Symplectic Integration Schemes Research Articles

Symplectic spatial integration schemes for systems of balance equations

A method to generate geometric pseudo-spectral spatial discretization schemes for hyperbolic or parabolic partial differential equations is presented. It applies to the spatial discretization of systems of conservation laws with boundary energy flows and/or distributed source terms. The symplecticity of the proposed spatial discretization schemes is defined with respect to the natural power pairing (form) used to define the port-Hamiltonian formulation for the considered systems of balance equations. The method is applied to the resistive diffusion model, a parabolic equation describing the plasma dynamics in tokamaks. A symplectic Galerkin scheme with Bessel conjugated bases is derived from the usual Galerkin method, using the proposed method. Besides the spectral and energetic properties expected from the symplecticity of the method, it is shown that more accurate approximation of eigenfunctions and reduced numerical oscillations result from this choice of conjugated approximation bases. Finally, the obtained numerical results are validated against experimental data from the tokamak Tore Supra facility.

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.

Variational integrators – A continuous time approach

SummaryThis article presents a family of variational integrators from a continuous time point of view. A general procedure for deriving symplectic integration schemes preserving an energy‐like quantity is shown, which is based on the principle of virtual work. The framework is extended to incorporate holonomic constraints without using additional regularization. In addition, it is related to well‐known partitioned Runge–Kutta methods and to other variational integration schemes. As an example, a concrete integration scheme is derived for the planar pendulum using both polar and Cartesian coordinates. Copyright © 2016 John Wiley & Sons, Ltd.

Symmetric integrator for nonintegrable Hamiltonian relativistic systems

By combining a standard symmetric, symplectic integrator with a new step size controller, we provide an integration scheme that is symmetric, reversible and conserves the values of the constants of motion. This new scheme is appropriate for long-term numerical integrations of geodesic orbits in spacetime backgrounds, whose corresponding Hamiltonian system is nonintegrable, and, in general, for any nonintegrable Hamiltonian system whose kinetic part depends on the position variables. We show by numerical examples that the new integrator is faster and more accurate (i) than the standard symplectic integration schemes with or without standard adaptive step size controllers and (ii) than an adaptive step Runge-Kutta scheme.

EFFICIENT INTEGRATION OF THE VARIATIONAL EQUATIONS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS: APPLICATION TO THE FERMI–PASTA–ULAM LATTICE

We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.

An energy-preserving Discrete Element Method for elastodynamics

We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical results are illustrated by numerical simulations of test cases involving large displacements.

Symplectic integration of space debris motion considering several Earth’s shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth’s gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth’s shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios.

Variational multirate integration of constrained dynamics

AbstractMechanical systems with dynamics on varying time scales, in particular those including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Small time steps increase integration times unnecessarily, especially for costly function evaluations. For systems comprising fast and slow dynamics, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. Main challenges are the identification of fast and slow parts (e.g. by separating the energy or by distinguishing sets of variables), the synchronisation of their dynamics and in particular the treatment of mixed parts that often appear when fast and slow dynamics are coupled by constraints. In this contribution, a multirate integrator is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Variational integrators (based on a discrete version of Hamilton's principle) lead to symplectic and momentum preserving integration schemes that also exhibit good energy behavior. The resulting multirate variational integrator has the same preservation properties. An example demonstrates the performance of the multirate integrator for constrained multibody dynamics. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hamiltonian higher-order nonlinear Schrödinger equations for broader-banded waves on deep water

Starting from the Hamiltonian formulation of the water wave problem and using the approach recently developed by Craig et al. (2010) [8], we derive Hamiltonian versions of the higher-order nonlinear Schrödinger equations for broader-banded deep-water waves, originally proposed by Trulsen and co-workers (1996, 2000) [4,5]. A Benjamin–Feir stability analysis is performed and shown to be in good agreement with previous work. Numerical simulations using a symplectic time integration scheme are also presented to illustrate the performance of these new models with regards to their conservative and Benjamin–Feir stability properties.

Using Symplectic Schemes for Nonlinear Dynamic Analysis of Flexible Beams Undergoing Overall Motions

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.

Symplectic integration of post-Newtonian equations of motion with spin

We present a noncanonically symplectic integration scheme tailored to numerically computing the post-Newtonian motion of a spinning black-hole binary. Using a splitting approach we combine the flows of orbital and spin contributions. In the context of the splitting, it is possible to integrate the individual terms of the spin-orbit and spin-spin Hamiltonians analytically, exploiting the special structure of the underlying equations of motion. The outcome is a symplectic, time-reversible integrator, which can be raised to arbitrary order by composition. A fourth-order version is shown to give excellent behavior concerning error growth and conservation of energy and angular momentum in long-term simulations. Favorable properties of the integrator are retained in the presence of weak dissipative forces due to radiation damping in the full post-Newtonian equations.

Static dipole polarizability of hydrogenlike ions in Debye plasmas

The scaled static dipole polarizabilities of $1s$ and $2s$ states of hydrogenlike ions embedded in a plasma environment are calculated as function of the interaction screening. The plasma screening of Coulomb interaction is described by the Debye-H\uckel potential. The electron energies and wave functions of bound and continuum states are calculated by solving numerically the Schr\odinger equation with this potential in a symplectic integration scheme. The screening of Coulomb interactions reduces the number of bound states, decreases the energies of bound states and broadens the radial extension of the wave functions of bound and continuum states. All these effects result in a dramatic increase in static polarizability when the interaction screening increases. Comparison of present results with those of other authors is made when available.

A new class of symplectic integration schemes based on generating functions

We present a new family of one-step symplectic integration schemes for Hamiltonian systems of the general form $${\dot y=J^{-1}\nabla H(y)^T}$$. Such a class of methods contains as particular cases the methods of Miesbach and Pesch (Numer Math 61:501–521, 1992), and also the family of symplectic Runge-Kutta methods. As in the case of the methods introduced in Miesbach and Pesch (Numer Math 61:501–521, 1992), the new integration methods are constructed by defining a generating function, which automatically determines a symplectic map. The resulting methods are implicit, and require the evaluation of the gradient of the Hamiltonian function as well as the Hessian times a vector.

Photoionization of Li and radiative recombination of Li+ in Debye plasmas

The photoionization cross sections in the photoelectron energy below 2 Ry are calculated for the ground and n≤4 excited states of Li embedded in plasma environments and the radiative-recombination (RR) rate coefficients for Li+ were presented for temperature T=100–32 000 K in a wide range of plasma parameters. The plasma screening interaction is described by the Debye–Hückel model and the energy levels and wave functions including both the bound and continuum states are calculated by solving the Schrödinger equation numerically in a symplectic integration scheme. The screening of Coulomb interactions remarkably changes the photoionization cross sections near the ionization threshold, and especially for the ns states, the Cooper minimum is uncovered and shifted to the higher energy as the screening interaction increases. The RR rate coefficients at low temperature have a complex variation on the Debye lengths; whereas at higher temperature the RR rate coefficients decrease with the increasing of screening effects. Comparison of present results with those of other authors when available is made.

An Averaging Technique for Highly Oscillatory Hamiltonian Problems

In this paper, we are concerned with the numerical solution of highly oscillatory Hamiltonian systems with a stiff linear part. We construct an averaged system whose solution remains close to the exact one over bounded time intervals, possesses the same adiabatic and Hamiltonian invariants as the original system, and is nonstiff. We then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in [E. Cances et al., Numer. Math., 100 (2005), pp. 211-232]. Eventually, we demonstrate the efficiency of our approach on two test problems with one or several frequencies.

