Structure-preserving algorithms for the Duffing equation

In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.     

Structure-Preserving Algorithms for the Lorenz System

Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville＇s formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge-Kutta （RK4） method and the fifth-order Runge-Kutta-Fehlberg （RKF45） method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space axe less than those of the Runge-Kutta methods.     

Numerical solution of Helmholtz equation of barotropic atmosphere using wavelets

The numerical solution of the Helmholtz equation for barotropic atmosphere is estimated by use of the wavelet-Galerkin method. The solution involves the decomposition of a circulant matrix consisting up of 2-term connection coefficients of wavelet scaling functions. Three matrix decompositions, i.e. fast Fourier transformation (FFT), Jacobian and QR decomposition methods, are tested numerically. The Jacobian method has the smallest matrix-reconstruction error with the best orthogonality while the FFT method causes the biggest errors. Numerical result reveals that the numerical solution of the equation is very sensitive to the decomposition methods, and the QR and Jacobian decomposition methods, whose errors are of the order of 10^{-3}, much smaller than that with the FFT method, are more suitable to the numerical solution of the equation. With the two methods the solutions are also proved to have much higher accuracy than the iteration solution with the finite difference approximation. In addition, the wavelet numerical method is very useful for the solution of a climate model in low resolution. The solution accuracy of the equation may significantly increase with the order of Daubechies wavelet.     

A conservative Fourier pseudospectral algorithm for the nonlinear Schrdinger equation

In this paper, we derive a new method for a nonlinear Schrodinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ｜｜·｜｜2 norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.     

Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure

Impulsively coupled systems are high-dimensional non-smooth systems that can exhibit rich and complex dynamics.This paper studies the complex dynamics of a non-smooth system which is unidirectionally impulsively coupled by three Duffing oscillators in a ring structure.By constructing a proper Poincare map of the non-smooth system,an analytical expression of the Jacobian matrix of Poincare map is given.Two-parameter Hopf bifurcation sets are obtained by combining the shooting method and the Runge-Kutta method.When the period is fixed and the coupling strength changes,the system undergoes stable,periodic,quasi-periodic,and hyper-chaotic solutions,etc.Floquet theory is used to study the stability of the periodic solutions of the system and their bifurcations.     

Data Preprocessing in Cluster Analysis of Gene Expression

Considering that the DNA microarray technology has generated explosive gene expression data and that it is urgent to analyse and to visualize such massive datasets with efficient methods,we investigate the data preprocessing methods used in cluster analysis,normalization or logarithm of the matrix,by using hierarchical clustering,principal component analysis(PCA) and self-organizing maps(SOMs).The results illustrate that when using the Euclidean distance as measuring metrics,logarithm of relative expression level is the best preprocessing method,while data preprocessed by normalization cannot attain the expected results because the data structure is ruined.If there are only a few principal components,the PCA is an effective method to extract the frame structure,while SOMs are more suitalbe for a specific structure.     

Variational-integral perturbation corrections of some lower excited states for hydrogen atoms in magnetic fields

A variational-integral perturbation method(VIPM) is established by combining the variational perturbation with the integral perturbation.The first-order corrected wave functions are constructed,and the second-order energy corrections for the ground state and several lower excited states are calculated by applying the VIPM to the hydrogen atom in a strong uniform magnetic field.Our calculations demonstrated that the energy calculated by the VIPM only shows a negative value,which indicates that the VIPM method is more accurate than the other methods.Our study indicated that the VIPM can not only increase the accuracy of the results but also keep the convergence of the wave functions.     

Tracing Nucleation and Growth on Atomic Level in Amorphous Sodium by Molecular Dynamics Simulation

Crystallization of amorphous Na is simulated by using the molecular dynamics method, and the evolutions of atomic clusters are traced by the cluster-type index method （CTIM） we proposed previously. It is demonstrated that the crystallization process exhibits three distinct stages of nucleation, subsequent growth of nuclei and coarsening of crystal grains. Both the size and the internal structure of a cluster play crucial roles in determining whether it is a nucleus. The cluster tracing analysis can identify different crystallization stages more clearly and accurately than other methods. Meanwhile, the simulation results can provide a reasonable explanation at an atomic level for the experimental phenomenon obtained previously.     

