Статті в журналах: "Explicit symplectic integrator"

1

Hu, Ai-Rong, and Guo-Qing Huang. "Application of Explicit Symplectic Integrators in the Magnetized Reissner–Nordström Spacetime." Symmetry 15, no. 5 (May 16, 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 extent of chaos can be strengthened when energy E, magnetic parameter B, or the charge q become larger. On the contrary, the occurrence of chaoticity is weakened with an increase of electric parameter Q and angular momentum L. The conclusion can also be supported by fast Lyapunov indicators.

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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 (August 25, 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 examples of the quadratic potential and the bounded orbits in the circular restricted three-body problem.

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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 (July 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 (July 4, 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. Dark matter acts as a gravitational force, so that chaotic motion can become more obvious as dark matter increases. The gravity of the black hole is weakened with an increasing positive cosmological parameter; therefore, the extent of chaos can be also strengthened. The proposed symplectic integrator is applied to a ray-tracing method and the study of such chaotic dynamics will be a possible reference for future studies of brane-world black hole shadows with chaotic patterns of self-similar fractal structures based on the Event Horizon Telescope data for M87* and Sagittarius A*.

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5

Huang, Zongqiang, Guoqing Huang, and Airong Hu. "Application of Explicit Symplectic Integrators in a Magnetized Deformed Schwarzschild Black Spacetime." Astrophysical Journal 925, no. 2 (February 1, 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 characteristic exponents, fast Lyapunov indicators, a smaller alignment index, and a generalized alignment index. It is found that the smaller alignment index and generalized alignment index are the fastest indicators for distinguishing between regular and chaotic cases. The smaller alignment index is applied to explore the effects of the parameters on the dynamical transition from order to chaos. When the positive deformation factor and angular momentum decrease, or when the energy, positive magnetic parameter, and the magnitude of the negative deformation parameter increase, chaos easily occurs for the appropriate choices of initial conditions and the other parameters.

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6

Wang 王, Long 龙. "New Insight of Time-transformed Symplectic Integrator. I. Hybrid Methods for Hierarchical Triples." Astrophysical Journal 978, no. 1 (December 26, 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 triple systems. The method can lead to an unphysical secular evolution of inner eccentricity in Kozal–Lidov triples, despite a small energy error. We demonstrate that hybrid methods, which apply LogH to the inner binary and alternative methods to the outer bodies, are significantly more effective, though not symplectic. Additionally, we introduce a more efficient hybrid method, BlogH, which eliminates the need for time synchronization and is time symmetric. The method is implemented in the few-body code SDAR. We explore suitable criteria for switching between the LogH and BlogH methods for general triple systems. These hybrid methods have the potential to enhance the integration performance of hierarchical triples.

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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 (February 19, 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 mathematically and numerically show that it can correctly reproduce the secular evolution like the orbit averaged method and also well conserve the angular momentum. For a weakly perturbed binary, the method is possible to provide a few orders of magnitude faster performance than the classical algorithm. A publicly available code written in the c++ language, sdar, is available on github. It can be used either as a standalone tool or a library to be plugged in other N-body codes. The high precision of the floating point to 62 digits is also supported.

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8

Pagliantini, Cecilia. "Dynamical reduced basis methods for Hamiltonian systems." Numerische Mathematik 148, no. 2 (June 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 a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.

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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 (May 28, 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 Hamiltonian normal-form theory, it has previously been shown that, if initial conditions for the system are chosen as image points of the map, then the fast components of the system have exponentially small magnitude for exponentially long times as ϵ →0, and this property is preserved if one uses a symplectic integrator for the numerical time stepping. The map may then be used to parametrize initial conditions near the slow manifold, allowing data assimilation to be performed without introducing any fast degrees of motion (more generally, the precise amount of fast motion can be selected).

