and nonlinear problems. To provide a flexible and convenient I/O interface, the libraries of Lua and Hdf5 are used. Following a three-step procedure, users can efficiently extend the libraries of electromagnetic configurations, external non-electromagnetic forces, particle pushers, and initialization approaches by use of the extendible module. APT has been used in simulations of key physical problems, such as runaway electrons in tokamaks and energetic particles in Van Allen belt. As an important realization, the APT SW version has been successfully distributed on the world's fastest computer, the Sunway TaihuLight supercomputer, by supporting master slave architecture of Sunway many-core processors. Based on large-scale simulations of a runaway beam under parameters of the ITER tokamak, it is revealed that the magnetic ripple field can disperse the pitch-angle distribution significantly and improve the confinement of energetic runaway beam on the same time. (C) 2017 Published by Elsevier B.V.

关键词： structure-preserving algorithms Plasma simulation Multi-timescale dynamics Large-scale simulation

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