Symplectic Scheme Research Articles

Structure-preserving methods for Marcus stochastic Hamiltonian systems with additive Lévy noise

A general structure-preserving method is proposed for a class of Marcus stochastic Hamiltonian systems driven by additive Lévy noise. The convergence of the symplectic Euler scheme for this systems is investigated by Generalized Milstein Theorem. Realizable numerical implementation of this scheme is also provided in details. Numerical experiments are presented to illustrate the effectiveness and superiority of the proposed scheme. Applications of the method to solve two mathematical physical problems are provided.

The Galactic population of canonical pulsar. II. Taking into account several evolutionary paths: Interstellar medium interaction, Galactic potential, and a death valley

Pulsars are highly magnetised rotating neutron stars, emitting in a broad electromagnetic energy range. These objects were discovered more than 55 years ago and are astrophysical laboratories for studying physics at extreme conditions. Reproducing the observed pulsar population helps refine our understanding of their formation and evolution scenarios, as well as their radiation processes and geometry. In this paper, we improve our previous population synthesis by focusing on both the radio and gamma -ray pulsar populations, investigating the impact of the Galactic gravitational potential and of the radio emission death line. To elucidate the necessity of a death line, we implemented our refined initial distributions of the spin period and spacial position at birth. This approach allowed us to elevate the sophistication of our simulations to the most recent state-of-the-art approaches. The motion of each individual pulsar was tracked in the Galactic potential by a fourth-order symplectic integration scheme. Our pulsar population synthesis took into account the secular evolution of the force-free magnetosphere and magnetic field decay simultaneously and self-consistently. Each pulsar was evolved from birth to the present time. The radio and gamma -ray emission locations were modelled by the polar cap geometry and striped wind model, respectively. By simulating ten million pulsars, we found that including a death line allows us to better reproduce the observational trend. However, when simulating one million pulsars, we obtained an even more realistic $P- P $ diagram, whether or not a death line was included. This suggests that the ages of the detected pulsars might be overestimated and so, it sets the need for a death line in pulsar population studies into question. Kolmogorov-Smirnov tests confirm the statistical similarity between the observed and simulated $P- P $ diagram. Additionally, simulations with increased gamma -ray telescope sensitivities hint at a significant contribution coming from the gamma -ray pulsars to the GeV excess in the Galactic centre.

Combined Compact Symplectic Schemes for the Solution of Good Boussinesq Equations

Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic structure. Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical tests indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation.

Symplectic Quantization III: Non-relativistic Limit

First of all we shortly illustrate how the symplectic quantization scheme (Gradenigo and Livi, Found Phys 51(3):66, 2021) can be applied to a relativistic field theory with self-interaction. Taking inspiration from the stochastic quantization method by Parisi and Wu, this procedure is based on considering explicitly the role of an intrinsic time variable, associated with quantum fluctuations. The major part of this paper is devoted to showing how the symplectic quantization scheme can be extended to the non-relativistic limit for a Schrödinger-like field. Then we also discuss how one can obtain from this non-relativistic theory a linear Schrödinger equation for the single-particle wavefunction. This further passage is based on a suitable coarse-graining procedure, when self-interaction terms can be neglected, with respect to interactions with any external field. In the Appendix we complete our survey on symplectic quantization by discussing how this scheme applies to a non-relativistic particle under the action of a generic external potential.

Numerical simulations of shallow-water sloshing coupled to horizontal vessel motion in the presence of a time-dependent porous baffle

Shallow-water fluid sloshing in the Lagrangian Particle Path formulation, with the addition of an energy-extracting porous baffle, is simulated numerically using a symplectic numerical scheme which captures, in an essential way, the energy exchange. The fluid motion in a rectangular vessel is dynamically coupled to a surface-piercing porous baffle. The fluid transmission through the baffle is characterized by a nonlinear Darcy–Forchheimer model equation. The numerical scheme is symplectic, based on the implicit-midpoint rule, and thus is strategically designed to maintain the energy partition between the fluid and vessel throughout numerous time steps. Our results demonstrate the non-conservative nature of the system, with the porous baffle effectively dissipating energy from the overall system. Furthermore, we present findings that demonstrate the role of time-periodic variations in baffle porosity on energy dissipation. By manipulating the frequency and magnitude of this time-dependent variability, it is established that a greater amount of energy can be extracted from the system compared with the optimal fixed porosity baffle. These results shed new light on potential strategies for enhancing energy dissipation in such configurations.

