Symplectic trigonometrically fitted partitioned Runge–Kutta methods

The numerical integration of Hamiltonian systems is considered in this Letter. Trigonometrically fitted symplectic partitioned Runge–Kutta methods of second, third and fourth orders are constructed. The methods are tested on the numerical integration of the harmonic oscillator, the two body problem and an orbital problem studied by Stiefel and Bettis.     

Nonlinear fluctuation,frequency and stability analyses in free vibration of circular sector oscillation systems

In the present paper, the modified homotopy perturbation method (MHPM) is employed to investigate about both nonlinear swinging oscillation and the stability of circular sector oscillation systems. The sensitivity study performed for frequency analysis of the mentioned oscillatory circular sector body shows that frequency of nonlinear oscillation depends on some specific parameters and can be optimized. Furthermore onset of the instability is dependent to angle α and initial amplitude.Comparisons made among the results of the present closed-form analytical solution and the traditional numerical iterative time integration solution confirms the accuracy and efficiency of the presented analytical solution.In contrast to the available numerical methods, the present analytical method is free from the numerical damping and the time integration accumulated errors. Moreover, in comparison with the traditional multistep numerical iterative time integration methods, a much less computational time is required for the present analytical method. Responses of the dynamical systems to some extent are affected by the natural frequencies. Results reveal that for nonlinear systems, the natural frequency is remarkably affected by the initial conditions.     

Algebraic stabilization of explicit numerical integration for extremely stiff reaction networks

In contrast to the prevailing view in the literature, it is shown that even extremely stiff sets of ordinary differential equations may be solved efficiently by explicit methods if limiting algebraic solutions are used to stabilize the numerical integration. The stabilizing algebra differs essentially for systems well-removed from equilibrium and those near equilibrium. Explicit asymptotic and quasi-steady-state methods that are appropriate when the system is only weakly equilibrated are examined first. These methods are then extended to the case of close approach to equilibrium through a new implementation of partial equilibrium approximations. Using stringent tests with astrophysical thermonuclear networks, evidence is provided that these methods can deal with the stiffest networks, even in the approach to equilibrium, with accuracy and integration timestepping comparable to that of implicit methods. Because explicit methods can execute a timestep faster and scale more favorably with network size than implicit algorithms, our results suggest that algebraically-stabilized explicit methods might enable integration of larger reaction networks coupled to fluid dynamics than has been feasible previously for a variety of disciplines.     

Analytical study of a flattened Gaussian beam through multi-apertured optical imaging systems of B = 0

By using the methods of the matrix decomposition and expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, the recurrence propagation expressions for a flattened Gaussian beam (FGB) through multi-apertured optical imaging systems of B = 0 are derived and illustrated with numerical examples. Comparisons with the straightforward numerical integration of the Collins formula and with the previous work are made. It is shown that the main advantages of our methods and results are the more accuracy and great reduction of computer time.     

Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations

The relation between the parameters of a differential equation and corresponding discrete maps is becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. An iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODEs with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is a polynomial in the state variables and if the ODE has a fixed point at the origin. For several nonlinear systems approximations of different orders are investigated.     

Many body forces between long conducting molecules

Non-equilibrium molecular dynamics is used to calculate the wavevector and strain rate dependence of shear viscosity in a soft sphere fluid. The calculations are consistent with a non-analytic functional dependence of viscosity with wavevector. A consequence of such non-analyticity would be that the linear and non-linear Burnett coefficients for viscosity would not exist for fluids in three dimensions. The calculations also show the numerical consistency of three different non-equilibrium simulation methods for calculating shear viscosity.     

Symplectic integration of nonlinear Hamiltonian systems

There exist several standard numerical methods for integrating ordinary differential equations. However, if one is interested in integration of Hamiltonian systems, these methods can lead to wrong results. This is due to the fact that these methods do not explicitly preserve the so-called ‘symplectic condition’ (that needs to be satisfied for Hamiltonian systems) at every integration step. In this paper, we look at various methods for integration that preserve the symplectic condition.     