Analysis and design of the taper in metal-grating periodic slow-wave structures for rectangular Cerenkov masers

The hybrid-mode dispersion equation of the metal-grating periodic slow-wave structure for a rectangular Cerenkov maser is derived by using the Borgnis function and field-matching methods.An equivalent-circuit model for the taper of the groove depth that matches the smooth waveguide to the metal-grating structure is proposed.By using the equivalentcircuit method,as well as the Ansoft high frequency structure simulator(HFSS) code,an appropriate electromagnetic mode for beam-wave interaction is selected and the equivalent-circuit analysis on the taper is given.The calculated results show that a cumulative reflection coefficient of 0.025 for the beam-wave interaction structure at a working frequency of 78.1 GHz can be reached by designing the exponential taper with a TE z10 rectangular waveguide mode as the input and the desired TE x10 mode as the output.It is worth pointing out that by using the equivalent-circuit method,the complex field-matching problems from the traditional field-theory method for taper design can be avoided,so the taper analysis process is markedly simplified.     

The Scaling Law in the Fine-Structure Splitting of ls2np States for the Lithium Isoelectronic Sequence

The ＇full-core plus correlation＇ （FCPC） and the ＇minimizing the expectation value of the Hamiltonian＇ methods are extended to calculate the fine-structure splitting of ls2np （n = 2-9） states for the lithium-like systems from Z = 21 to 30. The leading order relativistic effect is included by using first-order perturbation theory. The higher- order relativistic and the quantum-electrodynamics contributions to the fine-structure splitting are investigated under a hydrogenic approximation with effective nuclear charges, Our results are compared with other theoretical calculations and experimental results. It is shown that the FCPC method is also effective to obtain the ionic structure for high nuclear ions of lithium-like systems. By fitting our theoretical results, the scaling law of the fine-structure splitting of the lithium isoelectronie sequence behaves like a quartie function of the screened nuclear charge Z＊.     

Direct numerical simulation methods of hypersonic flat-plate boundary layer in thermally perfect gas

High-temperature effects alter the physical and transport properties of air such as vibrational excitation in a thermally perfect gas,and this factor should be considered in order to compute the flow field correctly.Herein,for the thermally perfect gas,a simple method of direct numerical simulation on flat-plat boundary layer is put forward,using the equivalent specific heat ratio instead of constant specific heat ratio in the N-S equations and flux splitting form of a calorically perfect gas.The results calculated by the new method are consistent with that by solving the N-S equations of a thermally perfect gas directly.The mean flow has the similarity,and consistent to the corresponding Blasius solution,which confirms that satisfactory results can be obtained basing on the Blasius solution as the mean flow directly in stability analysis.The amplitude growth curve of small disturbance is introduced at the inlet by using direct numerical simulation,which is consistent with that obtained by linear stability theory.It verified that the equation established and the simulation method is correct.     

Preconditioned pseudo-compressibility methods for incompressible Navier-Stokes equations

This paper investigates the pseudo-compressibility method for the incompressible Navier-Stokes equations and the preconditioning technique for accelerating the time marching for stiff hyperbolic equations,and derives and presents the eigenvalues and eigenvectors of the Jacobian matrix of the preconditioned pseudo-compressible Navier-Stokes equations in generally cur-vilinear coordinates.Based on the finite difference discretization the cored for efficiently solving incompressible flows numerically is established.The reliability of the procedures is demonstrated by the application to the inviscid flow past a circular cylinder,the laminar flow over a flat plate,and steady low Reynolds number viscous incompressible flows past a circular cylinder.It is found that the solutions to the present algorithm are in good agreement with the exact solutions or experimental data.The effects of the pseudo-compressibility factor and the parameter brought by preconditioning in convergence characteristics of the solution are investigated systematically.The results show that the upwind Roe’s scheme is superior to the second order central scheme,that the convergence rate of the pseudo-compressibility method can be effectively improved by preconditioning and that the self-adaptive pseudo-compressibility factor can modify the numerical convergence rate significantly compared to the constant form,without doing artificial tuning depending on the specific flow conditions.Further validation is also performed by numerical simulations of unsteady low Reynolds number viscous incompressible flows past a circular cylinder.The results are also found in good agreement with the existing numerical results or experimental data.     

