With the natural splitting of a Hamiltonian system into kinetic energy and potential energy,we construct two new optimal thirdorder force-gradient symplectic algorithms in each of which the norm of fourth-order truncation errors is minimized.They are both not explicitly superior to their no-optimal counterparts in the numerical stability and the topology structure-preserving,but they are in the accuracy of energy on classical problems and in one of the energy eigenvalues for one-dimensional time-independent Schrdinger equations.In particular,they are much better than the optimal third-order non-gradient symplectic method.They also have an advantage over the fourth-order non-gradient symplectic integrator.
LI Rong & WU Xin School of Science,Nanchang University,Nanchang 330031,China
The regular and chaotic dynamics of test particles in a superposed field between a pseudo-Newtonian Kerr black hole and quadrupolar halos is detailed.In particular,the dependence of dynamics on the quadrupolar parameter of the halos and the spin angular momentum of the rotating black hole is studied.It is found that the small quadrupolar moment,in contrast with the spin angular momentum,does not have a great effect on the stability and radii of the innermost stable circular orbits of these test particles.In addition,chaos mainly occurs for small absolute values of the rotating parameters,and does not exist for the maximum counter-rotating case under some certain initial conditions and parameters.This means that the rotating parameters of the black hole weaken the chaotic properties.It is also found that the counter-rotating system is more unstable than the co-rotating one.Furthermore,chaos is absent for small absolute values of the quadrupoles,and the onset of chaos is easier for the prolate halos than for the oblate ones.
The dependence of chaos on two parameters of the cosmological constant and the self-interacting coefficient in the imaginary phase space for a closed Friedman- Robertson-Walker （FRW） universe with a conformally coupled scalar field, as the full understanding of the dependence in real phase space, is investigated numerically. It is found that Poincar6 plots for the two parameters less than 1 are almost the same as those in the absence of the cosmological constant and self-interacting terms. For energies below the energy threshold of 0.5 for the imaginary problem in which there are no cosmological constant and self-interacting terms, an abrupt transition to chaos occurs when at least one of the two parameters is 1. However, the strength of the chaos does not increase for energies larger than the threshold. For other situations of the two parameters larger than 1, chaos is weaker, and even disappears as the two parameters increase.
By adding force gradient operators to symmetric compositions, we build a set of explicit fourth-order force gradient symplectic algorithms, including those of Chin and coworkers, for a separable Hamiltonian system with quadratic kinetic en- ergy T and potential energy V. They are extended to solve a gravitational n-body Hamiltonian system that can be split into a Keplerian part H0 and a perturbation part H1 in Jacobi coordinates. It is found that the accuracy of each gradient scheme is greatly superior to that of the standard fourth-order Forest-Ruth symplectic integra- tor in T ＋ V-type Hamiltonian decomposition, but they are both almost equivalent in the mean longitude and the relative position for H0 ＋//1-type decomposition. At the same time, there are no typical differences between the numerical performances of these gradient algorithms, either in the splitting of T ＋ V or in the splitting of H0 ＋//1. In particular, compared with the former decomposition, the latter can dra- matically improve the numerical accuracy. Because this extension provides a fast and high-precision method to simulate various orbital motions of n-body problems, it is worth recommending for practical computation.
An operator associated with third-order potential derivatives and a force gradient operator corresponding to second-order potential derivatives are used together to design a number of new fourth-order explicit symplectic integrators for the natural splitting of a Hamiltonian into both the kinetic energy with a quadratic form of momenta and the potential energy as a function of position coordinates.Numerical simulations show that some new optimal symplectic algorithms are much better than their non-optimal counterparts in terms of accuracy of energy and position calculations.
