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structure-preserving algorithm and Its Error Estimate for the Relativistic Charged-Particle Dynamics Under the Strong Magnetic Field

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JOURNAL OF SCIENTIFIC COMPUTING 2024年第3期100卷 70页

作者： Zhang, Ruili Liu, Tong Wang, Bin Liu, Jian Tang, Yifa Beijing Jiaotong Univ Sch Math & Stat Beijing 100044 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Shandong Univ SDU ANU Joint Sci Coll Weihai 264209 Peoples R China Chinese Acad Sci LSEC ICMSEC Acad Math & Syst Sci Beijing 100190 Peoples R China

This paper investigates the numerical algorithm and its error estimates for the dynamics of relativistic charged particles under a strong maximal ordering scaling magnetic field. To maintain the fundamental principles of relativistic dynamics, including energy conservation, volume preservation, and the Lorentz invariant property, we construct a structure-preserving algorithm using the splitting scheme. This algorithm ensures the preservation of volume, energy, and the Lorentz invariant property (VELPA) exactly. Specifically, we establish an uniform and optimal error bound in both 4-position and 4-velocity for VELPA. Numerical experiments are also presented to demonstrate the advantages of VELPA in both uniform error estimate and conservation of energy, compared to the implicit Euler method and traditional energy-preserving AVF method.

关键词： structure-preserving algorithm Strong magnetic field Relativistic charged-particle dynamics Splitting scheme Error estimate

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A structure-preserving algorithm for time-scale non-shiftedHamiltonian systems

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Theoretical & Applied Mechanics Letters 2022年第5期12卷 349-358页

作者： Xue Tian Yi Zhang School of Science Nanjing University of Science and TechnologyNanjing210094JiangsuChina School of Mathematics and Big Data Anhui University of Science and TechnologyHuainan232001AnhuiChina College of Civil Engineering Suzhou University of Science and TechnologySuzhou215011JiangsuChina

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational *** only can the combination ofand∇derivatives be beneficial to obtaining higher convergence order in numerical analysis,but also it prompts the timescale numerical computational scheme to have good properties,for instance,*** this letter,a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is *** using the time-scale discrete variational method and calculus theory,and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems,the corresponding discrete Hamiltonian principle can be ***,the time-scale discrete Hamilton difference equations,Noether theorem,and the symplectic scheme of discrete Hamiltonian systems are ***,taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples,they show that the time-scale discrete variational method is a structure-preserving *** new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.

关键词： Time-scale non-shifted system Hamiltonian system structure-preserving algorithm Noether conserved quantity

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Construction and Analysis of structure-preserving Numerical algorithm for Two-Dimensional Damped Nonlinear Space Fractional Schrödinger equation

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JOURNAL OF SCIENTIFIC COMPUTING 2024年第3期99卷 60-60页

作者： Ding, Hengfei Qu, Haidong Yi, Qian Guangxi Normal Univ Ctr Appl Math Guangxi Sch Math & Stat Guilin 541006 Peoples R China Guangxi Normal Univ Educ Dept Guangxi Zhuang Autonomous Reg Key Lab Math Model & Applicat Guilin 514006 Peoples R China Hanshan Normal Univ Dept Math Chaozhou 521041 Peoples R China Guangxi Normal Univ Sch Math & Stat Guilin 541006 Peoples R China

In this paper, we present a novel high-order structure-preserving numerical scheme for solving the damped nonlinear space fractional Schrodinger equation (DNSFSE) in two spatial dimensions. The main idea of constructing new algorithm consists of two parts. Firstly, we introduce an auxiliary exponential variable to transform the original DNSFSE into a modified one. The modified DNSFSE subjects to the conservation of mass and energy, which is crucial to develop structure-preserving numerical schemes. Secondly, we construct a high-order numerical differential formula to approximate the Riesz derivative in space, which contributes to a semi-discrete difference scheme for the modified DNSFSE. Combining the semi-discrete scheme with the variant Crank-Nicolson method in time, we can obtain the fully-discrete difference scheme for solving the modified DNSFSE. The advantage of the proposed scheme is that a fourth-order convergent accuracy can be achieved in space while maintaining the conservation of mass and energy. Subsequently, we conduct a detailed study on the boundedness, uniqueness, and convergence of solution for fully-discrete scheme. Furthermore, an improved efficient iterative algorithm is proposed for the fully-discrete scheme, which has the advantage of maintaining the same convergence order as the original difference scheme. Finally, extensive numerical results are reported to further verify the correctness of theoretical analysis and the effectiveness of the proposed numerical algorithm.

