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Journal articles on the topic 'Explicit symplectic integrator'

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Published: 8 February 2025

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1

Hu, Ai-Rong, and Guo-Qing Huang. "Application of Explicit Symplectic Integrators in the Magnetized Reissner–Nordström Spacetime." Symmetry 15, no. 5 (2023): 1094. http://dx.doi.org/10.3390/sym15051094.

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In recent works by Wu and Wang a class of explicit symplectic integrators in curved spacetimes was presented. Different splitting forms or appropriate choices of time-transformed Hamiltonians are determined based on specific Hamiltonian problems. As its application, we constructed a suitable explicit symplectic integrator for surveying the dynamics of test particles in a magnetized Reissner–Nordström spacetime. In addition to computational efficiency, the scheme exhibits good stability and high precision for long-term integration. From the global phase-space structure of Poincaré sections, the

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2

Tu, Xiongbiao, Qiao Wang, and Yifa Tang. "Highly Efficient Numerical Integrator for the Circular Restricted Three-Body Problem." Symmetry 14, no. 9 (2022): 1769. http://dx.doi.org/10.3390/sym14091769.

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The dynamic equation of a mass point in the circular restricted three-body problem is governed by Coriolis and centrifugal force, in addition to a co-rotating potential relative to the frame. In this paper, we provide an explicit, symmetric integrator for this problem. Such an integrator is more efficient than the symplectic Euler method and the Gauss Runge–Kutta method as regards this problem. In addition, we proved the integrator is symplectic by the discrete Hamilton’s principle. Several groups of numerical experiments demonstrated the precision and high efficiency of the integrator in the

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3

Nettesheim, Peter, Folkmar A. Bornemann, Burkhard Schmidt, and Christof Schütte. "An explicit and symplectic integrator for quantum-classical molecular dynamics." Chemical Physics Letters 256, no. 6 (1996): 581–88. http://dx.doi.org/10.1016/0009-2614(96)00471-x.

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4

Hu, Airong, and Guoqing Huang. "Chaos in a Magnetized Brane-World Spacetime Using Explicit Symplectic Integrators." Universe 8, no. 7 (2022): 369. http://dx.doi.org/10.3390/universe8070369.

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A brane-world metric with an external magnetic field is a modified theory of gravity. It is suitable for the description of compact sources on the brane such as stars and black holes. We design a class of explicit symplectic integrators for this spacetime and use one of the integrators to investigate how variations of the parameters affect the motion of test particles. When the magnetic field does not vanish, the integrability of the system is destroyed. Thus, the onset of chaos can be allowed under some circumstances. Chaos easily occurs when the electromagnetic parameter becomes large enough

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Huang, Zongqiang, Guoqing Huang, and Airong Hu. "Application of Explicit Symplectic Integrators in a Magnetized Deformed Schwarzschild Black Spacetime." Astrophysical Journal 925, no. 2 (2022): 158. http://dx.doi.org/10.3847/1538-4357/ac3edf.

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Abstract Following the latest work of Wu et al., we construct time-transformed explicit symplectic schemes for a Hamiltonian system on the description of charged particles moving around a deformed Schwarzschild black hole with an external magnetic field. Numerical tests show that such schemes have good performance in stabilizing energy errors without secular drift. Meantime, tangent vectors are solved from the variational equations of the system with the aid of an explicit symplectic integrator. The obtained tangent vectors are used to calculate several chaos indicators, including Lyapunov cha

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Wang 王, Long 龙. "New Insight of Time-transformed Symplectic Integrator. I. Hybrid Methods for Hierarchical Triples." Astrophysical Journal 978, no. 1 (2024): 65. https://doi.org/10.3847/1538-4357/ad98f3.

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Abstract Accurate N-body simulations of multiple systems such as binaries and triples are essential for understanding the formation and evolution of interacting binaries and binary mergers, including gravitational wave sources, blue stragglers, and X-ray binaries. The logarithmic time-transformed explicit symplectic integrator (LogH), also known as algorithmic regularization, is a state-of-the-art method for this purpose. However, we show that this method is accurate for isolated Kepler orbits because of its ability to trace Keplerian trajectories, but much less accurate for hierarchical tripl

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7

Wang, Long, Keigo Nitadori, and Junichiro Makino. "A slow-down time-transformed symplectic integrator for solving the few-body problem." Monthly Notices of the Royal Astronomical Society 493, no. 3 (2020): 3398–411. http://dx.doi.org/10.1093/mnras/staa480.