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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 (October 29, 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 radii of stable circular orbits on the equatorial plane also increase, whereas those of the innermost stable circular orbits decrease. On the other hand, the effects of small variations of the parameters on the orbital regular and chaotic dynamics of charged particles on the non-equatorial plane are traced by means of a time-transformed explicit symplectic integrator, Poincaré sections and fast Lyapunov indicators. It is shown that the dynamics sensitivity depends on small variations in the inductive charge parameter, magnetic field parameter, energy, and angular momentum. Chaos occurs easily as each of the inductive charge parameter, magnetic field parameter, and energy increases but is weakened as the angular momentum increases. When the dragging effects of the spacetime increase, the chaotic properties are not always weakened under some circumstances.

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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 (June 26, 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 occurrence of chaos. Employing an explicit symplectic integrator, we survey the contributions of the two parameters to the chaotic dynamical behavior. It is found that a small change of the parameter γ constrained by the shadow of M87* black hole has an almost negligible effect on the dynamical transition of particles from order to chaos. However, a small decrease in the parameter Ω leads to an enhancement in the strength of chaos from the global phase space structure. A theoretical interpretation is given to the different contributions. The term with the parameter Ω dominates the term with the parameter γ, even if the two parameters have same values. In particular, the parameter Ω acts as a repulsive force, and its decrease means a weakening of the repulsive force or equivalently enhancing the attractive force from the black hole. On the other hand, there is a positive Lyapunov exponent that is universally given by the surface gravity of the black hole when Ω ≥ 0 is small and the external magnetic field vanishes. In this case, the horizon would influence chaotic behavior in the motion of charged particles around the black hole surrounded by the external magnetic field. This point can explain why a smaller value of the renormalization group parameter would much easily induce chaos than a larger value.

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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 (July 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 and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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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 (May 25, 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 constructed by extending the original phase space and obtaining the augmented separable Hamiltonian, and then by using the splitting method and the midpoint permutation. The numerical experiments have shown that compared with the higher order implicit Runge-Kutta method, the explicit K-symplectic and explicit symplectic-like methods have obvious advantages in long-term energy conservation and higher computational efficiency. It is also shown that the influence of the parameter ε in the general strong magnetic field on the Runge-Kutta method is bigger than the two kinds of symplectic methods.

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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 (January 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 (December 1, 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 symplectic integrators, since their corresponding Hamiltonians are separable into more than two explicitly integrable pieces. Although some other curved spacetimes including the Kerr black hole do not have such multipart splits, their corresponding appropriate time-transformation Hamiltonians do. In fact, the key problem in obtaining symplectic analytically integrable decomposition algorithms is how to split these Hamiltonians or time-transformation Hamiltonians. Considering this idea, we develop explicit symplectic schemes in curved spacetimes. We introduce a class of spacetimes whose Hamiltonians are directly split into several explicitly integrable terms. For example, the Hamiltonian of a rotating black ring has a 13-part split. We also present two sets of spacetimes whose appropriate time-transformation Hamiltonians have the desirable splits. For instance, an eight-part split exists in a time-transformed Hamiltonian of a Kerr–Newman solution with a disformal parameter. In this way, the proposed symplectic splitting methods can be used widely for long-term integrations of orbits in most curved spacetimes we know of.

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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 (December 10, 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 symplectic methods are based on a time-transformed Hamiltonian split into seven parts, whose analytical solutions are explicit functions of new coordinate time. Numerical tests show that such explicit symplectic integrators for intermediate time steps perform well long-term when stabilizing Hamiltonian errors, regardless of regular or chaotic orbits. One of the explicit symplectic integrators with the techniques of Poincaré sections and fast Lyapunov indicators is applied to investigate the effects of the parameters, including the four free deformation parameters, on the orbital dynamical behavior. From the global phase-space structure, chaotic properties are typically strengthened under some circumstances, as the magnitude of the magnetic parameter or any one of the negative deformation parameters increases. However, they are weakened when the angular momentum or any one of the positive deformation parameters increases.