关键词： structure-preserving algorithm Numerical differential formula The damped nonlinear space fractional Schrodinger equations Iterative algorithm

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A complex structure-preserving algorithm for split quaternion matrix LDU decomposition in split quaternion mechanics

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CALCOLO 2021年第3期58卷 34-34页

作者： Wang, Gang Jiang, Tongsong Guo, Zhenwei Zhang, Dong North Eastern Fed Univ Inst Math & Informat Sci Yakutsk 677000 Russia Heze Univ Sch Math & Stat Heze 274015 Shandong Peoples R China Liaocheng Univ Sch Math Sci Liaocheng 252000 Shandong Peoples R China Shandong Univ Sci & Technol Coll Math & Syst Sci Qingdao 266590 Shandong Peoples R China

Matrix decompositions play a prominent role in the theoretical study and numerical calculation of split quaternion mechanics. This paper, by means of a complex representation of split quaternion matrices, introduces Gaussian elimination of split quaternion matrices, and obtains a complex structure-preserving algorithm for split quaternion matrix LDU decomposition. Numerical examples show that the complex structure-preserving algorithm is more efficient.

关键词： Split quaternion matrix Complex representation structure-preserving algorithm LDU decomposition

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A structure-preserving Jacobi algorithm for quaternion Hermitian eigenvalue problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2018年第3期75卷 809-820页

作者： Ma, Ru-Ru Jia, Zhi-Gang Bai, Zheng-Jian Xiamen Univ Sch Math Sci Xiamen 361005 Peoples R China Jiangsu Normal Univ Sch Math & Stat Xuzhou 221116 Peoples R China

A new real structure-preserving Jacobi algorithm is proposed for solving the eigenvalue problem of quaternion Hermitian matrix. By employing the generalized JRS-symplectic Jacobi rotations, the new quaternion Jacobi algorithm can preserve the symmetry and JRS-symmetry of the real counterpart of quaternion Hermitian matrix. Moreover, the proposed algorithm only includes real operations without dimension-expanding and is generally superior to the state-of-the-art algorithm. Numerical experiments are reported to indicate its efficiency and accuracy. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Real counterpart Quaternion Hermitian eigenvalue problem Jacobi rotation structure-preserving algorithm

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structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

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Plasma Science and Technology 2024年第6期26卷 88-102页

作者：张若涵王正汹肖建元王丰 Key Laboratory of Materials Modification by Laser Ion and Electron BeamsSchool of PhysicsDalian University of TechnologyDalian 116024People’s Republic of China Department of Plasma Physics and Fusion Engineering University of Science and Technology of ChinaHefei 230026People’s Republic of China

The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

关键词： structure-preserving algorithm averaged vector field classical Pauli particle guiding center dynamics

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New structure-preserving algorithms of Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems

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NUMERICAL algorithmS 2024年第3期95卷 1309-1323页

作者： Ding, Wenxv Liu, Zhihong Li, Ying Wei, Anli Zhang, Mingcui Liaocheng Univ Coll Math Sci Liaocheng 252000 Peoples R China

In this paper, we study the Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems Ax = b and obtain the structure-preserving algorithms of Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems Ax = b. The convergence and computational cost of these iteration methods are discussed. Numerical examples are given to demonstrate the efficiency of structure-preserving algorithms of Gauss-Seidel iteration and successive over-relaxation iteration methods. As an application, we apply two kinds of structure preserving iterative algorithms to solve elliptic biquaternion linear systems Ax = b.

