A brief introduction to symplectic integrators and recent results

The author begins with a brief synopsis about Hamiltonian systems and symplectic maps. A symplectic integrator is a symplectic map {phi}(q,p;t) that systematically approximates the time t flow of a Hamiltonian system. Systematic means: (1) in time step, t, i.e. the error should vanish as some power of the time step, and (2) in order of approximation, i.e. one would like a hierarchy of such {phi} that have errors that vanish as successively higher powers of the time step. At present the authors known two general types of symplectic integrators: (1) implicit integrators that are derived from a generating function or from algebraic conditions on Runge-Kutta schemes, and (2) explicit integrators that are derived from integrable Hamiltonians or from algebraic conditions on Runge-Kutta schemes

Two-Stream Instability Model With Electrons Trapped in Quadrupoles

We formulate the theory of the two-stream instability (e-cloud instability) with electrons trapped in quadrupole magnets. We show that a linear instability theory can be sensibly formulated and analyzed. The growth rates are considerably smaller than the linear growth rates for the two-stream instability in drift spaces and are close to those actually observed

Unequal relationships in high and low power distance societies: a comparative study of tutor - student role relations in Britain and China

This study investigated people's conceptions of an unequal role relationship in two different types of society: a high power distance society and a low power distance society. The study focuses on the role relationship of tutor and student. British and Chinese tutors and postgraduate students completed a questionnaire that probed their conceptions of degrees of power differential and social distance/closeness in this role relationship. ANOVA results yielded a significant nationality effect for both aspects. Chinese respondents judged the relationship to be closer and to have a greater power differential than did British respondents. Written comments on the questionnaire and interviews with 9 Chinese academics who had experienced both British and Chinese academic environments supported the statistical findings and indicated that there are fundamental ideological differences associated with the differing conceptions. The results are discussed in relation to Western and Asian concepts of leadership and differing perspectives on the compatibility/incompatibility of power and distance/closeness

Efficient numerical integrators for stochastic models

The efficient simulation of models defined in terms of stochastic differential equations (SDEs) depends critically on an efficient integration scheme. In this article, we investigate under which conditions the integration schemes for general SDEs can be derived using the Trotter expansion. It follows that, in the stochastic case, some care is required in splitting the stochastic generator. We test the Trotter integrators on an energy-conserving Brownian model and derive a new numerical scheme for dissipative particle dynamics. We find that the stochastic Trotter scheme provides a mathematically correct and easy-to-use method which should find wide applicability.Comment: v

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

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.Comment: Submitted to Phys. Rev. E, 29 Page

Symplectic integrators with adaptive time steps

In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, \Delta=\Delta(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of \Delta(t) and the orbital motion. In the second category the time step is a function of phase space variables \Delta=\Delta(q,p). In this case, the system of equations to be solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p) d\tau. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant \Delta. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio

Climate variability and ice-sheet dynamics during the last three glaciations

AbstractA composite North Atlantic record from DSDP Site 609 and IODP Site U1308 spans the past 300,000 years and shows that variability within the penultimate glaciation differed substantially from that of the surrounding two glaciations. Hematite-stained grains exhibit similar repetitive down-core variations within the Marine Isotope Stage (MIS) 8 and 4–2 intervals, but little cyclic variability within the MIS 6 section. There is also no petrologic evidence, in terms of detrital carbonate-rich (Heinrich) layers, for surging of the Laurentide Ice Sheet through the Hudson Strait during MIS 6. Rather, very high background concentration of iceberg-rafted debris (IRD) indicates near continuous glacial meltwater input that likely increased thermohaline disruption sensitivity to relatively weak forcing events, such as expanded sea ice over deepwater formation sites. Altered (sub)tropical precipitation patterns and Antarctic warming during high orbital precession and low 65°N summer insolation appear related to high abundance of Icelandic glass shards and southward sea ice expansion. Differing European and North American ice sheet configurations, perhaps aided by larger variations in eccentricity leading to cooler summers, may have contributed to the relative stability of the Laurentide Ice Sheet in the Hudson Strait region during MIS 6

Explicit Lie-Poisson integration and the Euler equations

We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho

Detailed paleomagnetic and rock magnetic variability within three high-resolution study intervals from site 1233

We carried out a detailed rock magnetic and paleomagnetic study of deep-sea sediments from selected intervals of Site 1233 in order to assess whether they contain reproducible evidence for submillennial-scale environmental, climatic, and geomagnetic field variability. Three 1.5-m sediment intervals from oxygen isotope Stages 1, 2, and 3 were sampled using continuous U-channels; replicate sediment sequences were collected from two or three holes in each interval to test for reproducibility. Rock magnetic and paleomagnetic measurements were made at 1- cm intervals. Rock magnetic results identify distinctive centennial (~150–300 yr)- and millennial-scale variability that is probably continuous over the entire Site 1233 record. They also show evidence for intermittent multidecadal-scale rock magnetic variability. The rock magnetic variability allows us to modify the meters composite depth (mcd) correlation in each of the three high-resolution intervals to permit correlations between holes at ~5-cm resolution. Paleomagnetic results indicate that there is also a reproducible pattern of paleomagnetic secular variation at the centennial to millennial scale. The centennial-scale variability is dominated by ~5°–10° oscillations in both inclination and declination with an average interval of ~200–300 yr. Deconvolution studies and paleomagnetic secular variation (PSV) correlations between holes both suggest that the PSVs are real and not caused by similar-scale rock magnetic variability. The paleomagnetic results also document distinctive, larger-amplitude multicentennial- and millennial-scale directional variability that is consistent with preliminary shipboard paleomagnetic studies