Symplectic Integration Algorithm Research Articles

Molecular Dynamics of Artificially Pair-Decoupled Systems: An Accurate Tool for Investigating the Importance of Intramolecular Couplings.

We propose a numerical technique to accurately simulate the vibrations of organic molecules in the gas phase, when pairs of atoms (or, in general, groups of degrees of freedom) are artificially decoupled, so that their motion is instantaneously decorrelated. The numerical technique we have developed is a symplectic integration algorithm that never requires computation of the force but requires estimates of the Hessian matrix. The theory we present to support our technique postulates a pair-decoupling Hamiltonian function, which parametrically depends on a decoupling coefficient α ∈ [0, 1]. The closer α is to 0, the more decoupled the selected atoms. We test the correctness of our numerical method on small molecular systems, and we apply it to study the vibrational spectroscopic features of salicylic acid at the Density Functional Theory ab initio level on a fitted potential. Our pair-decoupled simulations of salicylic acid show that decoupling hydrogen-bonded atoms do not significantly influence the frequencies of stretching modes, but enhance enormously the out-of-plane wagging and twisting motions of the hydroxyl and carboxyl groups to the point that the carboxyl and hydroxyl groups may overcome high potential energy barriers and change the salicylic acid conformation after a short simulation time. In addition, we found that the acidity of salicylic acid is more influenced by the dynamical couplings of the proton of the carboxylic group with the carbon ring than with the hydroxyl group.

Adjustment of Force–Gradient Operator in Symplectic Methods

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.

Massively parallel symplectic algorithm for coupled magnetic spin dynamics and molecular dynamics

A parallel implementation of coupled spin–lattice dynamics in the LAMMPS molecular dynamics package is presented. The approach is very general, and can be applied to simple ferromagnets, magnetic alloys, or amorphous magnetic materials. Equations of motion for both spin only and coupled spin–lattice dynamics are first reviewed, including a detailed account of how magneto-mechanical potentials can be used to perform a proper coupling between spin and lattice degrees of freedom. A symplectic numerical integration algorithm is then presented which combines the Suzuki–Trotter decomposition for non-commuting variables and conserves the geometric properties of the equations of motion. The numerical accuracy of the serial implementation was assessed by verifying that it conserves the total energy and the norm of the total magnetization up to second order in the timestep size. Finally, a very general parallel algorithm is proposed that allows large spin–lattice systems to be efficiently simulated on large numbers of processors without degrading its mathematical accuracy. Its correctness as well as scaling efficiency were tested for realistic coupled spin–lattice systems, confirming that the new parallel algorithm is both accurate and efficient.

Simulations of the Fomalhaut system within its local galactic environment

Fomalhaut A is among the most well-studied nearby stars and has been discovered to possess a putative planetary object as well as a remarkable eccentric dust belt. This eccentric dust belt has often been interpreted as the dynamical signature of one or more planets that elude direct detection. However, the system also contains two other stellar companions residing ~100,000 AU from Fomalhaut A. We have designed a new symplectic integration algorithm to model the evolution of Fomalhaut A's planetary dust belt in concert with the dynamical evolution of its stellar companions to determine if these companions are likely to have generated the dust belt's morphology. Using our numerical simulations, we find that close encounters between Fomalhaut A and B are expected, with a ~25% probability that the two stars have passed within at least 400 AU of each other at some point. Although the outcomes of such encounter histories are extremely varied, these close encounters nearly always excite the eccentricity of Fomalhaut A's dust belt and occasionally yield morphologies very similar to the observed belt. With these results, we argue that close encounters with Fomalhaut A's stellar companions should be considered a plausible mechanism to explain its eccentric belt, especially in the absence of detected planets capable of sculpting the belt's morphology. More broadly, we can also conclude from this work that very wide binary stars may often generate asymmetries in the stellar debris disks they host.

Solving the spin-2 Gross-Pitaevskii equation using exact nonlinear dynamics and symplectic composition.

We develop numerical methods for solving the spin-2 Gross-Pitaevskii equation. The basis of our work is a two-way splitting of this evolution equation that leads to two exactly solvable subsystems. Utilizing second-order and fourth-order composition schemes we realize two fully symplectic integration algorithms, the first such algorithms for evolving spin-2 condensates. We demonstrate the accuracy of these algorithms against other methods on application to an exact continuous wave solution that we derive.

Fast Ion Trajectory Calculations in Tokamak Magnetic Configuration Using Symplectic Integration Algorithm

A numerical scheme based on Symplectic Integration Algorithm (SIA) has been used to develop an orbit following code to calculate fast ion trajectories in tokamak magnetic configuration. For the purpose of demonstrating the expediency of symplectic schemes, the algorithm has been applied to the H´enon-Heiles system and compared with non- symplectic Runge–Kutta Algorithm (RKA) for numerically integrating the Hamiltonian equations. In contrast to RKA, the long-time stability of SIA has been highlighted. Furthermore, the SIA has been used to find the exact trajectories of, trapped and passing, fast ions in tokamak. In particular, the effect of an intrinsic magnetic field perturbation has been investigated, i.e. toroidal field ripples. This perturbation is toroidal field ripple (TFR) which are there due the discrete number to toroidal coils around the torus. The numerical scheme used shows excellent conservation of particle energy as well as of angular momentum (in axi-symmetric case). The effect of TFR on these trajectories has been simulated and it is shown that the resonance between toroidal precession of bananas and the field ripples results in spread of the trajectories for banana particles, whereas the passing fast ions are unaffected.

