Symplectic Integrators Research Articles

Formula omitted]osmo[formula omitted]attice: A modern code for lattice simulations of scalar and gauge field dynamics in an expanding universe

This paper describes CosmoLattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoLattice incorporates a series of features that makes it very versatile and powerful: i) it is written in C++ fully exploiting the object oriented programming paradigm, with a modular structure and a clear separation between the physics and the technical details, ii) it is MPI-based and uses a discrete Fourier transform parallelized in multiple spatial dimensions, which makes it specially appropriate for probing scenarios with well-separated scales, running very high resolution simulations, or simply very long ones, iii) it introduces its own symbolic language, defining field variables and operations over them, so that one can introduce differential equations and operators in a manner as close as possible to the continuum, iv) it includes a library of numerical algorithms, ranging from O(δt2) to O(δt10) methods, suitable for simulating global and gauge theories in an expanding grid, including the case of ‘self-consistent’ expansion sourced by the fields themselves. Relevant observables are provided for each algorithm (e.g. energy densities, field spectra, lattice snapshots) and we note that, remarkably, all our algorithms for gauge theories (Abelian or non-Abelian) always respect the Gauss constraint to machine precision. Program summaryProgram Title::CosmoLatticeCPC Library link to program files:https://doi.org/10.17632/44vr5xssc6.1Developer's repository link:http://github.com/cosmolattice/cosmolatticeLicensing provisions: MITProgramming language: C++, MPINature of problem: The phenomenology of high energy physics in the early universe is typically characterized by non-linear dynamics, which cannot be captured accurately with analytical techniques. In order to fully understand the non-linearities developed in a given scenario, one needs to carry out lattice simulations. A number of public packages for lattice simulations have appeared over the years, but most of them are only capable of simulating scalar fields. However, realistic models of particle physics do contain other kind of field species, such as (Abelian or non-Abelian) gauge fields, whose non-linear dynamics can also play a relevant role in the early universe. Tensor modes representing gravitational waves are also naturally expected in many scenarios.Solution method:CosmoLattice represents a modern code for lattice simulations of scalar-gauge field theories in an expanding universe. It allows for the simulation of the evolution of interacting (singlet) scalar fields, charged scalar fields under U(1) and/or SU(2) gauge groups, and the corresponding associated Abelian and/or non-Abelian gauge fields. From version 1.1 onward, CosmoLattice also allows to simulate the production of gravitational waves. Simulations can be done either in a flat space-time background, or in a homogeneous and isotropic (spatially flat) expanding FLRW background. CosmoLattice provides symplectic integrators, with accuracy ranging from O(δt2) up to O(δt10), to simulate the non-linear dynamics of the appropriate fields in comoving three-dimensional lattices. The code is parallelized with MPI, and uses a discrete Fourier Transform parallelized in multiple spatial dimensions, which makes it a very powerful code for probing physical problems with well-separated scales. Moreover, the code has been designed as a ‘platform’ to implement any system of dynamical equations suitable for discretization on a lattice.

Geodesic motion and phase-space evolution of massive neutrinos

The non-trivial phase-space distribution of relic neutrinos is responsible for the erasure of primordial density perturbations on small scales, which is one of the main cosmological signatures of neutrino mass. In this paper, we present a new code, fastdf, for generating 1%-accurate particle realisations of the neutrino phase-space distribution using relativistic perturbation theory. We use the geodesic equation to derive equations of motion for massive particles moving in a weakly perturbed spacetime and integrate particles accordingly. We demonstrate how to combine geodesic-based initial conditions with the δf method to minimise shot noise and clarify the definition of the neutrino momentum, finding that large errors result if the wrong parametrisation is used. Compared to standard Lagrangian methods with ad-hoc thermal motions, fastdf achieves substantial improvements in accuracy. We outline the approximation schemes used to speed up the code and to ensure symplectic integration that preserves phase-space density. Finally, we discuss implications for neutrino particles in cosmological N-body simulations. In particular, we argue that particle methods can accurately describe the neutrino distribution from z = 109, when neutrinos are linear and ultra-relativistic, down to z = 0, when they are nonlinear and non-relativistic. fastdf can be used to set up accurate initial conditions (ICs) for N-body simulations and has been integrated into the higher-order IC code monofonic.

