Delay-dependent stability of linear multistep methods for differential systems with distributed delays

This paper deals with the stability of linear multistep methods for multidimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.

NGP_G-STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF GENERALIZED NEUTRAL DELAY DIFFERENTIAL EQUATIONS

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations, After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP(G)-stable if and only if it is A-stable.

IMPLICIT-EXPLICIT MULTISTEP FINITE ELEMENT-MIXED FINITE ELEMENT METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by implicit-explicit multistep finite element methods. The schemes are very efficient. The optimal order error estimates both in time and space are derived.

Stability of linear multistep methods for delay differential equations in the light of Kreiss resolvent condition

This paper deals with the stability analysis of the linear multistep (LM) methods in the numerical solution of delay differential equations. Here we provide a qualitative stability estimates, pertiment to the classical scalar test problem of the form y′(t)=λy(t)+μy(t-τ) with τ>0 and λ,μ are complex, by using (vartiant to) the resolvent condition of Kreiss. We prove that for A stable LM methods the upper bound for the norm of the n th power of square matrix grows linearly with the order of the matrix.

Geometric representation for numerical stability region of linear multistep methods

Studies the numerical stability region of linear multistep(LM) methods applied to linear test equation of the form y′(t)=ay(t)+by(t-1), t>0, y(t)=g(t)-1≤t≤0, a,b∈R, proves through delay dependent stability analysis that the intersection of stability regions of the equation and the method is not empty, in addition to approaches to the boundary of the delay differential equation(DDEs) in the limiting case of step size boundary of the stability region of linear multistep methods.

Algebraic Stability of Multistep Runge-Kutta Methods

A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.

Numerical Integration of Forced and Damped Oscillators through a New Multistep Method

Forced and damped oscillators appear in the mathematical modelling of many problems in pure and applied sciences such as physics, engineering and celestial mechanics among others. Although the accuracy of the T-functions series method is high, the calculus of their coefficients needs specific recurrences in each case. To avoid this inconvenience, the T-functions series method is transformed into a multistep method whose coefficients are calculated using recurrence procedures. These methods are convergent and have the same properties to the T-functions series method. Numerical examples already used by other authors are presented, such as a stiff problem, a Duffing oscillator and an equatorial satellite problem when the perturbation comes from zonal harmonics J2.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.

Stability analysis of linear multistep methods for neutral delay differential equations

The stability analysis of linear multistep (LM) methods is carried out under Kreiss resolvent condition when they are applied to neutral delay differential equations of the form y′(t)=ay(t)+by(t-τ)+ cy′(t- τ) y(t)=g(t) -τ≤t≤0 with τ>0 and a, b and c∈, and it is proved that the ‖B n‖ is suitably bounded, where B is the companion matrix.

Dissipativity of Multistep Runge-Kutta Methods for Nonlinear Neutral Delay-Integro-Differential Equations with Constrained Grid

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.We investigate the dissipativity properties of-algebraically stable multistep Runge-Kutta methods with constrained grid.The finite-dimensional and infinite-dimensional dissipativity results of-algebraically stable multistep Runge-Kutta methods are obtained.

航天器气动耦合六自由度姿轨动力学多步法积分预报误差分析

航天器轨降过程中的姿态估计是载人航天领域中的重要一环.随着近些年测站精度的提高,任务要求的增加,且研究表明姿态会影响航天器轨降过程中受到的气动力,进而对轨道产生影响,因此发展高精度姿轨耦合预报对航天器状态实时测控至关重要.本文以“天宫一号”航天器轨降过程中姿轨耦合沿弹道联合预报为背景,研究线性多步法积分误差对大型航天器姿轨预报精度的影响.具体包括Adams-Bashforth法、Adams-Moulton法、预估校正法等,为大型航天器轨降过程中的姿轨耦合预报以及落点预报提供数据参考.

STABILITY AND CONVERGENCE OF STEPSIZE-DEPENDENT LINEAR MULTISTEP METHODS FOR NONLINEAR DISSIPATIVE EVOLUTION EQUATIONS IN BANACH SPACE

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by ω-dissipative vector fields in Banach space.To break through the order barrier p≤1 of unconditionally contractive linear multistep methods for dissipative systems,strongly dissipative systems are introduced.By employing the error growth function of the methods,new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems(ω<0)and strongly dissipative systems.Some applications of the main results to several linear multistep methods,including the trapezoidal rule,are supplied.The theoretical results are also illustrated by a set of numerical experiments.