Renormalization of position space amplitudes in a massless QFT

Ultraviolet renormalization of position space massless Feynman amplitudes has been shown to yield associate homogeneous distributions. Their degree is determined by the degree of divergence while their order—the highest power of logarithm in the dilation anomaly—is given by the number of (sub)divergences. In the present paper we review these results and observe that (convergent) integration over internal vertices does not alter the total degree of (superficial) ultraviolet divergence. For a conformally invariant theory internal integration is also proven to preserve the order of associate homogeneity. The renormalized 4-point amplitudes in the φ⁴ theory (in four space-time dimensions) are written as (non-analytic) translation invariant functions of four complex variables with calculable conformal anomaly.Our conclusion concerning the (off-shell) infrared finiteness of the ultraviolet renormalized massless φ⁴ theory agrees with the old result of Lowenstein and Zimmermann [23].     

Random dynamics of a nonlinear spur gear pair in probabilistic domain

This paper investigates the response of a spur gear pair subjected to both deterministic and random loads. Backlash nonlinearity and time-varying mesh stiffness in gear systems are considered in the model. Path integration is adopted to capture the random response in probabilistic domain. In the path integration algorithm, the transition probability density function (PDF) within a short time interval is assumed as Gaussian. Then the mean and variance of the responses are calculated and expressed as closed forms for two different cases in gear systems, which are further used to construct the transition PDF. The simulation results are compared with that from Monte Carlo (MC) simulation and deterministic numerical integration. Good agreements are shown between these results. In addition, the multi-solutions feature characterizing the nonlinear gear system is also captured.     

Novel Gear-like predictor–corrector integration methods for molecular dynamics

A new class of predictor–corrector integration methods intended for molecular dynamics simulation is presented. The methods are derived from the original Gear methods by analysing the numerical solution of the harmonic oscillator. The corrector coefficients are chosen to improve the time-reversibility while keeping maximum stability. Tests performed on systems with forces not dependent on velocities (classical two-body problem and three simple atomistic systems) show good energy conservation and reduced deviations of measured quantities in comparison with the Verlet method. As regards the systems with velocity-dependent right-hand sides, a moderate improvement is found for the Nosé–Hoover thermostat but no improvement for fixed bonds constrained by the general constraint dynamics based on Lagrange multipliers.     

A conservative discretization of the Kepler problem based on the L-transformations

Conservative numerical schemes for the two- and three-dimensional Kepler motion were recently proposed in [Y. Minesaki, Y. Nakamura, Phys. Lett. A 306 (2–3) (2002) 127; Y. Minesaki, Y. Nakamura, Phys. Lett. A 324 (4) (2004) 282]. They are based on Levi-Civita and Kustaanheimo–Stiefel transformations respectively. The schemes preserve all first integrals of Kepler motion: the Hamiltonian function, the angular momentum and Runge–Lenz vector. In the present Letter we extend this approach to the L-transformations (a generalization of Levi-Civita and Kustaanheimo–Stiefel transformations). Thus, we can consider more general family of the conservative numerical schemes for the two- and three-dimensional Kepler motion as well as construct conservative schemes for higher-dimensional Kepler problems. Conservation of the first integrals is proved with the help of L-matrix identities. The proposed numerical scheme permits explicit implementation.     

Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems

This paper focuses on studying a new energy-work relationship numerical integration scheme of nonholonomic Hamiltonian systems. The signal-stage numerical, multi-stage and parallel composition numerical integration schemes are presented. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multi-stage schemes of order 2, its order of accuracy is 2n. The connection, which is discrete analogue of usual case, between the change of energy and work of nonholonomic constraint forces is obtained for nonholonomic Hamiltonian systems. This paper also gives that there is smaller error of the scheme when taking a large number of stages than a less one. Finally, an applied example is discussed to illustrate these results.     

Off-perturbative states in disordered systems

The systematic approach for the off-perturbative calculations in disordered systems is developed. The proposed scheme is applied for the random temperature and the random field ferromagnetic Ising models. It is shown that away from the critical point, in the paramagnetic phase of the random temperature model, and in the ferromagnetic phase of the random field one, the free energy contains non-analytic contributions which have the form of essential singularities. It is demonstrated that these contributions appear due to localized in space instanton-like excitations.     