Shear Force Detection Using Single—Tine Oscillating Tuning Fork for Scanning Near—Field Optical Microscopy

We propose a new method to detect near-field by using a single-tine oscillating tuning fork with mechanically asymmetric excitation that exhibits the sensitivity and stability better than that by using a double-tine oscillating one.Comparison of shear forces for the two methods demonstrate that the single-tine oscillating tuning fork provides a simpler and more sensitive method for near-field measurements.A theoretical analysis is presented for explanation to the greater sensitivity,The method is demonstrated by imaging a sparse-packed layer of micro-spheres in size of 200nm.     

New transfer matrix method for long-period fiber gratings with coupled multiple cladding modes

A new transfer matrix method for long-period fiber gratings with coupled multiple cladding modes is proposed and numerically characterized. The transmission spectra of uniform and non-uniform longperiod fiber gratings are numerically characterized. The theoretical results excellently agree with the experimental measurements. Compared with commonly used methods,such as using the fourth-order adaptive step size control of the Runge-Kutta algorithm in solving the coupled mode equation,the new transfer matrix method exhibits a faster calculation speed.     

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schroedinger system

We derive a new method for a coupled nonlinear Schr/Sdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.     

Soliton-like pulse timing jitter in dispersion-managed systems

In this paper, the timing jitter in dispersion-managed soliton-like systems with the Gaussian pulse is studied by using two methods. Firstly, the derivation of the dynamic equations for the evolution of soliton-like parameters and the timing jitter expressions for the dispersion-managed soliton-like systems are carried out by the perturbed variational method. By analysing and simulating these timing jitter expressions, one can find that the timing jitter is induced by the amplified spontaneous emission noise and the frequency shift, etc. Nonlinear gain can suppress the timing jitter. The chirp sign and the filters action have also effects on the total timing jitter. Secondly, the timing jitter is calculated and analysed by using the moment method. The results of the two methods prove to be consistent with each other.     

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrdinger system

We derive a new method for a coupled nonlinear Schrdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative.We prove the proposed method preserves the charge and energy conservation laws exactly.In numerical tests,we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions.Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws.These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.     

Method for fitting crystal field parameters and the energy level fitting for Yb3+ in crystal Sc2O3

<正>A method to compute the numerical derivative of eigenvalues of parameterized crystal field Hamiltonian matrix is given,based on the numerical derivatives the general iteration methods such as Levenberg-Marquardt,Newton method, and so on,can be used to solve crystal field parameters by fitting to experimental energy levels.With the numerical eigenvalue derivative,a detailed iteration algorithm to compute crystal field parameters by fitting experimental energy levels has also been described.This method is used to compute the crystal parameters of Yb~(3+) in Sc_2O_3 crystal, which is prepared by a co-precipitation method and whose structure was refined by Rietveld method.By fitting on the parameters of a simple overlap model of crystal field,the results show that the new method can fit the crystal field energy splitting with fast convergence and good stability.     

An Improved CE/SE Scheme and Its Application to Detonation Propagation

An improved two-dimensional space-time conservation element and solution element （CE/SE） method with second-order accuracy is proposed, examined and extended to simulate the detonation propagations using detailed chemical reaction models. The numerical results of planar and cellular detonation are compared with corresponding results by the Chapman-Jouguet theory and experiments, and prove that the method is a new reliable way for numerical simulations of detonation propagation.     

Comparison of calculation methods for the tunnel splitting at excited states of biaxial spin models

We present the calculation and comparison of tunnel splitting at excited levels of biaxial spin models by various methods, including the generalized instanton method, the generalized path integral method for coherent spin states, the perturbation method, and the exact method by numerical diagonalization of the Hamiltonian. It is found that, for integer spin with spin number around 10, tunnel splitting predicted by the generalized path integral for coherent spin states is about 10^{-n} times of the exact numerical result for the nth excited level, while the ratio of the results of the perturbation method and the exact numerical method diverges in the large spin limit. We thus conclude that the generalized instanton method is the best approximate way for calculating tunnel splitting in spin models.