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ABSTRACT An accurate and efficient method dealing with the few-body dynamics is important for simulating collisional N-body systems like star clusters and to follow the formation and evolution of compact binaries. We describe such a method which combines the time-transformed explicit symplectic integrator and the slow-down method. The former conserves the Hamiltonian and the angular momentum for a long-term evolution, while the latter significantly reduces the computational cost for a weakly perturbed binary. In this work, the Hamilton equations of this algorithm are analysed in detail. We mat

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8

Pagliantini, Cecilia. "Dynamical reduced basis methods for Hamiltonian systems." Numerische Mathematik 148, no. 2 (2021): 409–48. http://dx.doi.org/10.1007/s00211-021-01211-w.

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AbstractWe consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by

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9

Cotter, Colin. "Data assimilation on the exponentially accurate slow manifold." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1991 (2013): 20120300. http://dx.doi.org/10.1098/rsta.2012.0300.

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I describe an approach to data assimilation making use of an explicit map that defines a coordinate system on the slow manifold in the semi-geostrophic scaling in Lagrangian coordinates, and apply the approach to a simple toy system that has previously been proposed as a low-dimensional model for the semi-geostrophic scaling. The method can be extended to Lagrangian particle methods such as Hamiltonian particle–mesh and smooth-particle hydrodynamics applied to the rotating shallow-water equations, and many of the properties will remain for more general Eulerian methods. Making use of Hamiltoni

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10

Sun, Xin, Xin Wu, Yu Wang, Chen Deng, Baorong Liu, and Enwei Liang. "Dynamics of Charged Particles Moving around Kerr Black Hole with Inductive Charge and External Magnetic Field." Universe 7, no. 11 (2021): 410. http://dx.doi.org/10.3390/universe7110410.

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We mainly focus on the effects of small changes of parameters on the dynamics of charged particles around Kerr black holes surrounded by an external magnetic field, which can be considered as a tidal environment. The radial motions of charged particles on the equatorial plane are studied via an effective potential. It is found that the particle energies at the local maxima values of the effective potentials increase with an increase in the black hole spin and the particle angular momenta, but decrease with an increase of one of the inductive charge parameter and magnetic field parameter. The r

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11

Lu, Junjie, and Xin Wu. "Effects of Two Quantum Correction Parameters on Chaotic Dynamics of Particles Near Renormalized Group Improved Schwarzschild Black Holes." Universe 10, no. 7 (2024): 277. http://dx.doi.org/10.3390/universe10070277.

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A renormalized group improved Schwarzschild black hole spacetime contains two quantum correction parameters. One parameter γ represents the identification of cutoff of the distance scale, and another parameter Ω stems from nonperturbative renormalization group theory. The two parameters are constrained by the data from the shadow of M87* central black hole. The dynamics of electrically charged test particles around the black hole are integrable. However, when the black hole is immersed in an external asymptotically uniform magnetic field, the dynamics are not integrable and may allow for the o

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12

Deng, Jian, Cristina Anton, and Yau Shu Wong. "High-Order Symplectic Schemes for Stochastic Hamiltonian Systems." Communications in Computational Physics 16, no. 1 (2014): 169–200. http://dx.doi.org/10.4208/cicp.311012.191113a.

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AbstractThe construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence

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13

Lu, Yulan, Junbin Yuan, Haoyang Tian, Zhengwei Qin, Siyuan Chen, and Hongji Zhou. "Explicit K-Symplectic and Symplectic-like Methods for Charged Particle System in General Magnetic Field." Symmetry 15, no. 6 (2023): 1146. http://dx.doi.org/10.3390/sym15061146.

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We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical and a non-canonical Hamiltonian system. If the three components of the magnetic field can be integrated in closed forms, we construct explicit K-symplectic methods for the non-canonical charged particle system; otherwise, explicit symplectic-like methods can be constructed for the canonical charged particle system. The symplectic-like methods are con

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14

Yonglei Fang and Qinghong Li. "A CLASS OF EXPLICIT RATIONAL SYMPLECTIC INTEGRATORS." Journal of Applied Analysis & Computation 2, no. 2 (2012): 161–71. http://dx.doi.org/10.11948/2012012.