Stability Analysis on the Moon’s Rotation in a Perturbed Binary Asteroid

Numerical calculation provides essential tools for deep space exploration, which are indispensable to mission design and planetary research. In a specific case of binary asteroid defense such as the DART mission, an accurate understanding of the possible dynamical responses and the system’s stability and engineers’ prerequisite. In this paper, we discuss the numeric techniques for tracking the year-long motion of the secondary after being perturbed, based upon which its rotational stability is analyzed. For long-term simulations, we compared the performances of several integrating schemes in the scenario of a post-impact full two-body system, including low- and high-order Runge–Kutta methods, and a symplectic integrator that combines the finite element method with a symplectic integral format. For rotational stability analysis of the secondary, we focus on the rotation of the secondary around its long-axis. We calculated a linearised error propagation matrix and found that, in the case of tidal locking of the secondary to the primary, the rotation is stable; as the perturbation amplitude of the spin angular velocity of the secondary increases, the rotation will lose stability and will be prone to being unstable in the long-axis direction of the secondary. Furthermore, we investigated the one-year-long influences of the non-spherical perturbations of the primary and the secondary.

Symplectic model reduction methods for the Vlasov equation

AbstractParticle‐based simulations of the Vlasov equation typically require a large number of particles, which leads to a high‐dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work, we compare several model reduction techniques and demonstrate their applicability to numerical simulations of the Vlasov equation. The necessity of symplectic model reduction algorithms is illustrated with a simple numerical experiment.

A Hamiltonian Formulation on Manifolds for Dynamic Modeling of Constrained Mechanisms and Energy-Preserving Integration

Abstract The dynamic modeling of multibody system is crucial in motion simulations, design, and control of mechanisms. This paper proposes a Hamiltonian formulation on manifolds for mechanism modeling which involves three key steps: (1) the local parameterization of regular configuration space; (2) the coordinate formulation of the Legendre transformation; and (3) the derivation of Hamiltonian equations. Geometric numerical integrators can be naturally deployed on the proposed formulation and achieve a long-time energy-preserving integration. Based on parametric symplectic integrators and the chart-splicing technique, a novel energy-preserving scheme is proposed. Simulations on two constrained mechanisms verify our claims.

Numerical methods, energy conservation, and a new method for particle motion in magnetic fields

This article mathematically examines the energy conservation of explicit Runge–Kutta methods and basic symplectic integration methods for a simple harmonic oscillator and a particle in a magnetic field. With the demonstration of the failure of these methods to conserve the energy of a particle in a magnetic field, a new integration method is presented for this type of system where energy conservation is dictated by the type of floating point variable used, not the size of the time step or the method order. Higher order versions of the method improve the accuracy in the calculation of the particle position and allow for more accurate calculations of the velocity in a spatially varying field. Furthermore, the new method is demonstrated to have improved phase accuracy when compared to several common implicit methods.

Semiexplicit symplectic integrators for non-separable Hamiltonian systems

We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao’s explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems.

Functional Equivariance and Conservation Laws in Numerical Integration

Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs. We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.

An explicit symplectic approach to solving the wave equation in moving media

AbstractAn explicit approach using symplectic time integration in conjunction with traditional finite difference spatial derivatives to solve the wave equation in moving media is presented. A simple operator split of this second order wave equation into two coupled first order equations is performed, allowing these split equations to be solved symplectically. Orders of symplectic time integration ranging from first to fourth along with orders of spatial derivatives ranging from second to sixth are explored. The case of cylindrical acoustic spreading in air under a constant velocity in a 2D square structured domain is considered. The variation of the computed time‐of‐flight, frequency, and wave length are studied with varying grid resolution and the deviations from the analytical solutions are determined. It was found that symplectic time integration interferes with finite difference spatial derivatives higher than second order causing unexpected results. This is actually beneficial for unstructured finite volume tools like OpenFOAM where second order spatial operators are the state‐of‐the art. Cylindrical acoustic spreading is simulated on an unstructured 2D triangle mesh showing that symplectic time integration is not limited to the spatial discretization paradigm and overcomes the numerical diffusion arising with the in‐built numerical methods which hinder wave propagation.

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

A manifold of planar triangular meshes with complete Riemannian metric

Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes forms a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.