关键词： Quaternion linear systems Gauss-Seidel iteration Successive over-relaxation iteration structure-preserving algorithm

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H∞ norm computation of linear continuous-time periodic systems by a structure-preserving algorithm

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INTERNATIONAL JOURNAL OF CONTROL 2014年第1期87卷 131-142页

作者： Peng, Haijun Wu, Zhigang Zhong, Wanxie Dalian Univ Technol Dept Engn Mech State Key Lab Struct Anal Ind Equipment Dalian 116024 Peoples R China Dalian Univ Technol Sch Aeronaut & Astronaut Dalian 116024 Peoples R China

A new reliable structure-preserving algorithm for computing H norm of linear continuous-time periodic systems is proposed in this paper. In the computation of the H norm, no Riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic Hamiltonian system will be evaluated. First, the monodromy matrix is expressed as the product of state transition matrices of the Hamiltonian system. Second, these state transition matrices, which have been proved to be symplectic matrices, are evaluated by a structure-preserving Magnus series method. Then, in order to preserve the standard symplectic form of the monodromy matrix, the structure-preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix. At last, the effectiveness and the high accuracy of the proposed structure-preserving algorithm are demonstrated by numerical examples.

关键词： H-infinity norm continuous-time periodic system monodromy matrix Hamiltonian system structure-preserving algorithm

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Auxiliary relaxation method to derive thermodynamically consistent phase field models with constraints and structure preserving numerical approximations

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JOURNAL OF COMPUTATIONAL PHYSICS 2025年 522卷

作者： Hong, Qi Zhang, Zengyan Zhao, Jia Nanjing Univ Aeronaut & Astronaut Sch Math Nanjing 211106 Peoples R China MIIT Key Lab Math Modelling & High Performance Comp Air Nanjing 211106 Peoples R China Nanjing Normal Univ Key Lab Numer Simulat Large Scale Complex Syst Minist Educ Nanjing 210023 Peoples R China Binghamton Univ Dept Math & Stat Binghamton NY 13850 USA

In this paper, we introduce a novel approach for formulating phase field models with constraints. The main idea is to introduce auxiliary variables that regularize and gradually dissipate constraint deviations of the phase variables, which we name the auxiliary relaxation method. It integrates seamlessly with the energy variational framework to ensure thermodynamic consistency in the resulting phase field models. Unlike traditional penalty methods, which introduce high stiffness due to large penalty parameters to enforce constraints in phase field models, our approach reduces system stiffness, allowing larger time step sizes when solving phase field models with constraints numerically, thus improving numerical accuracy and efficiency. We demonstrate the effectiveness and robustness of the proposed auxiliary relaxation method by applying it across several scenarios to derive thermodynamically consistent phase field models with constraints. Furthermore, we introduce a general second-order implicit-explicit Crank-Nicolson scheme, combining the relaxed scalar auxiliary variable method with a stabilization technique to solve these models. Through extensive numerical tests, we validate the capability of our modeling and numerical framework to reliably simulate complex dynamics governed by phase field equations with constraints.

关键词： Onsager principle Thermodynamic consistency Phase field model Auxiliary relaxation method structure-preserving algorithm Energy stable

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structure-preserving exponential time differencing methods for modeling Josephson Junctions

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APPLIED MATHEMATICS LETTERS 2025年 162卷

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

Explicit, conformal symplectic, exponential time differencing (ETD) methods have numerous advantages over other well-known and commonly used methods, including structure-preservation, high stability, ease of implementation, and computational efficiency. Such methods are constructed with second and fourth order accuracy through composition techniques using a simple first order scheme. For modeling Josephson Junctions, these ETD schemes regularly exhibit the best balance of efficiency and accuracy when compared to other commonly used methods.

关键词： Conformal symplectic structure-preserving algorithm Time-dependent damping

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