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15

López-Marcos, M. A., J. M. Sanz-Serna, and Robert D. Skeel. "Explicit Symplectic Integrators Using Hessian--Vector Products." SIAM Journal on Scientific Computing 18, no. 1 (1997): 223–38. http://dx.doi.org/10.1137/s1064827595288085.

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16

Wu, Xin, Ying Wang, Wei Sun, Fu-Yao Liu, and Wen-Biao Han. "Explicit Symplectic Methods in Black Hole Spacetimes." Astrophysical Journal 940, no. 2 (2022): 166. http://dx.doi.org/10.3847/1538-4357/ac9c5d.

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Abstract Many Hamiltonian problems in the solar system are separable into two analytically solvable parts, and thus serve as a great chance to develop and apply explicit symplectic integrators based on operator splitting and composing. However, such constructions are not in general available for curved spacetimes in general relativity and modified theories of gravity because these curved spacetimes correspond to nonseparable Hamiltonians without the two-part splits. Recently, several black hole spacetimes such as the Schwarzschild black hole were found to allow for the construction of explicit

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17

Zhang, Hongxing, Naying Zhou, Wenfang Liu, and Xin Wu. "Charged Particle Motions near Non-Schwarzschild Black Holes with External Magnetic Fields in Modified Theories of Gravity." Universe 7, no. 12 (2021): 488. http://dx.doi.org/10.3390/universe7120488.

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A small deformation to the Schwarzschild metric controlled by four free parameters could be referred to as a nonspinning black hole solution in alternative theories of gravity. Since such a non-Schwarzschild metric can be changed into a Kerr-like black hole metric via a complex coordinate transformation, the recently proposed time-transformed, explicit symplectic integrators for the Kerr-type spacetimes are suitable for a Hamiltonian system describing the motion of charged particles around the non-Schwarzschild black hole surrounded with an external magnetic field. The obtained explicit symple

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18

Zhang, Lina, Xin Wu, and Enwei Liang. "Adjustment of Force–Gradient Operator in Symplectic Methods." Mathematics 9, no. 21 (2021): 2718. http://dx.doi.org/10.3390/math9212718.

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Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems H=K(p,q)+V(q) with integrable part K(p,q)=∑i=1n∑j=1naijpipj+∑i=1nbipi, where aij=aij(q) and bi=bi(q) are functions of coordinates q. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or

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19

Blanes, Sergio, and Arieh Iserles. "Explicit adaptive symplectic integrators for solving Hamiltonian systems." Celestial Mechanics and Dynamical Astronomy 114, no. 3 (2012): 297–317. http://dx.doi.org/10.1007/s10569-012-9441-z.

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20

Wu, Xin, Ying Wang, Wei Sun, Fuyao Liu, and Dazhu Ma. "Explicit Symplectic Integrators with Adaptive Time Steps in Curved Spacetimes." Astrophysical Journal Supplement Series 275, no. 2 (2024): 31. http://dx.doi.org/10.3847/1538-4365/ad8351.

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Abstract Recently, our group developed explicit symplectic methods for curved spacetimes that are not split into several explicitly integrable parts but are via appropriate time transformations. Such time-transformed explicit symplectic integrators should have employed adaptive time steps in principle, but they are often difficult in practical implementations. In fact, they work well if time transformation functions cause the time-transformed Hamiltonians to have the desired splits and approach 1 or constants for sufficiently large distances. However, they do not satisfy the requirement of ste

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21

Ahmad, Junaid, Yousaf Habib, Shafiq Rehman, Azqa Arif, Saba Shafiq, and Muhammad Younas. "Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems." Symmetry 11, no. 2 (2019): 142. http://dx.doi.org/10.3390/sym11020142.

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A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system confirming the efficiency of the approach is also provided with good energy conservation.

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22

VAN DE VYVER, HANS. "FOURTH ORDER SYMPLECTIC INTEGRATION WITH REDUCED PHASE ERROR." International Journal of Modern Physics C 19, no. 08 (2008): 1257–68. http://dx.doi.org/10.1142/s0129183108012844.