Applying SPH-DEM theory to the motion of particulate solid-liquid flow on continuous screening

Challenges in current research on fine mesh wet screening (≥50 mesh) include representing fine wire boundaries, contentious porous media handling, and low computational efficiency. To address these challenges, this study employs an averaged SPH-DEM method for modeling solid-liquid flow on screen. It integrates the Symplectic scheme and quaternion method to enhance computational accuracy and scalability. The fine wire boundaries are discretized into spheres to facilitate Hertz contact of DEM particles, and a novel screen aperture-fluid resistance model is introduced to calculate the resistance exerted on SPH particles by screen surface. The simulation results of dam break and periodic screening exhibit robust consistency with experimental data and other models, effectively validating the feasibility and accuracy of this methodology. Moreover, simulations of continuous screening using a 0.2 mm wire demonstrate that liquid enhances particle passage, further substantiating the practical viability of employing this model for simulating fine mesh wet screening.

Анализ эффективности симплектических схем высокого порядка точности на примере задачи о соударении наночастицы с преградой

Работа посвящена применению метода молекулярной динамики (МД) для численного моделирования высокоскоростного взаимодействия твердых тел. Численно смоделировано явление отскока медной наночастицы от алюминиевой пластины. Для схемы Верле и новой симплектической схемы FR50, полученной ранее автором, найдены интервалы временных шагов, такие, что при задании временного шага в этих интервалах не происходит дрейфа полной энергии. При этом дрейф полной энергии возникает в случае схемы FR50 при гораздо больших шагах, чем в случае метода Верле. На основе анализа результатов численного моделирования задачи о соударении наночастицы с преградой показана необходимость введения в рассмотрение нескольких критериев подобия по аналогии с механикой сплошных сред. Предложен быстрый простой компьютерный тест для отбраковки ЕАМ-потенциалов до начала решения основной МД-задачи. The work is devoted to the application of the molecular dynamics (MD) method for numerical simulation of high-velocity interaction of solids. The phenomenon of the copper nanoparticle rebound from the aluminum plate has been simulated numerically. For the Verlet scheme and the new symplectic scheme FR50 obtained recently by the author, the intervals of time steps were found such that at the specification of the time step within these intervals, no drift of the total energy occurs. The total energy drift arises in the case of the FR50 scheme at much larger time steps than in the case of the Verlet method. The need in introducing several similarity criteria by analogy with the continuum mechanics has been shown by analyzing the results of the numerical simulation of problems involving the nanoparticle collision with an obstacle. A rapid and simple computer test has been proposed for the rejection of the EAM potentials before starting to solve the main MD problem.

A nonsmooth modified symplectic integration scheme for frictional contact dynamics of rigid–flexible multibody systems

Constructing an efficient high-order nonsmooth time-stepping integration scheme for rigid–flexible multibody systems with frictional contacts and impacts is a challenging task. This paper presents a nonsmooth modified symplectic integration scheme that can automatically achieve second-order convergence if possible. This scheme is obtained in three main steps. Firstly, the position coordinates are used to interpolate midpoint approximated integrals, which include velocity jumps implicitly; Secondly, a two-timestep central difference scheme is formulated to approximate the measure differential inclusions; Thirdly, composing two half-condensed two-timestep schemes leads to a one-timestep scheme, which retains the structure of a second-order scheme. Furthermore, while bilateral constraints are satisfied at the position level, unilateral constraints at the velocity level are subject to constraint drifts. To alleviate this issue, a classified cone complementarity problem is formulated to restrain penetration from deteriorating and sticking contacts from fictitious slipping. After time discretization, a set of nonlinear equations with cone complementarity conditions at each timestep is solved by a two-layer iterative computational strategy, where the outer layer is Newton iteration and the inner layer is a primal–dual interior point method. Several numerical examples demonstrate the high accuracy attained by the proposed method. Nonsmooth dynamics of a rigid–flexible robot involving high-speed and high-friction impacts can be simulated faster than real-time.

A symplectic fission scheme for the association scheme of rectangular matrices and its automorphisms

A symplectic fission scheme for the association scheme of rectangular matrices and its automorphisms

Hamiltonian formulation and symplectic split-operator schemes for time-dependent density-functional-theory equations of electron dynamics in molecules

We revisit Kohn–Sham time-dependent density-functional theory (TDDFT) equations and show that they derive from a canonical Hamiltonian formalism. We use this geometric description of the TDDFT dynamics to define families of symplectic split-operator schemes that accurately and efficiently simulate the time propagation for certain classes of DFT functionals. We illustrate these with numerical simulations of the far-from-equilibrium electronic dynamics of a one-dimensional carbon chain. In these examples, we find that an optimized 4th order scheme provides a good compromise between the numerical complexity of each time step and the accuracy of the scheme. We also discuss how the Hamiltonian structure changes when using a basis set to discretize TDDFT and the challenges this raises for using symplectic split-operator propagation schemes.