Application research of symplectic Runge-Kutta method of solving Lagrange-Maxwell equation

In this paper, we show the numerical integration method of solving Lagrange-Maxwell equation by using the symplectic Runge-Kutta (R-K) method, and numerically study the motion of the plate in an RLC circuit spring coupled system and the current changes. Its result is consistent with that obtained by the traditional R-K method, which demonstrates symplectic integration algorithm is reasonable and effective in studying the electro-mechanical systems. And on this basis, the form invariance of Noether sense is studied by using the symplectic Runge-Kutta method.

Symplectic Simulation of Fast Alpha Particle Radial Transport in Tokamaks in the Presence of TF Ripples and a Neoclassical Tearing Mode

A Hamiltonian guiding centre drift orbit code based on a symplectic integration algorithm, which enables the efficient calculation of particle trajectories and diffusion coefficients, is applied to fast alpha particle motion in magnetically perturbed tokamak plasmas. In particular, fast ion drift motion is examined in the presence of a stationary, low mode-number MHD magnetic perturbation in a toroidally rippled tokamak with circular flux surface. The main focus of our study is to investigate the dependence of the radial diffusion coefficient of energetic ions on their energy, on the perturbation strength and the localization of the perturbation. As expected, the resonance between bounce motion and toroidal field ripples plays a significant role in this context. For an ensemble of fast ions uniformly distributed in toroidal angle but with a given poloidal starting position their radial transport coefficient takes on higher values in the neighbourhood of resonance speeds and can exhibit there local minima, i.e. it shows an M-shaped speed dependence around resonances for sufficiently strong ripple perturbations. Expectedly, the addition of a modelled low-mode number neoclassical tearing mode perturbation will modify the pure ripple resonance structure of the radial diffusion coefficient. Depending on the strength and localization of the MHD mode it can cause enhancement or degradation of the radial ripple diffusion coefficient.

1815 Compact差分法とSymplectic積分法の波動音響に対する適用検討(OS18.流体の数値計算手法と数値シミュレーション(4),OS・一般セッション講演)

1815 Compact差分法とSymplectic積分法の波動音響に対する適用検討(OS18.流体の数値計算手法と数値シミュレーション(4),OS・一般セッション講演)

Planetary stability zones in hierarchical triple star systems

A symplectic integrator algorithm suitable for hierarchical triple systems is formulated and tested. The positions of the stars are followed in hierarchical Jacobi coordinates, whilst the planets are referenced purely to their primary. The algorithm is fast, accurate and easily generalised to incorporate collisions. There are five distinct cases -- circumtriple orbits, circumbinary orbits and circumstellar orbits around each of the stars in the hierarchical triple -- which require a different formulation of the symplectic integration algorithm. As an application, a survey of the stability zones for planets in hierarchical triples is presented, with the case of a single planet orbiting the inner binary considered in detail. Fits to the inner and outer edges of the stability zone are computed. Considering the hierarchical triple as two decoupled binary systems, the earlier work of Holman & Wiegert on binaries is shown to be applicable to triples, except in the cases of high eccentricities and close or massive stars. Application to triple stars with good data in the multiple star catalogue suggests that more than 50 per cent are unable to support circumbinary planets, as the stable zone is almost non-existent.

Review of the Anisotropic Interface Ray Propagator: Symplecticity, Eigenvalues, Invariants and Applications

A review of the 6 × 6 anisotropic interface ray propagator matrix in Cartesian coordinates and within the framework of the Hamiltonian formalism shows that there is one unique propagator satisfying the symplectic property. This is essential, since the symplecticity furnishes an exact inverse, while an eigenvalue analysis indicates that the propagator may be arbitrarily ill-conditioned. As such, the symplectic interface propagator naturally connects to symplectic ray integration algorithms for smooth media, designed to maintain accuracy. Moreover, several ray invariants for smooth media remain invariant across interfaces. It is straightforward to derive expressions for the interface propagator, both explicit and implicit. Symplecticity is equivalent to the condition that the propagator preserves the eikonal constraint across the interface. The symplectic interface propagator complies with phase matching of the incident and reflected/transmitted ray field, and is therefore in accordance with the earlier derived 4 × 4 matrix in ray-centred coordinates. The symplectic property is related to the symmetry of the second derivative matrix of the reflected/transmitted traveltime field. Thanks to the analytic expression of the symplectic interface propagator, relating interface curvature directly to second derivatives of traveltimes observed at a datum level, numerous applications are available in the area of processing and inversion.

