We compare the performances of symplectic and non-symplectic integrators for the computation of normal geodesics and conjugate points in sub-Riemannian geometry at the example of the Martinet case. For this case study we consider first the flat metric, and then a one parameter perturbation leading to non integrable geodesics. From our computations we deduce that near the abnormal directions a symplectic method is much more efficient for this optimal control problem. The explanation relies on the theory of backward error analysis in geometric numerical integration.
This paper studies explicit symplectic adapted exponential integrators for solving charged-particle dynamics in a strong and constant magnetic field. We first formulate the scheme of adapted exponential integrators and then derive its symplecticity conditions. Based on the symplecticity conditions, we propose five practical explicit symplectic adapted exponential integrators. Two numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new methods.
A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems
Qufu Normal University（曲阜师范大学）；University of Tübingen（蒂宾根大学）；Nanjing University（南京大学）
HNets is a class of neural networks on grounds of physical prior for learning Hamiltonian systems. This paper explains the influences of different integrators as hyper-parameters on the HNets through error analysis. If we define the network target as the map with zero empirical loss on arbitrary training data, then the non-symplectic integrators cannot guarantee the existence of the network targets of HNets. We introduce the inverse modified equations for HNets and prove that the HNets based on symplectic integrators possess network targets and the differences between the network targets and t…
It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators based on Type II or Type III generating functions is required instead of the more traditional L…
Preservation of stability properties near fixed points of linear hamiltonian systems by symplectic integrators
Harbin Institute of Technology（哈尔滨工业大学）；University of Washington（华盛顿大学）；Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods
College of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）；LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Science, East China Jiaotong University（华东交通大学理学院）
In this paper we propose and investigate a general approach to constructing local energy-preserving algorithms which can be of arbitrarily high order in time for solving Hamiltonian PDEs. This approach is based on the temporal discretization using continuous Runge-Kutta-type methods, and the spatial discretization using pseudospectral methods or Gauss--Legendre collocation methods. The local energy conservation law of our new schemes is analyzed in detail. The effectiveness of the novel local energy-preserving integrators is demonstrated by coupled nonlinear Schrödinger equations and 2D nonlin…
Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized algorithms and from choosing the best ordering of terms. The cost in programming and execution time is minimal, while the performance improvements can be large.
Center of Mathematics for Artificial Intelligence and Data Science（人工智能与数据科学数学中心）；Chubu University（中部大学）；The Institute of Statistical Mathematics（统计数学研究所）；RIKEN AIP（理化学研究所先进智能项目中心）
In this paper, explicit stable integrators based on symplectic and contact geometries are proposed for a non-autonomous ordinarily differential equation (ODE) found in improving convergence rate of Nesterov's accelerated gradient method. Symplectic geometry is known to be suitable for describing Hamiltonian mechanics, and contact geometry is known as an odd-dimensional counterpart of symplectic geometry. Moreover, a procedure, called symplectization, is a known way to construct a symplectic manifold from a contact manifold, yielding Hamiltonian systems from contact ones. It is found in this pa…
Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka--Volterra model
School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）；School of Science, East China Jiaotong University（华东交通大学理学院）
We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their …
Multi-symplectic Preserving Integrator for the Schrödinger Equation with Wave Operator
Jiangxi Normal University（江西师范大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, CAS（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
The kinetic term of the $N$-body Hamiltonian system defined on the surface of the sphere is non-separable. As a result, standard explicit symplectic integrators are inapplicable. We exploit an underlying hierarchy in the structure of the kinetic term to construct an explicit time-reversible symplectic scheme of second order. We use iterative applications of the method to construct a fourth order scheme and demonstrate its efficiency.
Exponential Stability Estimate of Symplectic Integrators for Integrable Hamiltonian Systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）；South China University of Technology（华南理工大学）
In this article we introduce a low order implicit symplectic integrator designed to follow the Hamiltonian flow as close as possible. This integrator is obtained by the method of Liouvillian forms and does not require particular hypotheses on the Hamiltonian. The numerical scheme introduced in this paper is a modification of the symplectic mid-point rule, it is symmetric and it is obtained by an isotopy of the deformation of the exact Hamiltonian flow to the straight line passing by two consecutive points of the discretized flow. This isotopy generates an alternative vector field on the flow l…
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 integrat…
In this paper, we are concerned with the construction and analysis of a new class of methods obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that the new integrators are symmetric and symplectic up to high orders if one uses a symmetric and symplectic basic method. In terms of efficiency, the aforementioned technique requires fewer stages than standard compositions of the same orders and is thus expected to lead to faster methods.
High precision Symplectic Integrators for the Solar System
Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC（巴黎天文动力学与系统研究所）；Instituto de Matemática Multidisciplinar, Universitat Politécnica de València（瓦伦西亚理工大学多学科数学研究所）；Institut de Matemàtiques i Aplicacions de Castelló and Departament de Matemàtiques, Universitat Jaume I（卡斯特利翁数学与应用研究所及马德里卡洛斯三世大学数学系）；Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, EHU/UPV（巴斯克大学计算机学院）
Numerical Construction of Quasi-Periodic Solutions Beyond Symplectic Integrators
University of Chinese Academy of Sciences（中国科学院大学）；Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）
Institute of Fundamental Sciences, Massey University（马斯特里赫特大学基础科学研究所）；Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）；Mathematics and Mathematical Statistics, Umeå Universitet（于默奥大学数学与数理统计系）
Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be appr…
We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible-irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g., Verlet) with degenerate variables being left unchanged, for which an associate…
Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems
Inner Mongolia University（内蒙古大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院华罗庚数学重点实验室）
