High order symplectic integrators for perturbed Hamiltonian systems

Abstract

We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O({\tau }^p\epsilon +{\tau }^2{\epsilon }^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O({\tau }^p\epsilon +{\tau }^4{\epsilon }^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.

24 pages, 6 figurres

References in corpus (1)

• Pseudo-High-Order Symplectic Integrators

Cited by in corpus (157)

• La2010: A new orbital solution for the long term motion of the Earth
• A dynamically-packed planetary system around GJ667C with three super-Earths in its habitable zone
• REBOUNDx: A Library for Adding Conservative and Dissipative Forces to Otherwise Symplectic N-body Integrations
• Delocalization of wave packets in disordered nonlinear chains
• A pair of planets around HD 202206 or a circumbinary planet?
• TTVFast: An efficient and accurate code for transit timing inversion problems
• The HARPS search for southern extra-solar planets XIX. Characterization and dynamics of the GJ876 planetary system
• Chaotic diffusion in the Solar System
• The HARPS search for southern extra-solar planets. XXVII. Up to seven planets orbiting HD 10180: probing the architecture of low-mass planetary systems
• Constraining the cometary flux through the asteroid belt during the late heavy bombardment
• The HARPS search for southern extra-solar planets. IX. mu Ara, a system with four planets
• Habitable-zone super-Earth candidate in a six-planet system around the K2.5V star HD 40307
• The crossover from strong to weak chaos for nonlinear waves in disordered systems
• Intermittent many-body dynamics at equilibrium
• New families of symplectic splitting methods for numerical integration in dynamical astronomy
• Injection of Oort Cloud Comets: The Fundamental Role of Stellar Perturbations
• HD45364, a pair of planets in a 3:2 mean motion resonance
• Numerical integration of variational equations
• Multiple mean motion resonances in the HR 8799 planetary system
• Spreading of wave packets in disordered systems with tunable nonlinearity
• Asteroid families in the first order resonances with Jupiter
• Rapid dynamical chaos in an exoplanetary system
• The HARPS search for southern extra-solar planets. XXXIV. A planetary system around the nearby M dwarf GJ163, with a super-Earth possibly in the habitable zone
• Splitting and composition methods in the numerical integration of differential equations
• The resonant structure of Jupiter's trojan asteroids-II. What happens for different configurations of the planetary system
• High precision Symplectic Integrators for the Solar System
• Nonequilibrium chaos of disordered nonlinear waves
• Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients
• Dynamical stability analysis of the HD202206 system and constraints to the planetary orbits
• A Double Planetary System around the Evolved Intermediate-Mass Star HD 4732
• Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes
• Nonlinear waves in disordered chains: probing the limits of chaos and spreading
• On post-Newtonian orbits and the Galactic-center stars
• Radial-velocity discovery of a second planet in the TOI-1338/BEBOP-1 circumbinary system
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• A dynamical analysis of the Kepler-11 planetary system
• Long term stability and dynamical environment of the PSR 1257+12 planetary system
• The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects
• An anisotropic distribution of spin vectors in asteroid families
• HD60532, a planetary system in a 3:1 mean motion resonance
• Nonlinear lattice waves in heterogeneous media
• 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport
• Frequency map analysis and quasiperiodic decompositions
• High order symplectic integrators for planetary dynamics and their implementation in REBOUND
• Symplectic integration of space debris motion considering several Earth's shadowing models
• Superdiffusive Transport and Energy Localization in Disordered Granular Crystals
• The Eos family halo
• Eurybates - the only asteroid family among Trojans?
• Delocalization and spreading in a nonlinear Stark ladder
• Wave chaos in the non-equilibrium dynamics of the Gross-Pitaevskii equation
• Dynamical glass in weakly non-integrable Klein-Gordon chains
• Superdiffusive dispersals impart the geometry of underlying random walks
• Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice
• Subdiffusion of nonlinear waves in quasiperiodic potentials
• The SOPHIE search for northern extrasolar planets. X. Detection and characterization of giant planets by the dozen
• Energy transport in one-dimensional disordered granular solids
• The Reversibility Error Method (REM): a new, dynamical fast indicator for planetary dynamics
• On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems
• Triggering waves in nonlinear lattices: Quest for anharmonic phonons and corresponding mean free paths
• Detectability of quasi-circular co-orbital planets. Application to the radial velocity technique
• Co-orbital exoplanets from close period candidates: The TOI-178 case
• A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes
• A Symplectic Integrator for Hill's Equations
• Can GJ 876 host four planets in resonance?
• Non-Reciprocal Geometric Wave Diode by Engineering Asymmetric Shapes of Nonlinear Materials
• Long-term dynamics of the solar system inner planets
• High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation
• The orbital architecture and debris disks of the HR 8799 planetary system
• A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems
• HD 202206 : A Circumbinary Brown Dwarf System
• Hektor - an exceptional D-type family among Jovian Trojans
• Neptune Trojans and Plutinos: colors, sizes, dynamics, and their possible collisions
• TRACE: a code for Time-Reversible Astrophysical Close Encounters
• Revisiting high-order Taylor methods for astrodynamics and celestial mechanics
• Renormalized phonons in nonlinear lattices: A variational approach
• Quasiperiodic driving of Anderson localized waves in one dimension
• Behavior and Breakdown of Higher-Order Fermi-Pasta-Ulam-Tsingou Recurrences
• Dynamics of charged particles moving around Kerr black hole with inductive charge and external magnetic field