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In this paper we introduce a symplectic explicit RKN method for Hamiltonian systems with periodical solutions. The method has algebraic order four and phase-lag order six at a cost of four function evaluations per step. Numerical experiments show the relevance of the developed algorithm. It is found that the new method is much more efficient than the standard symplectic fourth-order method.

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23

Cary, J. R., and I. Doxas. "An Explicit Symplectic Integration Scheme for Plasma Simulations." Journal of Computational Physics 107, no. 1 (1993): 98–104. http://dx.doi.org/10.1006/jcph.1993.1127.

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24

VIGO-AGUIAR, JESÚS, T. E. SIMOS, and A. TOCINO. "AN ADAPTED SYMPLECTIC INTEGRATOR FOR HAMILTONIAN PROBLEMS." International Journal of Modern Physics C 12, no. 02 (2001): 225–34. http://dx.doi.org/10.1142/s0129183101001626.

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In this paper, a new procedure for deriving efficient symplectic integrators for Hamiltonian problems is introduced. This procedure is based on the combination of the trigonometric fitting technique and symplecticness conditions. Based on this procedure, a simple modified Runge–Kutta–Nyström second algebraic order trigonometrically fitted method is developed. We present explicity the symplecticity conditions for the new modified Runge–Kutta–Nyström method. Numerical results indicate that the new method is much more efficient than the "classical" symplectic Runge–Kutta–Nyström second algebraic

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25

Cheng, Jian Lian, and Tie Shuan Zhao. "Using Symplectic Schemes for Nonlinear Dynamic Analysis of Flexible Beams Undergoing Overall Motions." Advanced Materials Research 156-157 (October 2010): 854–61. http://dx.doi.org/10.4028/www.scientific.net/amr.156-157.854.

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In this paper, we developed a high-fidelity model to handle large overall motion of multi-flexible bodies. As a demonstration, the model is applied to a planar flexible beam system. An explicit expression of the kinetic energy is derived for the planar beams. The elastic strain energy is described via an accurate beam finite element formulation. The Hamilton equations are integrated by a symplectic integration scheme for enhanced accuracy and guaranteed numerical stability. The Hamilton and the corresponding Hamilton’s equations of beam vibration problems are formulated. It appears that the pr

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26

Zhou, Naying, Hongxing Zhang, Wenfang Liu, and Xin Wu. "A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes." Astrophysical Journal 927, no. 2 (2022): 160. http://dx.doi.org/10.3847/1538-4357/ac497f.

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Abstract In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic o

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27

Wang, Fang Zong, Yi Fan He, and Jing Ye. "Transient Stability Simulation by Explicit and Symplectic Runge-Kutta-Nyström Method." Advanced Materials Research 383-390 (November 2011): 1960–64. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.1960.

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The symplectic algorithm is a kind of new numerical integration methods. This paper proposes the application of the explicit and symplectic Runge-Kutta-Nyström method to solve the differential equations encountered in the power system transient stability simulation. The proposed method achieves significant improvement both in speed and in calculation precision as compared to the conventional Runge-Kutta method which is widely used for power system transient stability simulation. The proposed method is applied to the IEEE 145-bus system and the results are reported.

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28

Modin, K. "On explicit adaptive symplectic integration of separable Hamiltonian systems." Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 222, no. 4 (2008): 289–300. http://dx.doi.org/10.1243/14644193jmbd171.

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29

Zhu, Huajun, Songhe Song, and Yaming Chen. "Multi-Symplectic Wavelet Collocation Method for Maxwell’s Equations." Advances in Applied Mathematics and Mechanics 3, no. 6 (2011): 663–88. http://dx.doi.org/10.4208/aamm.11-m1183.

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AbstractIn this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell’s equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit

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30

Luo, Junjie, Weipeng Lin, and Lili Yang. "Explicit symplectic-like integration with corrected map for inseparable Hamiltonian." Monthly Notices of the Royal Astronomical Society 501, no. 1 (2020): 1511–19. http://dx.doi.org/10.1093/mnras/staa3745.

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ABSTRACT Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However

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31

Quispel, G. R. W., and D. I. McLaren. "Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows." Journal of Computational Physics 186, no. 1 (2003): 308–16. http://dx.doi.org/10.1016/s0021-9991(03)00068-8.