An efficient method of finding new symplectic schemes for Hamiltonian mechanics problems with the aid of parametric Gröbner bases

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using parametric Gröbner bases, resultants, permutations of variables, and optimization techniques is proposed. The implementation of the method is described by the example of four-stage partitioned Runge–Kutta (PRK) schemes of the Forest–Ruth and Yoshida families. 97 new PRK schemes of Forest–Ruth family have been obtained. The error functionals for five new schemes FR47, FR50, FR51, FR52, and FR59 are smaller by more than two orders of magnitude than in the case of the fourth-order schemes, which were obtained previously in the explicit form in this family. In the family of Yoshida schemes, 115 new schemes have been found. Verification of the schemes was carried out on the Kepler's problem, which has the exact solution. Calculations of this task performed at large time steps showed that the FR50 scheme significantly exceeds in terms of accuracy the four-stage Forest–Ruth schemes known in the literature, the Verlet scheme, schemes from the Yoshida family Y7, Y95, Y110, and a six-stage fourth-order scheme.

Application of Symmetric Explicit Symplectic Integrators in Non-Rotating Konoplya and Zhidenko Black Hole Spacetime

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.

Symplectic numerical integration for Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus

In this paper, we propose a symplectic numerical integration method for a class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus. We first construct a general symplectic Euler scheme for these equations, then we prove its convergence. In addition, we provide realizable numerical implementations for the proposed symplectic Euler scheme in detail. Some numerical experiments are conducted to demonstrate the effectiveness and superiority of the proposed method by the simulations of its orbits, Hamiltonian and convergence order over a long time interval. The results show the applicability of the methods considered.

Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory

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.

Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems

We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.

Symplectic Integration Schemes for Systems With Nonlinear Dissipation

In previous work a variational integration scheme for mechanical systems containing linear dissipation was developed (Limebeer et al., 2020). The key idea is to use a transmission line as the solution to a power-balancing control problem—in the physics literature controllers of this type are known as “heat baths.” The power-balanced closed-loop system is thereby made amenable to analysis with Hamilton's principle. As a consequence, variational principles can be used to find symplectic integration schemes for dissipative systems. The outcome of this work is a coherent physical explanation as to why variational/symplectic integration algorithms for the open dissipative system can perform as well as their conservative counterparts. An important application of these integration algorithms is the solution of long-time-horizon optimal control problems. In this article, these ideas are extended to systems with smooth memoryless nonlinear dissipation. This extension employs a nonlinear lossless transmission line, which is again the solution to a power-balancing control problem. As in the case of linear dissipation, the resulting variational integration schemes are demonstrated to have excellent numerical properties, which are significantly less demanding computationally than their implicit nonvariational counterparts.

Meshless symplectic and multi-symplectic scheme for the coupled nonlinear Schrödinger system based on local RBF approximation

Meshless symplectic and multi-symplectic scheme for the coupled nonlinear Schrödinger system based on local RBF approximation

Two Novel Classes of Arbitrary High-Order Structure-Preserving Algorithms for Canonical Hamiltonian Systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

Explicit Symplectic Methods in Black Hole Spacetimes

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.

Symplectic central difference scheme for quasi-linear autonomous Birkhoffian systems

In this paper, a symplectic central difference scheme (SCDS) for the quasi-linear autonomous Birkhoffian system is proposed. Firstly, the definition of the quasi-linear autonomous Birkhoffian system in this paper is given. Then, by performing central difference discretization for the quasi-linear autonomous Birkhoffian equation, the SCDS is derived and proved to be strictly symplectic structure-preserving. Based on the precise integration method, an iterative process is used to obtain the transfer matrix of SCDS, which improves the accuracy of SCDS without significantly increasing the computational effort. In addition, the flexibility of SCDS for the quasi-linear autonomous Birkhoffian equation is discussed: the first is the applicability to odd-dimensional generalized Birkhoffian systems which appear widely in engineering problems; the second is the feasibility in structural dynamic response problems, for which the analytical procedure for applying SCDS based on the framework of the perturbation method is given. Several numerical examples verify the excellent performance of the proposed method.