A Dissipative Mapping Technique for theN-Body Problem Incorporating Radiation Pressure, Poynting-Robertson Drag, and Solar Wind Drag

By implementing a version of the dissipative mapping technique introduced by R. Malhotra, we have developed a new integration code for the N-body problem that incorporates the effects of radiation pressure, Poynting-Robertson (P-R) drag, and solar wind drag. The advantage of employing the dissipative mapping technique is that it modifies the basic N-body symplectic integration algorithm developed by Wisdom & Holman to allow certain nongravitational effects to be modeled and therefore retains the speed of execution common to codes based upon this algorithm. To achieve this, we have adapted the dissipative mapping technique to the requirements of the forces being modeled. We present the results of tests that demonstrate the suitability of this new dissipative integration code for investigating the dynamical behavior of micron-sized dust particles in heliocentric orbits in the solar system and, more generally, of particles in exosolar planetary systems where the dominant nongravitational perturbations to the particles' astrocentric orbits are due to the effects of radiation pressure, P-R drag, and solar wind drag.

POLYNOMIAL MAP FACTORIZATION OF SYMPLECTIC MAPS

Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the Hamiltonian system is refactorized using polynomial symplectic maps. This method is analyzed for the three degree of freedom case. Finally, we apply this algorithm to study a large particle storage ring.

Symplectic integration of Hamiltonian systems using polynomial maps

In order to perform numerical studies of long-term stability in nonlinear Hamiltonian systems, one needs a numerical integration algorithm which is symplectic. Further, this algorithm should be fast and accurate. In this Letter, we propose such a symplectic integration algorithm using polynomial map refactorization of the symplectic map representing the Hamiltonian system. This method should be particularly useful in long-term stability studies of particle storage rings in accelerators.

Symplectic algorithm for constant-pressure molecular dynamics using a Nosé–Poincaré thermostat

We present a new algorithm for isothermal–isobaric molecular-dynamics simulation. The method uses an extended Hamiltonian with an Andersen piston combined with the Nosé–Poincaré thermostat, recently developed by Bond, Leimkuhler, and Laird [J. Comp. Phys. 151, 114 (1999)]. This Nosé–Poincaré–Andersen (NPA) formulation has advantages over the Nosé-Hoover-Andersen approach in that the NPA is Hamiltonian and can take advantage of symplectic integration schemes, which lead to enhanced stability for long-time simulations. The equations of motion are integrated using a generalized leapfrog algorithm (GLA) and the method is easy to implement, symplectic, explicit, and time reversible. To demonstrate the superior stability of the method we show results for test simulations using a model for aluminum and compare it to a recently developed time-reversible algorithm for Nosé–Hoover–Anderson. In addition, an extension of the NPA to multiple time steps is outlined and a symplectic and time-reversible integration algorithm, based on the GLA, is given.

Symplectic integration using solvable maps

In this paper, a symplectic integration algorithm using solvable maps is investigated. Examples are studied in two and four dimensions. The method is shown to give good results.

A Class of Symplectic Integrators with Adaptive Time Step for Separable Hamiltonian Systems

Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian systems, because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive time step control is added to a symplectic integrator. We describe an adaptive time step, symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required time step depends only on the potential energy (e.g., test-particle integrations in fixed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential; this integrator has a time step proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force field with only "along-track" errors; i.e., the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(N-2), where N is the number of steps per period.

Split Integration Symplectic Method for Molecular Dynamics Integration

An explicit Split Integration Symplectic Method (SISM) for molecular dynamics (MD) simulations is described. This work is an extension of an efficient symplectic integration algorithm introduced by Janežic and Merzel (J. Chem. Inf. Comput. Sci. 1995, 35, 321−326). SISM is based on splitting of the total Hamiltonian of the system into a harmonic part and the remaining part in such a way that both parts can be efficiently computed. The Hamilton equations of motion are then solved using the second order generalized leap-frog integration scheme in which the high-frequency motions are treated analytically by the normal mode analysis which is carried out only once, at the beginning of the calculation. SISM requires only one force evaluation per integration step; the computation cost per integration step is approximately the same as that of the standard leap-frog-Verlet (LFV) method, and it allows an integration time step up to an order of magnitude larger than can be used by other methods of the same order and ...

Interaction of nonlinear Schrodinger solitons with an external potential

Employing a particularly suitable higher-order symplectic integration algorithm, we integrate the one-dimensional nonlinear Schrodinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential. We find that the soliton can break up into two solitons, eventually accompanied by radiation of non-solitary waves. Reflection coefficients and inelasticities are computed as functions of the height of the potential step and of its steepness.

An Efficient Symplectic Integration Algorithm for Molecular Dynamics Simulations

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTAn Efficient Symplectic Integration Algorithm for Molecular Dynamics SimulationsDusanka Janezic and Franci MerzelCite this: J. Chem. Inf. Comput. Sci. 1995, 35, 2, 321–326Publication Date (Print):March 1, 1995Publication History Published online1 May 2002Published inissue 1 March 1995https://doi.org/10.1021/ci00024a022RIGHTS & PERMISSIONSArticle Views254Altmetric-Citations30LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (562 KB) Get e-Alerts