Bound-bound transitions in hydrogenlike ions in Debye plasmas

Plasma screening effects on the properties of bound-bound transitions of hydrogenlike ions imbedded in Debye plasmas are investigated. The electron eigenenergies and wave functions are determined by numerically solving the scaled Schr\odinger equation with a Debye potential by the fourth-order symplectic integration scheme. The scaled spectral properties of hydrogenlike ions in the plasma, including the transition frequencies, absorption oscillator strengths, radiative transition probabilities, as well as the line intensities of the Lyman and Balmer series, are presented for a wide range of plasma screening parameters. While for the $\ensuremath{\Delta}n\ensuremath{\ne}0$ transitions the oscillator strengths and spectral line intensities decrease with increasing the plasma screening, those for the $\ensuremath{\Delta}n=0$ transitions rapidly increase. The lines associated with the $\ensuremath{\Delta}n\ensuremath{\ne}0$ transitions are redshifted, whereas those for $\ensuremath{\Delta}n=0$ transitions are blueshifted. Comparison of present results with those of other authors, when available, is made. The results reported here should be useful in the interpretation of spectral properties of hydrogenlike ions in laboratory and astrophysical Debye plasmas.

Extended Born-Oppenheimer Molecular Dynamics

A Lagrangian generalization of time-reversible Born-Oppenheimer molecular dynamics Niklasson et al. [Phys. Rev. Lett. 97, 123001 (2006)10.1103/PhysRevLett.97.123001] is proposed. The formulation enables the application of higher-order symplectic or geometric integration schemes that are stable and energy conserving even under incomplete self-consistency convergence. It is demonstrated how the accuracy is improved by over an order of magnitude compared to previous formulations at the same level of computational cost. The proposed Lagrangian includes extended electronic degrees of freedom as auxiliary dynamical variables in addition to the nuclear coordinates and momenta. While the nuclear degrees of freedom propagate on the Born-Oppenheimer potential energy surface, the extended auxiliary electronic degrees of freedom evolve as a harmonic oscillator centered around the adiabatic propagation of the self-consistent ground state.

Fluid–shell structure interaction analysis by coupled particle and finite element method

This paper presents a coupled particle and finite element method for fluid–shell structure interaction analysis. The Moving Particle Semi-Implicit (MPS) method is used to analyze fluid flow and the MITC4 shell element is used in the FEM analysis of the structure. This paper considers partitioned coupling between the fluid and structural solvers. In order to satisfy compatibility in the employed partitioned coupling scheme, the Neumann–Dirichlet condition is applied to both the fluid and the structure. A symplectic time integration scheme is used to preserve energy when analyzing the shell structure. If the frequencies of the shell analysis are much higher than those of the MPS fluid solver, its time integration scheme is sub-cycled. When the presented coupling scheme was applied to simulate the sloshing phenomenon in an elastic thin shell structure, fluid fragmentation and large structural deformations were observed.

Dérivation de schémas numériques symplectiques pour des systèmes hamiltoniens hautement oscillants

We introduce here a class of symplectic schemes for the numerical integration of highly oscillatory Hamiltonian systems. The bottom line for the approach is to exploit the Hamilton–Jacobi form of the equations of motion. Because we perform a two-scale expansion of the solution of the Hamilton–Jacobi equations itself, we readily obtain, after an appropriate discretization, a symplectic integration scheme. An example of such an integration scheme, following the general approach, is presented here on a specific commonly used test case. The efficiency of the approach is demonstrated. Further developments will be presented elsewhere. To cite this article: C. Le Bris, F. Legoll, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Hamiltonian map description of electron dynamics in gyrotrons

Electron dynamics in gyrotron resonators are described in terms of a Hamiltonian map. This map incorporates the dependency of electron dynamics on the parameters of the interacting radio-frequency (RF) field and it can be used for trajectory calculations through successive iteration, resulting in a symplectic integration scheme. The direct relation of the map to the physics of the model, along with its canonical form (phase space volume preserving) and the significant reduction of the number of iteration steps required for acceptable accuracy, are the main advantages of this method in comparison with standard methods such as Runge-Kutta. The general form of the Hamiltonian map allows for wide applications as a part of several numerical algorithms which incorporate CPU-consuming electron trajectories calculations