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18

Zhang, Lina, Xin Wu, and Enwei Liang. "Adjustment of Force–Gradient Operator in Symplectic Methods." Mathematics 9, no. 21 (October 27, 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 adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.

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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 (August 25, 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 (November 19, 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 step-size selections in this case. Based on the step-size control technique proposed by Preto & Saha, the nonadaptive time-step time-transformed explicit symplectic methods are slightly adjusted as adaptive ones. The adaptive methods have only two additional steps and a negligible increase in computational cost compared with the nonadaptive ones. Their implementation is simple. Several dynamical simulations of particles and photons near black holes have demonstrated that the adaptive methods typically improve the efficiency of the nonadaptive methods. Because of the desirable property, the new adaptive methods are applied to investigate the chaotic dynamics of particles and photons outside the horizon in a Schwarzschild–Melvin spacetime. The new methods are widely applicable to all curved spacetimes corresponding to Hamiltonians or time-transformed Hamiltonians with the expected splits. In addition, application to the backward ray-tracing method for studying the motion of photons and shadows of black holes is possible.

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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 (January 28, 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 (August 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 (July 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 (February 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 order method introduced in Ref. 1.

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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 proposed symplectic finite elements are capable of providing accurate and robust simulation in the dynamic modeling of multi-flexible bodies systems with large overall motions.

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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 (March 1, 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 orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.

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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 (December 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 (December 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 splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.

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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 (December 3, 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, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.

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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 (March 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 the separable K-canonical governing equation for a single plate. Then, to obtain the high accurate numerical results, a multi-scale iterative technique is also presented. Finally, based on the interlaminar compatibility condition (displacements and stresses), the excellent performance of the method presented in this paper is demonstrated by several numerical experiments of the static responses of laminated plates.

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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>A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in [<xref ref-type="bibr" rid="b1">1</xref>] and [<xref ref-type="bibr" rid="b2">2</xref>]. 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 time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in [<xref ref-type="bibr" rid="b3">3</xref>] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.</abstract>

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34

Sasa, Narimasa. "Momentum Conservation Law in Explicit Symplectic Integrators for Nonlinear Wave Equations." Journal of the Physical Society of Japan 83, no. 5 (May 15, 2014): 054004. http://dx.doi.org/10.7566/jpsj.83.054004.

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35

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 (May 4, 2017): 3031–41. http://dx.doi.org/10.1093/mnras/stx1059.

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36

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 (January 3, 2017): 64. http://dx.doi.org/10.3847/1538-4357/834/1/64.

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37

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 (May 1, 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 scheme so that it preserves the Hamiltonian structure of the wave equation in long-term simulations. Consequently, the symplectic IPDG method combines the advantages of discontinuous Galerkin method and the symplectic time integration. We investigate the properties of the method in detail for high-order spatial basis functions, including the stability criteria, numerical dispersion and dissipation relationships, and numerical errors. The analyses indicate that the symplectic SIPG method is nondissipative and retains low numerical dispersion. We also find that different symplectic IPDG methods have different convergence behaviors. It is indicated that using coarse meshes with a high-order method produces smaller errors and retains high accuracy. We have applied our method to simulate the scalar wavefields for different models, including layered models, a rough topography model, and the Marmousi model. The numerical results show that the symplectic IPDG method can suppress numerical dispersion effectively and provide accurate information on the wavefields. We also conduct a long-term experiment that verifies the capability of symplectic IPDG method for long-time simulations.

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38

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 (January 29, 2021): 66. http://dx.doi.org/10.3847/1538-4357/abcb8d.

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39

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 (June 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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41

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 (February 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 modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

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42

Oh, Thae, Ji Choe, and Jin Kim. "Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order." Engineering Mathematics 7, no. 1 (July 3, 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 preserved the original property of the systems. So, study of the symplectic Runge-Kutta methods have performed. The typical symplectic Runge-Kutta method is the Gauss-Legendre method, whose drawback is that it is a general implicit scheme and is too computationally expensive. Despite these drawbacks, the study of the diagonally implicit symplectic Runge-Kutta methods that preserves symplecticity has attracted much attention. The symplectic Runge-Kutta method has been studied up to sixth order in the past and efforts to obtain higher order conditions and algorithms are being intensified. In many applications such as molecular dynamics as well as in space science, such as satellite relative motion studies, this method is very effective and its application is wider. In this paper, it is presented the 7th order condition and derive the corresponding optimized method. So the diagonally implicit symplectic eleven-stages Runge-Kutta method with algebraic order 7 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

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43

Zhang, Lina, Wenfang Liu, and Xin Wu. "Study of Chaos in Rotating Galaxies Using Extended Force-Gradient Symplectic Methods." Symmetry 15, no. 1 (December 26, 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 the standard fourth-order symplectic method but inferior to the optimized extended force-gradient fourth-order symplectic method in accuracy. The optimized extended algorithm with symmetry is used to explore the dynamical features of regular and chaotic orbits in these rotating galaxy models. The gravity effects and the degree of chaos increase with an increase in the number of radial terms in the series expansions of the potential. There are similar dynamical structures of regular and chaotical orbits in the three types of models for the same number of radial terms in the series expansions, energy and initial conditions.

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44

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 (September 30, 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 results indicate that the smaller alignment index obtained by using a global symplectic algorithm is the fastest method for distinguishing between regular and chaotic cases. The smaller alignment index is used to study the effects of parameters on the dynamic transition from order to chaos. If initial conditions and other parameters are appropriately chosen, we observe that an increase in energy E or the deformation parameter η can easily lead to chaos. Similarly, chaos easily occurs when the angular momentum L is small enough or the magnetic parameter Q stays within a suitable range. By varying the initial conditions of the particles, a distribution plot of the smaller alignment in the X–Z plane of the black hole is obtained. It is found that the particle orbits exhibit a remarkably rich structure. Researching the motion of charged particles around a black hole contributes to our understanding of the mechanisms behind black hole accretion and provides valuable insights into the initial formation process of an accretion disk.

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45

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 (March 1, 2021): 22. http://dx.doi.org/10.3847/1538-4357/abd701.

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46

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 (May 15, 2013): 053001. http://dx.doi.org/10.7566/jpsj.82.053001.

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47

Liu, Caiyu, and Xin Wu. "Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory." Universe 9, no. 8 (August 7, 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 dynamics of charged particles because it exhibits excellent long-term performance in conserving the Hamiltonian. No universal rule can be given to the dependence of regular and chaotic dynamics on varying one or two parameters c13 and c14 in the two cases of 0≤2c13<c14<2 and 0≤c14<2c13<2. The distributions of order and chaos in the binary parameter space (c13,c14) rely on different combinations of the other parameters and the initial conditions.

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48

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 (June 8, 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 static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincaré sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can have chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos.

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49

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 (September 1994): 4062–72. http://dx.doi.org/10.1063/1.467523.

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50

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 (October 18, 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 schemes generically exploit symmetries in the applied external forces, which often strongly suppress integration errors. Furthermore, we demonstrate that so-called ‘symplectic correctors’ (which reduce energy errors by orders of magnitude at fixed computational cost) apply equally well to weakly dissipative systems and can thus be more generally thought of as ‘weak splitting correctors’. Finally, we show that previously advocated approaches of incorporating additional forces into symplectic methods work well for dissipative forces, but give qualitatively wrong answers for conservative but velocity-dependent forces like post-Newtonian corrections. We release REBOUNDx, an open-source C library for incorporating additional effects into REBOUNDN-body integrations, together with a convenient python wrapper. All effects are machine independent and we provide a binary format that interfaces with the SimulationArchive class in REBOUND to enable the sharing and reproducibility of results. Users can add effects from a list of pre-implemented astrophysical forces, or contribute new ones.

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