• On the coplanar eccentric non restricted co-orbital dynamics
• Computational efficiency of symplectic integration schemes: Application to multidimensional disordered Klein-Gordon lattices
• Chaotic diffusion of the Vesta family induced by close encounters with massive asteroids
• Two Super-Earths in the 3:2 MMR around KOI-1599
• Are long-term $N$-body simulations reliable?
• Temperature dependence of thermal conductivities of coupled rotator lattice and the momentum diffusion in standard map
• Stability analysis of the Martian obliquity during the Noachian era
• Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance
• Thermalization slowing down in multidimensional Josephson junction networks
• Stepsize errors in the $N$-body problem: discerning Mercury's true possible long-term orbits
• On the accuracy of symplectic integrators for secularly evolving planetary systems
• GRIT: a package for structure-preserving simulations of gravitationally interacting rigid-bodies
• Do nonlinear waves in random media follow nonlinear diffusion equations?
• Nonlinear Lattice Waves in Random Potentials
• The architecture and formation of the Kepler-30 planetary system
• Heat conduction and energy diffusion in momentum-conserving 1D full lattice ding-a-ling model
• A possible misaligned orbit for the young planet AU Mic c
• Physical and dynamical characterization of the Euphrosyne asteroid Family
• Simple physics and integrators accurately reproduce Mercury instability statistics
• The $\beta$ Fermi-Pasta-Ulam-Tsingou Recurrence Problem
• A study of 3-dimensional shapes of asteroid families with an application to Eos
• Mechanic: the MPI/HDF code framework for dynamical astronomy
• Enhancement of chaotic subdiffusion in disordered ladders with synthetic gauge fields
• High order regularised symplectic integrator for collisional planetary systems
• Statistics of Vector Manakov Rogue Waves
• BEBOP VII. SOPHIE discovery of BEBOP-3b, a circumbinary giant planet on an eccentric orbit
• Long-term orbital and rotational motions of Ceres and Vesta
• The Scaling, Splitting and Squaring Method for the Exponential of Perturbed Matrices
• Non-singular recursion formulas for third-body perturbations in mean vectorial elements
• Elliptic tori in FPU non-linear chains with a small number of nodes
• Splitting and composition methods with embedded error estimators
• Stretch diffusion and heat conduction in 1D nonlinear lattices
• Variational Integrators for Almost-Integrable Systems
• Variational approach to renormalized phonon in momentum-nonconserving nonlinear lattices
• Two neighbours of the ultra-short-period Earth-sized planet K2-157 b in the warm Neptunian savanna
• Analysis of ballistic transport and resonance in the $\alpha$-Fermi-Pasta-Ulam-Tsingou model
• The K2-24 planetary system revisited by CHEOPS
• A pair of Jovian Trojans at the L4 Lagrange point
• Dynamical chaos in the integrable Toda chain induced by time discretization
• Secular orbital dynamics of the innermost exoplanet of the $\upsilon$-Andromedæ system
• Forward Symplectic Integrators for Solving Gravitational Few-Body Problems
• Symplectic integration of deviation vectors and chaos determination. Application to the Hénon-Heiles model and to the restricted three-body problem
• The adjustment-stabilization method for constrained systems
• Chaotic dynamics of a Bose-Einstein condensate coupled to a qubit
• Hamiltonian formulation of X-point collapse in an extended magnetohydrodynamics framework
• A semi-relativistic approach to the circular restricted three body problem and a numerical experiment around the 3:2 resonance
• Quench dynamics in disordered two-dimensional Gross-Pitaevskii Lattices
• Fourth-order leapfrog algorithms for numerical time evolution of classical and quantum systems
• Close-in compact super-Earth systems emerging from resonant chains: slow destabilization by unseen remnants of formation
• Dynamics of trans-Neptunian objects near the 3/1 mean-motion resonance with Neptune
• High order explicit symplectic integrators for the Discrete Non Linear Schrödinger equation
• A numerical criterion evaluating the robustness of planetary architectures; applications to the $\upsilon$ Andromedæ system
• A joint effort to discover and characterize two resonant mini Neptunes around TOI-1803 with TESS, HARPS-N and CHEOPS
• New insights on numerical error in symplectic integration
• High Order Three Part Split Symplectic Integration Schemes
• Simulating DNLS models
• Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas
• Existence proof of librational invariant tori in an averaged model of HD60532 planetary system
• Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat
• Dynamics of co-orbital exoplanets
• Evolution of a migrating giant planet in the presence of an inclined binary companion
• Comparing the efficiency of numerical techniques for the integration of variational equations
• Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials
• Significant improvement in planetary system simulations from statistical averaging
• An efficient integrator for stellar dynamics in effective gravity fields based on the isochrone potential
• Generalized flow-composed symplectic methods for post-Newtonian Hamiltonian systems
• Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions
• On the symplectic integration of the Klein Gordon lattice model
• Trotter transition in BCS pairing dynamics
• Variational integrators for perturbed non-canonical Hamiltonian systems
• Improved characterization of the TOI-2141 system: a dense sub-Neptune with non-transiting inner and outer companions
• Galactic tides in the Solar System within a non-axisymmetric Milky Way model adjusted to Gaia data
• Extended phase-space symplectic integration for electron dynamics
• Chaotic behaviour of disordered nonlinear lattices
• On the use of the MEGNO indicator with the Global Symplectic Integrator
• The orbital architecture and stability of the $\mu$ Arae planetary system
• Numerical integrators for continuous disordered nonlinear Schrödinger equation
• Clean Numerical Simulation: A New Strategy to Obtain Reliable Solutions of Chaotic Dynamic Systems
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems