Symplectic Integrators Research Articles

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.

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.

EXPLICIT SYMPLECTIC-LIKE INTEGRATORS WITH MIDPOINT PERMUTATIONS FOR SPINNING COMPACT BINARIES

ABSTRACT We refine the recently developed fourth-order extended phase space explicit symplectic-like methods for inseparable Hamiltonians using Yoshida’s triple product combined with a midpoint permuted map. The midpoint between the original variables and their corresponding extended variables at every integration step is readjusted as the initial values of the original variables and their corresponding extended ones at the next step integration. The triple-product construction is apparently superior to the composition of two triple products in computational efficiency. Above all, the new midpoint permutations are more effective in restraining the equality of the original variables and their corresponding extended ones at each integration step than the existing sequent permutations of momenta and coordinates. As a result, our new construction shares the benefit of implicit symplectic integrators in the conservation of the second post-Newtonian Hamiltonian of spinning compact binaries. Especially for the chaotic case, it can work well, but the existing sequent permuted algorithm cannot. When dissipative effects from the gravitational radiation reaction are included, the new symplectic-like method has a secular drift in the energy error of the dissipative system for the orbits that are regular in the absence of radiation, as an implicit symplectic integrator does. In spite of this, it is superior to the same-order implicit symplectic integrator in accuracy and efficiency. The new method is particularly useful in discussing the long-term evolution of inseparable Hamiltonian problems.

Spectral-collocation variational integrators

Spectral methods are a popular choice for constructing numerical approximations for smooth problems, as they can achieve geometric rates of convergence and have a relatively small memory footprint. In this paper, we introduce a general framework to convert a spectral-collocation method into a shooting-based variational integrator for Hamiltonian systems. We also compare the proposed spectral-collocation variational integrators to spectral-collocation methods and Galerkin spectral variational integrators in terms of their ability to reproduce accurate trajectories in configuration and phase space, their ability to conserve momentum and energy, as well as the relative computational efficiency of these methods when applied to some classical Hamiltonian systems. In particular, we note that spectrally-accurate variational integrators, such as the Galerkin spectral variational integrators and the spectral-collocation variational integrators, combine the computational efficiency of spectral methods together with the geometric structure-preserving and long-time structural stability properties of symplectic integrators.

Lorentz covariant canonical symplectic algorithms for dynamics of charged particles

In this paper, the Lorentz covariance of algorithms is introduced. Under Lorentz transformation, both the form and performance of a Lorentz covariant algorithm are invariant. To acquire the advantages of symplectic algorithms and Lorentz covariance, a general procedure for constructing Lorentz covariant canonical symplectic algorithms (LCCSAs) is provided, based on which an explicit LCCSA for dynamics of relativistic charged particles is built. LCCSA possesses Lorentz invariance as well as long-term numerical accuracy and stability, due to the preservation of a discrete symplectic structure and the Lorentz symmetry of the system. For situations with time-dependent electromagnetic fields, which are difficult to handle in traditional construction procedures of symplectic algorithms, LCCSA provides a perfect explicit canonical symplectic solution by implementing the discretization in 4-spacetime. We also show that LCCSA has built-in energy-based adaptive time steps, which can optimize the computation performance when the Lorentz factor varies.

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.

Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems

An explicit high-order noncanonical symplectic algorithm for ideal two-fluid systems is developed. The fluid is discretized as particles in the Lagrangian description, while the electromagnetic fields and internal energy are treated as discrete differential form fields on a fixed mesh. With the assistance of Whitney interpolating forms [H. Whitney, Geometric Integration Theory (Princeton University Press, 1957); M. Desbrun et al., Discrete Differential Geometry (Springer, 2008); J. Xiao et al., Phys. Plasmas 22, 112504 (2015)], this scheme preserves the gauge symmetry of the electromagnetic field, and the pressure field is naturally derived from the discrete internal energy. The whole system is solved using the Hamiltonian splitting method discovered by He et al. [Phys. Plasmas 22, 124503 (2015)], which was been successfully adopted in constructing symplectic particle-in-cell schemes [J. Xiao et al., Phys. Plasmas 22, 112504 (2015)]. Because of its structure preserving and explicit nature, this algorithm is especially suitable for large-scale simulations for physics problems that are multi-scale and require long-term fidelity and accuracy. The algorithm is verified via two tests: studies of the dispersion relation of waves in a two-fluid plasma system and the oscillating two-stream instability.

Symplectic fourth-order maps for the collisionalN-body problem

We study analytically and experimentally certain symplectic and time-reversible N-body integrators which employ a Kepler solver for each pair-wise interaction, including the method of Hernandez & Bertschinger (2015). Owing to the Kepler solver, these methods treat close two-body interactions correctly, while close three-body encounters contribute to the truncation error at second order and above. The second-order errors can be corrected to obtain a fourth-order scheme with little computational overhead. We generalise this map to an integrator which employs a Kepler solver only for selected interactions and yet retains fourth-order accuracy without backward steps. In this case, however, two-body encounters not treated via a Kepler solver contribute to the truncation error.

Hamiltonian discontinuous Galerkin FEM for linear, stratified (in)compressible Euler equations: internal gravity waves

The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. A discontinuous Galerkin finite element method has been developed in which this Hamiltonian structure is discretized, resulting in conservation of discrete analogs of phase space and energy. This required (i) the discretization of the Hamiltonian structure using alternating flux functions and symplectic time integration, (ii) the discretization of a divergence-free velocity field using Dirac's theory of constraints and (iii) the handling of large-scale computational demands due to the 3-dimensional nature of internal gravity waves and, in confined, symmetry-breaking fluid domains, possibly its narrow zones of attraction.

Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance.

Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. This article proposes for arbitrary Hamiltonians similar integrators, which are explicit, of any even order, symplectic in an extended phase space, and with pleasant long time properties. They are based on a mechanical restraint that binds two copies of phase space together. Using backward error analysis, Kolmogorov-Arnold-Moser theory, and additional multiscale analysis, an error bound of O(Tδ^{l}ω) is established for integrable systems, where T,δ,l, and ω are, respectively, the (long) simulation time, step size, integrator order, and some binding constant. For nonintegrable systems with positive Lyapunov exponents, such an error bound is generally impossible, but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear Schrödinger equation.

Symplectic integrators for the matrix Hill equation

We consider the numerical integration of the matrix Hill equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, Hill’s equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this work we present new sixth-and eighth-order symplectic exponential integrators that are tailored to Hill’s equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. The proposed methods can also be used for solving general second order linear differential equations where their performance will depend on how the methods are finally adapted to each particular problem or the qualitative properties one is interested to preserve. Several numerical examples illustrate the performance of the new methods.

Higher order variational integrators in optimal control theory

AbstractHigher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well‐known integrator is equivalent to a symplectic partitioned Runge‐Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge‐Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Explicit high-order symplectic integrators for charged particles in general electromagnetic fields

This article considers non-relativistic charged particle dynamics in both static and non-static electromagnetic fields, which are governed by nonseparable, possibly time-dependent Hamiltonians. For the first time, explicit symplectic integrators of arbitrary high-orders are constructed for accurate and efficient simulations of such mechanical systems. Performances superior to the standard non-symplectic method of Runge–Kutta are demonstrated on two examples: the first is on the confined motion of a particle in a static toroidal magnetic field used in tokamak; the second is on how time-periodic perturbations to a magnetic field inject energy into a particle via parametric resonance at a specific frequency.

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.

On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder

We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrodinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the system’s Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to N ≈ 70 sites. An advantage of the latter methods is the better conservation of the system’s second integral, i.e. the wave packet’s norm.

A meshless symplectic method for two-dimensional Schrödinger equation with radial basis functions

In this study we propose a meshless symplectic algorithm for two-dimensional time-dependent Schrödinger equation with radial basis functions (RBFs) approximation theory. The truncation error and global error are also investigated. The numerical experiments show that the scheme is higher-order accuracy, easy to implement with the scattered knots. Furthermore, the algorithm possesses a long-time tracking capability.

Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

In this article, we propose a kind of numerical methods based on the Padé approximations, for two kinds of stochastic Hamiltonian systems. For the general linear stochastic Hamiltonian systems, it is shown that the applied Padé approximations P(k,k) produce numerical solutions that are symplectic, and the proposed numerical schemes based on P(r,s) are of root-mean-square convergence order r+s2. For a special kind of linear stochastic Hamiltonian systems with additive noises, the numerical methods using two kinds of Padé approximations, P(rˆ,sˆ) and P(ř,1), possess root-mean-square convergence order ř+2 when rˆ+sˆ=ř+3, and are symplectic if rˆ=sˆ. These generalize the Padé approximation approaches for symplectic integration of linear Hamiltonian systems to the stochastic context.

Development of a symplectic and phase error reducing perturbation finite-difference advection scheme

ABSTRACTThe aim of this work is to develop a new scheme for solving the pure advection equation. This scheme formulated within the perturbation finite-difference context not only conserves symplecticity but also preserves the numerical dispersion relation equation. The employed symplectic integrator of second-order accuracy in time enables calculation of a long-time accurate solution in the sense that the Hamiltonian is conserved at all times. The generalized high-order spatially accurate perturbation difference scheme optimizes numerical phase accuracy through the minimization of the difference between the numerical and exact dispersion relation equations. Our proposed new class of phase error reducing perturbation difference schemes can in addition locally capture discontinuities underlying the concept of applying a shope/flux limiter. The high-order spatial accuracy can be recovered in a smooth region. Besides the Fourier analysis of the discretization errors, anisotropy and dispersion analyses are both conducted on the dispersion-relation and symplecticity-preserving pure advection scheme to shed light on the distinguished nature of the proposed scheme. Numerical tests are carried out and the results compare well with the exact solutions, demonstrating the efficiency, accuracy, and the discontinuity-resolving ability using the proposed class of high-resolution perturbation finite-difference schemes.

Hamilton–Jacobi theory, symmetries and coisotropic reduction

Reduction theory has played a major role in the study of Hamiltonian systems. Whilst the Hamilton–Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators relies on approximations of a complete solution of the Hamilton–Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton–Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton–Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reduction of the Hamilton–Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge–Marsden reduction procedure [18] and the results in [17]. Quite surprisingly, the classical ansätze available in the literature to solve the Hamilton–Jacobi equation (see [2,19]) are also particular instances of our framework.

Explicit symplectic algorithms based on generating functions for charged particle dynamics.

Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term accuracy and fidelity. For long-term simulations with high efficiency, explicit symplectic algorithms are desirable. However, it is generally believed that explicit symplectic algorithms are only available for sum-separable Hamiltonians, and this restriction limits the application of explicit symplectic algorithms to charged particle dynamics. To overcome this difficulty, we combine the familiar sum-split method and a generating function method to construct second- and third-order explicit symplectic algorithms for dynamics of charged particle. The generating function method is designed to generate explicit symplectic algorithms for product-separable Hamiltonian with form of H(x,p)=p_{i}f(x) or H(x,p)=x_{i}g(p). Applied to the simulations of charged particle dynamics, the explicit symplectic algorithms based on generating functions demonstrate superiorities in conservation and efficiency.