Localization in the Non-analytic Quantum Kicked Systems

Numerical investigations on non-analytic quantum kicked systems are presented. A new type of localization - power-law localization is found to be universal in the nonanalytic systems. With increasing the perturbation strength, a transition from perturbative localization to pseudo-integrable system, to dynamical localization and to complete extension is clearly demonstrated. The dependence of the localization length on perturbation is given in different parameter regimes.     

A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres

A recently developed matrix formulation of the discrete ordinate method is extended for application to an inhomogeneous atmosphere. The solution yields fluxes, as well as the complete azimuthal dependence of the intensity at any level in the atmosphere. The numerical aspects of the solution are discussed and numerical verification is provided by comparing computed results with those obtained by other methods. In particular, it is shown that a simple scaling scheme, which removes the positive exponentials in the coefficient matrix when solving for the constants of integration, provides unconditionally stable solutions for arbitrary optical thicknesses. An assessment of the accuracy to be expected is also provided, and it is shown that low-order discrete ordinate approximations yield very accurate flux values.     

BEPCII横向束流反馈系统前端电子学

选择了一台商业化的前端电子学RFTF（radio frequency front-end for transverse feedback system）用于BEPCII横向束流反馈系统, 这是跟进世界先进加速器技术的具体实施。在BEPC同步运行模式下对RFTF的性能进行了测试实验，结果表明使用RFTF可以得到理想的波形，能够满足BEPCII横向反馈系统的需要，但是，当储存环出现不稳性时，系统无法正常工作，这就要求BEPCII纵向不稳定很小或者用纵向反馈系统来抑制存在的纵向不稳定。     

Numerical methods for solving one-dimensional cochlear models in the time domain

In this article, a robust numerical solution method for one-dimensional (1-D) cochlear models in the time domain is presented. The method has been designed particularly for models with a cochlear partition having nonlinear and active mechanical properties. The model equations are discretized with respect to the spatial variable by means of the principle of Galerkin to yield a system of ordinary differential equations in the time variable. To solve this system, several numerical integration methods concerning stability and computational performance are compared. The selected algorithm is based on a variable step size fourth-order Runge-Kutta scheme; it is shown to be both more stable and much more efficient than previously published numerical solution techniques.     

ANALYSIS OF THE MULTIPLE-SOLUTION RESPONSE OF A FLEXIBLE ROTOR SUPPORTED ON NON-LINEAR SQUEEZE FILM DAMPERS

The multiple-solution response of rotors supported on squeeze film dampers is a typical non-linear phenomenon. The behaviour of the multiple-solution response in a flexible rotor supported on two identical squeeze film dampers with centralizing springs is studied by three methods: synchronous circular centred-orbit motion solution, numerical integration method and slow acceleration method using the assumption of a short bearing and cavitated oil film; the differences of computational results obtained by the three different methods are compared in this paper. It is shown that there are three basic forms for the multiple-solution response in the flexible rotor system supported on the squeeze film dampers, which are the resonant, isolated bifurcation and swallowtail bifurcation multiple solutions. In the multiple-solution speed regions, the rotor motion may be subsynchronous, super-subsynchronous, almost-periodic and even chaotic, besides synchronous circular centred, even if the gravity effect is not considered. The assumption of synchronous circular centred-orbit motion for the journal and rotor around the static deflection line can be used only in some special cases; the steady state numerical integration method is very useful, but time consuming. Using the slow acceleration method, not only can the multiple-solution speed regions be detected, but also the non-synchronous response regions.     

A multirate time integrator for regularized Stokeslets

The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid–structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the use of standard implicit time integration methods for the method of regularized Stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems. Here, an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method. The multirate methods are higher-order methods that treat different portions of the ODE explicitly with different time steps depending on the stiffness of each component. Numerical examples on two nontrivial three-dimensional problems demonstrate the increased efficiency of the multi-explicit approach with no significant increase in numerical error.