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32

Qing, Guang Hui, Liang Wang, and Li Zhong Shi. "Separable K-Canonical Formulation of Rectangular Element and Symplectic Integration Method for Analysis of Laminated Plates." Advanced Materials Research 194-196 (February 2011): 1496–505. http://dx.doi.org/10.4028/www.scientific.net/amr.194-196.1496.

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In the state space framework, a separable K-canonical formulation of rectangular element and explicit symplectic schemes for the static responses analysis of three-dimensional (3D) laminated plates are proposed in this paper. Firstly, the modified Hellinger-Reissner (H-R) variational principle for linear elastic solid is simply mentioned. Secondly, the separable J-canonical system with Hamiltonian H and the separable K-canonical formulation of rectangular element are constructed. Thirdly, on the basis of the symplectic difference schemes, the explicit symplectic schemes are employed to solve t

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33

Duruisseaux, Valentin, and Melvin Leok. "Time-adaptive Lagrangian variational integrators for accelerated optimization." Journal of Geometric Mechanics 15, no. 1 (2023): 224–55. http://dx.doi.org/10.3934/jgm.2023010.

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<abstract><p>A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> and <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable ti

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Sasa, Narimasa. "Momentum Conservation Law in Explicit Symplectic Integrators for Nonlinear Wave Equations." Journal of the Physical Society of Japan 83, no. 5 (2014): 054004. http://dx.doi.org/10.7566/jpsj.83.054004.

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Li, Dan, and Xin Wu. "Modification of logarithmic Hamiltonians and application of explicit symplectic-like integrators." Monthly Notices of the Royal Astronomical Society 469, no. 3 (2017): 3031–41. http://dx.doi.org/10.1093/mnras/stx1059.

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Luo, Junjie, Xin Wu, Guoqing Huang, and Fuyao Liu. "EXPLICIT SYMPLECTIC-LIKE INTEGRATORS WITH MIDPOINT PERMUTATIONS FOR SPINNING COMPACT BINARIES." Astrophysical Journal 834, no. 1 (2017): 64. http://dx.doi.org/10.3847/1538-4357/834/1/64.

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He, Xijun, Dinghui Yang, Xiao Ma, and Yanjie Zhou. "Symplectic interior penalty discontinuous Galerkin method for solving the seismic scalar wave equation." GEOPHYSICS 84, no. 3 (2019): T133—T145. http://dx.doi.org/10.1190/geo2018-0492.1.

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To improve the computational accuracy and efficiency of long-time wavefield simulations, we have developed a so-called symplectic interior penalty discontinuous Galerkin (IPDG) method for 2D acoustic equation. For the symplectic IPDG method, the scalar wave equation is first transformed into a Hamiltonian system. Then, the high-order IPDG formulations are introduced for spatial discretization because of their high accuracy and ease of dealing with computational domains with complex boundaries. The time integration is performed using an explicit third-order symplectic partitioned Runge-Kutta sc

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Wang, Ying, Wei Sun, Fuyao Liu, and Xin Wu. "Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes." Astrophysical Journal 907, no. 2 (2021): 66. http://dx.doi.org/10.3847/1538-4357/abcb8d.

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Wu, Xin, Ying Wang, Wei Sun, and Fuyao Liu. "Construction of Explicit Symplectic Integrators in General Relativity. IV. Kerr Black Holes." Astrophysical Journal 914, no. 1 (2021): 63. http://dx.doi.org/10.3847/1538-4357/abfc45.

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40

Tao, Molei. "Explicit high-order symplectic integrators for charged particles in general electromagnetic fields." Journal of Computational Physics 327 (December 2016): 245–51. http://dx.doi.org/10.1016/j.jcp.2016.09.047.

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Shang, Xiaocheng, and Hans Christian Öttinger. "Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2234 (2020): 20190446. http://dx.doi.org/10.1098/rspa.2019.0446.

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We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated

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Oh, Thae, Ji Choe, and Jin Kim. "Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order." Engineering Mathematics 7, no. 1 (2024): 19–28. http://dx.doi.org/10.11648/j.engmath.20230701.12.

