Average vector field methods for the coupled Schrdinger KdV equations

The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.

Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

A high order energy preserving scheme for the strongly coupled nonlinear Schr¨odinger system

A high order energy preserving scheme for a strongly coupled nonlinear Schrōdinger system is roposed by using the average vector field method. The high order energy preserving scheme is applied to simulate the soliton evolution of the strongly coupled Schrōdinger system. Numerical results show that the high order energy preserving scheme can well simulate the soliton evolution, moreover, it preserves the discrete energy of the strongly coupled nonlinear Schrōdinger system exactly.

Modeling and forecasting time series of precious metals:a new approach to multifractal data

We introduce a novel approach to multifractal data in order to achieve transcended modeling and forecasting performances by extracting time series out of local Hurst exponent calculations at a specified scale.First,the long range and co-movement dependencies of the time series are scrutinized on time-frequency space using multiple wavelet coherence analysis.Then,the multifractal behaviors of the series are verified by multifractal de-trended fluctuation analysis and its local Hurst exponents are calculated.Additionally,root mean squares of residuals at the specified scale are procured from an intermediate step during local Hurst exponent calculations.These internally calculated series have been used to estimate the process with vector autoregressive fractionally integrated moving average(VARFIMA)model and forecasted accordingly.In our study,the daily prices of gold,silver and platinum are used for assessment.The results have shown that all metals do behave in phase movement on long term periods and possess multifractal features.Furthermore,the intermediate time series obtained during local Hurst exponent calculations still appertain the co-movement as well as multifractal characteristics of the raw data and may be successfully re-scaled,modeled and forecasted by using VARFIMA model.Conclusively,VARFIMA model have notably surpassed its univariate counterpart(ARFIMA)in all efficacious trials while re-emphasizing the importance of comovement procurement in modeling.Our study’s novelty lies in using a multifractal de-trended fluctuation analysis,along with multiple wavelet coherence analysis,for forecasting purposes to an extent not seen before.The results will be of particular significance to finance researchers and practitioners.

Novel energy dissipative method on the adaptive spatial discretization for the Allen–Cahn equation

We propose a novel energy dissipative method for the Allen–Cahn equation on nonuniform grids.For spatial discretization,the classical central difference method is utilized,while the average vector field method is applied for time discretization.Compared with the average vector field method on the uniform mesh,the proposed method can involve fewer grid points and achieve better numerical performance over long time simulation.This is due to the moving mesh method,which can concentrate the grid points more densely where the solution changes drastically.Numerical experiments are provided to illustrate the advantages of the proposed concrete adaptive energy dissipative scheme under large time and space steps over a long time.

A LINEARLY-FITTED CONSERVATIVE （DISSIPATIVE） SCHEME FOR EFFICIENTLY SOLVING CONSERVATIVE （DISSIPATIVE） NONLINEAR WAVE PDES

The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative （dissipative） nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.

High Order Energy-Preserving Method of the “Good” Boussinesq Equation

The fourth order average vector field(AVF)method is applied to solve the“Good”Boussinesq equation.The semi-discrete system of the“good”Boussi-nesq equation obtained by the pseudo-spectral method in spatial variable,which is a classical finite dimensional Hamiltonian system,is discretizated by the fourth order average vector field method.Thus,a new high order energy conservation scheme of the“good”Boussinesq equation is obtained.Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the“good”Boussinesq equation exactly and simulate evolution of different solitary waves well.

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preservmg formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq＂ （t）＋ Bq（t） = f（q（t））, where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f（q） = -VqV（q） for a real-valued function V（q）. The energy-preserving formulae can exactly preserve the Hamiltonian H（q＇,q） = 1/2q＇^TAq＇＋ 1/2q^TBq - V（q）. We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.

An Energy-Preserving Scheme for the Coupled Gross-Pitaevskii Equations

An energy-preserving scheme is proposed for the coupled Gross-Pitaevskii equations.The scheme is constructed by high order compact method in the spatial direction and average vector field method in the temporal direction,respectively.The scheme is energy-preserving,stable,and of sixth order in space and of second order in time.Numerical experiments verify the theoretical results.The dynamic behavior modeled by the coupled Gross-Pitaevskii equations is also numerically investigated.