We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.
By introducing an integration factor to the differential one-form of contact dynamics, equations of motion are derived variationally, and contact Poisson bracket and contact Lagrangian are formulated. Discrete symplectic integrator, named hybrid leap-frog method, is found to be a numerical solver of equations of motion for contact dissipative systems.
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.
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…
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）
Collective Lie-Poisson integrators on $\mathbf{R}^{3}$
Institute of Fundamental Sciences, Massey University（梅西大学基础科学研究所）；Department of Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）；Department of Mathematics, University of Bergen（卑尔根大学数学系）
Butcher series for Hamiltonian Poisson integrators through symplectic groupoids
Univ Rennes, INRIA (Research team MINGuS), IRMAR (CNRS UMR 6625) and ENS Rennes（雷恩大学、INRIA（MINGuS研究团队）、IRMAR（CNRS UMR 6625）和雷恩高等师范学院）；Göttingen Mathematisches Institut, Georg-August-Universität Göttingen（哥廷根数学研究所，格奥尔格-奥古斯特哥廷根大学）