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The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. Since Hamiltonian systems have good properties such as symplecticity, numerical methods that preserve these properties have attracted the great attention. In fact, the explicit Runge-Kutta methods have used due to that schemes are very simple and its computational amounts are very small. However, the explicit schemes aren’t stable so the implicit Runge-Kutta methods have widely studied. Among those implicit schemes, symplectic numerical methods were interested. It is because it has preserve

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Zhang, Lina, Wenfang Liu, and Xin Wu. "Study of Chaos in Rotating Galaxies Using Extended Force-Gradient Symplectic Methods." Symmetry 15, no. 1 (2022): 63. http://dx.doi.org/10.3390/sym15010063.

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We take into account the dynamics of three types of models of rotating galaxies in polar coordinates in a rotating frame. Due to non-axisymmetric potential perturbations, the angular momentum varies with time, and the kinetic energy depends on the momenta and spatial coordinate. The existing explicit force-gradient symplectic integrators are not applicable to such Hamiltonian problems, but the recently extended force-gradient symplectic methods proposed in previous work are. Numerical comparisons show that the extended force-gradient fourth-order symplectic method with symmetry is superior to

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He, Guandong, Guoqing Huang, and Airong Hu. "Application of Symmetric Explicit Symplectic Integrators in Non-Rotating Konoplya and Zhidenko Black Hole Spacetime." Symmetry 15, no. 10 (2023): 1848. http://dx.doi.org/10.3390/sym15101848.

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In this study, we construct symmetric explicit symplectic schemes for the non-rotating Konoplya and Zhidenko black hole spacetime that effectively maintain the stability of energy errors and solve the tangent vectors from the equations of motion and the variational equations of the system. The fast Lyapunov indicators and Poincaré section are calculated to verify the effectiveness of the smaller alignment index. Meanwhile, different algorithms are used to separately calculate the equations of motion and variation equations, resulting in correspondingly smaller alignment indexes. The numerical

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Wang, Ying, Wei Sun, Fuyao Liu, and Xin Wu. "Construction of Explicit Symplectic Integrators in General Relativity. II. Reissner–Nordström Black Holes." Astrophysical Journal 909, no. 1 (2021): 22. http://dx.doi.org/10.3847/1538-4357/abd701.

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Sasa, Narimasa. "Momentum Conservation Law in Explicit Symplectic Integrators for a Nonlinear Schrödinger-Type Equation." Journal of the Physical Society of Japan 82, no. 5 (2013): 053001. http://dx.doi.org/10.7566/jpsj.82.053001.

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Liu, Caiyu, and Xin Wu. "Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory." Universe 9, no. 8 (2023): 365. http://dx.doi.org/10.3390/universe9080365.

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There are two free coupling parameters c13 and c14 in the Einstein–Æther metric describing a non-rotating black hole. This metric is the Reissner–Nordström black hole solution when 0≤2c13<c14<2, but it is not for 0≤c14<2c13<2. When the black hole is immersed in an external asymptotically uniform magnetic field, the Hamiltonian system describing the motion of charged particles around the black hole is not integrable; however, the Hamiltonian allows for the construction of explicit symplectic integrators. The proposed fourth-order explicit symplectic scheme is used to investigate the

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Yang, Daqi, Wenfu Cao, Naying Zhou, Hongxing Zhang, Wenfang Liu, and Xin Wu. "Chaos in a Magnetized Modified Gravity Schwarzschild Spacetime." Universe 8, no. 6 (2022): 320. http://dx.doi.org/10.3390/universe8060320.

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Based on the scalar–tensor–vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is stat

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Gray, Stephen K., Donald W. Noid, and Bobby G. Sumpter. "Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods." Journal of Chemical Physics 101, no. 5 (1994): 4062–72. http://dx.doi.org/10.1063/1.467523.

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Tamayo, Daniel, Hanno Rein, Pengshuai Shi, and David M. Hernandez. "REBOUNDx: a library for adding conservative and dissipative forces to otherwise symplectic N-body integrations." Monthly Notices of the Royal Astronomical Society 491, no. 2 (2019): 2885–901. http://dx.doi.org/10.1093/mnras/stz2870.

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ABSTRACT Symplectic methods, in particular the Wisdom–Holman map, have revolutionized our ability to model the long-term, conservative dynamics of planetary systems. However, many astrophysically important effects are dissipative. The consequences of incorporating such forces into otherwise symplectic schemes are not always clear. We show that moving to a general framework of non-commutative operators (dissipative or not) clarifies many of these questions, and that several important properties of symplectic schemes carry over to the general case. In particular, we show that explicit splitting

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