Nonlinear state equation and adaptive symplectic algorithm for the control rod drop

Control rod drop is the key part of ensuring the safety of nuclear reactor shutdown. Aiming at the control rod drop problem, we established a physical model of the control rod drive line. The proposed model considers the influence of the absorbing rod’s position and hence evaluates the resultant force accurately on the control rod during the whole dropping process. Based on the physical model, a unified coupling state equation on the control rod drop and fluid flow is developed by integrating the motion variables of the control rod and the fluid into a state vector. A time step adaptive symplectic (TSAS) algorithm is proposed to solve the proposed state equation. The TSAS algorithm efficiently and stably predicts the motion states of the control rod and the fluid synchronously and accurately captures the sudden change of flow state. Numerical examples confirm the correctness of the proposed model and the TSAS algorithm. Sensitivity analysis is also conducted to discuss the effects of different parameters on the control rod drop, which provides reference and help for the design and manufacturing of the control rod drive line.

Equivalence between two charged black holes in dynamics of orbits outside the event horizons

Using the Fermi–Dirac distribution function, Balart and Vagenas gave a charged spherically symmetric regular black hole, which is a solution of Einstein field equations coupled to a nonlinear electrodynamics. In fact, the regular black hole is a Reissner–Nordström (RN) black hole when a metric function is given a Taylor expansion to first order approximations. It does not have a curvature singularity at the origin, but the RN black hole does. Both black hole metrics have horizons and similar asymptotic behaviors, and satisfy the weak energy conditions everywhere. They are almost the same in photon effective potentials, photon circular orbits and photon spheres outside the event horizons. Due to the approximately same photon spheres, the two black holes have no explicit differences in the black hole shadows and constraints of the black hole charges based on the First Image of Sagittarius A $$^{*}$$ . There are relatively minor differences between effective potentials, stable circular orbits and innermost stable circular orbits of charged particles outside the event horizons of the two black holes immersed in external magnetic fields. Although the two magnetized black holes allow different construction methods of explicit symplectic integrators, they exhibit approximately consistent results in the regular and chaotic dynamics of charged particles outside the event horizons. Chaos gets strong as the magnetic field parameter or the magnitude of negative Coulomb parameter increases, but becomes weak when the black hole charge or the positive Coulomb parameter increases. A variation of dynamical properties is not sensitive dependence on an appropriate increase of the black hole charge. The basic equivalence between the two black hole gravitational systems in the dynamics of orbits outside the event horizons is due to the two metric functions having an extremely small difference. This implies that the RN black hole is reasonably replaced by the regular black hole without curvature singularity in many situations.

Resonance capture and long-term evolution of planets in binary star systems

Aims. The growing population of planets discovered in orbit around one stellar component of a binary star raises the question of the influence of the binary companion on the formation process of planetary systems. The aim of this work is to study the impact of a binary companion on the evolution of two-planet systems during both the Type-II migration phase and their long-term evolution after the dissipation of the protoplanetary disk. Methods. We used the symplectic integrator SyMBA, modified to include a wide binary companion. We also included the Type-II migration of giant planets during the protoplanetary disk phase with suitable eccentricity and inclination damping as well as the gravitational potential acting on the planets due to the disk and the nodal precession of the disk induced by the binary companion. We considered various inclinations, eccentricities, and separations of the binary companion. Results. Disk migration allows for the formation of planet pairs in mean-motion resonances despite the presence of the binary companion. When the binary separation is wide (1000 au), the timescale of the perturbations that it raises on the planets is longer than the disk’s lifetime and resonant pairs are routinely formed in the 2:1, 5:2, and 3:1 commensurabilities. Provided the planet-planet interaction timescale is smaller than the timescale of binary perturbations, these systems can remain in resonance long after the disk has dissipated. When the binary separation is smaller (250 au), only planets in the 2:1 resonance tend to remain in a resonant state and more chaotic evolutions are observed, as well as more ejections. After those ejections, the remaining planet can become eccentric due to the perturbations from the binary companion in addition, for strongly inclined binary companions, captures in the von Ziepel-Lidov-Kozai resonance can occur. Whereas in systems with two planets, this mechanism is quenched by planet-planet interactions. Our simulations reveal that the interplay between planet-disk, planet-planet, and planet-binary interactions can lead to the formation of resonant pairs of planets which remain stable over timescales much longer than the disk’s lifetime.

