Construction And Application Of High Order Explicit Symplectic-like Algorithms

Pihajoki proposed the extended phase-space second-order explicit symmetric leapfrog methods for inseparable Hamiltonian systems.On the basis of this work,we survey a critical problem on how to mix the variables in the extended phase space.Numerical tests show that sequent permutations of coordinates and momenta can make the leapfrog-like methods yield the most accurate results and the optimal long-term stabilized error behaviour.We also present a novel method to construct many fourth-order extended phase-space explicit symmetric integration schemes.Each scheme represents the symmetric production of six usual second-order leapfrogs without any permutations.This construction consists of four segments: the permuted coordinates,triple product of the usual second-order leapfrog without permutations,the permuted momenta and the triple product of the usual second-order leapfrog without permutations.Similarly,extended phase-space sixth,eighth and other higher order explicit symmetric algorithms are available.We used several inseparable Hamiltonian examples,such as the post-Newtonian approach of non-spinning compact binaries,to show that one of the proposed fourth-order methods is more efficient than the existing methods;examples include the fourth-order explicit symplectic integrators of Chin and the fourth-order explicit and implicit mixed symplectic integrators of Zhong et al.Given a moderate choice for the related mixing and projection maps,the extended phase-space explicit symplectic-like methods are well suited for various inseparable Hamiltonian problems.Samples of these problems involve the algorithmic regular ization of gravitational systems with velocity-dependent perturbations in the Solar system and post-Newtonian Hamiltonian formulations of spinning compact objectswe discuss the geodesic motions of test particles in the intermediate vacuum between a monopolar core and an exterior shell of dipoles,quadrupoles and octopoles.The radii of the innermost stable circular orbits at the equatorial plane depend only on the quadrupoles.A given oblate quadrupolar leads to the existence of two innermost stable circular orbits,and their radii are larger than in the Schwarzschild spacetime.However,a given prolate quadrupolar corresponds to only one innermost stable circular orbit,and its radius is smaller than in the Schwarzschild spacetime.As to the general geodesic orbits,one of the recently developed extended phase space fourth order explicit symplectic-like methods is efficiently applicable to them although the Hamiltonian of the relativistic core-shell system is not separable.With the aid of both such a fast integrator without secular growth in the energy errors and gauge invariant chaotic indicators,the effect of these shell multipoles on the geodesic dynamics of order and chaos is estimated numerically.