Graveyard orbits for future Mars missions

Mars is expected to become a focal point of exploration (human and robotic) in the coming century, with a very likely need for a robust space infrastructure. Be it communication and navigation satellite constellations or scientific missions in low Mars orbits (LMO) and Areosynchronous orbits (ASO), every individual satellite will have a definitive period of operation after which it becomes derelict. At the end-of-life (EOL) the satellite shall be proactively dealt with in a sustainable manner to protect our access to the space environment of Mars and opportunities to use this. Clearly, impacting Mars or escaping Mars’ gravity are no viable options. This paper aims at identifying graveyard orbit solutions in circummartian space for future Mars space debris. Orbital stability for a period of 200 years is studied for Martian orbits using the symplectic integration technique. Extensive validations are performed and propagation and integration settings are tuned to suit a variety of configurations. A plethora of candidate graveyard orbit solutions are found and presented for orbits in the ASO and LMO regimes. For example, it is found that transferring an ASO satellite to 400 km below the nominal orbit altitude would ensure a stability margin of ±25 km for at least 200 years. Multiple orbital geometry characteristics (combinations of semi-major axis, inclination, right ascension of ascending node), satellite geometries (various values of area-to-mass ratio) and uncertainties are studied to produce a comprehensive analysis of long-term stability of potential graveyard orbits around Mars, making them attractive for such purposes. The protected zones are found to be safe from debris even for an uncertainty in initial eccentricity of 0.01 and variations in cross-sectional area due to uncontrolled tumbling. The overall objective of this paper is to make designers of future missions to Mars aware of the EOL aspects and include this in their mission design proposals at an early stage already.

Meshless symplectic and multi-symplectic algorithm for Klein–Gordon–Schrödinger system with local RBF collocation

This work proposes a novel meshless symplectic and multi-symplectic approximation for Klein–Gordon–Schrödinger system. The main features of this method are: (I) the application of local radial basis function (RBF) collocation method (LRBFCM) in spatial discretization; (II) the application of symplectic integrator in time discretization. LRBFCM method can deal with the ill-conditioned problem and shape-parameter sensitivity of global RBF method, and is simple and effective. The conservatism of this method is discussed and its accuracy is evaluated. Numerical simulations are given for one dimension (1D)/2D with regular and irregular domains to verify the effectiveness of our new method.

Chaos in a Magnetized Brane-World Spacetime Using Explicit Symplectic Integrators

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*.

A meshless multi-symplectic local radial basis function collocation scheme for the “good” Boussinesq equation

A novel meshless multi-symplectic scheme is proposed for the “good” Boussinesq equation. The scheme consists of a local radial basis function (RBF) collocation method (LRBFCS) in space and a symplectic integrator in time. The LRBFCS is simple and efficient, since only a sparse banded linear system has to be solved. Moreover, it can avoid the ill-conditioned problem and shape-parameter-sensitivity of the global RBF method. The multi-symplectic LRBFCS (MLRBFCS) is more accurate than traditional methods. Numerical experiments with uniform knots and random knots are designed to verify the effectiveness of the method.

Dynamic analysis on flexible hub-beam with step-variable cross-section

Decreasing the vibration amplitude of the manipulator in the Space Manipulator Systems (SMS) with a given mass limitation is an effective way to reduce the control costs for the SMS. In this paper, the dynamic response of the SMS modeled by a flexible hub-beam with a step-variable cross-section excited by an external torque is investigated employing a new complex structure-preserving method. The dynamic model of the flexible hub-beam with a step-variable cross-section is deduced firstly. Then, a new structure-preserving method connecting the symplectic precise integration method and the generalized multi-symplectic method is proposed, in which, the time-consuming iteration is avoided and the simulation time is saved. In the numerical simulations, several simple cases for the step-variable cross-section are considered and the stable vibration amplitudes of the flexible beam at the free end are reported. The main contribution of this work is proposing a structure-preserving method for the solving the coupling dynamic problems of the hub-beam system to provide some suggestions on the structure design for the SMS directly.

Hamiltonian preserving nonlinear optics

In this paper we present a method of constructing a discrete nonlinear accelerator lattice that has an approximate integral of motion that is given upfront. The integral under consideration is a Hamiltonian in normalized (canonical) coordinates that is preserved by a lattice with a given accuracy. We establish a connection between the integrator of a Hamiltonian in normalized coordinates and a real lens arrangement. We consider accelerator as an analog computer and apply known algorithms of high-order symplectic integrators, to produce several nonlinear lattices. Following this concept we introduce new lattice design based on Ruth and Yoshida integrator that could be experimentally tested at existing accelerator facilities and Paul traps.