基于线性多步方法的二阶动力系统的模型识别

目的 针对二阶动力系统的识别问题,提出一种基于线性多步方法的稀疏识别方法。方法 首先,构造一个包含几乎所有可能基函数的基函数库,用于近似潜在的二阶动力系统;然后,利用线性多步方法离散近似后的二阶动力系统;接着,在广义最小二乘原理的指导下,选取一个合适的噪声残差项近似协方差矩阵,再利用该矩阵对上述过程得到的最小化问题进行加权,从而降低噪声对模型识别过程的影响;最后,使用稀疏回归算法从基函数库中挑选出最有意义的最小特征项,并通过稀疏迭代求解其对应系数。结果 比较了不同时间步长和不同噪声水平下,使用提出的线性多步稀疏识别方法挖掘潜在二阶动力系统的数值实验,通过这些实验可以得出:所提出的方法用于识别未知的二阶动力系统具有较高的精度和较好的鲁棒性。结论 通过数值实验,验证了算法的有效性。

APPLICATION OF THE DIGITAL MOIRE METHOD IN FRACTURE ANALYSIS OF A CRACKED RUBBER SHEET

This paper deals with the mode I crack problem of a cracked rubber sheet under plane stress condition using the delicate digital moiré technique. Through the four step phaseshifting method of automated fringe analysis, the displacement fields in the Cartesian coordinate system are given. By the coordinate-transform equation, the radial and circular displacement distributions in the polar coordinate system are obtained. The displacement isoline distributions and displacement vector distributions near the crack tip are discussed. The strain isoline distributions near the crack tip are also analyzed in this paper. Finally, the distribution rules for the mechanical fields near the crack tip are discussed with the sector division method.

Method of Lines for the Chiral Nonlinear Schrödinger Equation

In this paper, we solve chiral nonlinear Schrodinger equation (CNSE) numerically. Two numerical methods are derived using the explicit Runge-Kutta method of order four and the linear multistep method (Predictor-Corrector method of fourth order). The resulting schemes of fourth order accuracy in spatial and temporal directions. The CNSE is non-integrable and has two kinds of soliton solutions: bright and dark soliton. The exact solutions and the conserved quantities of CNSE are used to display the efficiency and robustness of the numerical methods we derived. Interaction of two bright solitons for different parameters is also displayed.

A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations

In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.

基于扩展卡尔曼滤波的储能电池能量和功率状态联合估计方法

电池储能是碳中和目标的有力抓手,准确估计其能量状态(state of energy,SOE)和峰值功率状态(state of power,SOP)是电池储能高效可靠运行的关键和基础。由于电池的电化学反应过程十分复杂,作为隐性状态量的SOE和SOP精确值难以获得。为此,本工作提出了一种基于模型SOE和SOP联合估计方法。应用Thevenin等效电路模型,采用递归最小二乘法建立了在线参数辨识算法,获得准确的模型参数。为解决恒定功率需求下的功率预测难题,提出了多步功率预测法,提高了SOP的预测精度,并结合扩展卡尔曼滤波算法,进一步提出了多状态联合估计方法。实验验证了算法的可行性,结果表明,在存在较大初始误差的情况下,所提出的方法电压、SOE最大预测误差均<2%,实现了准确的SOP预测。

非定常Stokes/Darcy模型一种新的time filter算法的分析

首先,在非定常Stokes/Darcy模型的线性多步法的一阶θ-格式的基础上,该文结合time filter算法在几乎不增加计算量的情况下有效地将线性多步法的收敛阶由一阶提高到二阶,从而提出一种新的高效数值算法.其次,该文分别对耦合和解耦的线性多步法加time filter算法的稳定性和误差估计进行了理论分析.最后,数值实验进一步展示了耦合和解耦算法的有效性,收敛性和高效性.

预测铣削稳定性的Hamming线性多步法

针对铣削加工过程中产生的振动现象,提出了一种Hamming线性多步法(HAMM)来预测铣削加工过程中的稳定性。考虑再生颤振的铣削加工动力学方程可以表示为时滞线性微分方程,将刀齿周期划分为自由振动阶段和强迫振动阶段,对强迫振动阶段进行离散,运用HAMM方法构建状态传递矩阵,利用Floquet理论,判定系统的稳定性,获得系统的稳定性叶瓣图。Matlab软件仿真结果表明,HAMM方法是预测铣削稳定性的一种有效方法。随着离散数的增加,HAMM方法的收敛速度要快于一阶半离散法(1st-SDM)和二阶全离散法(2nd-FDM),离散数较少的HAMM方法能达到和离散数较多的1st-SDM方法和2nd-FDM方法的局部离散误差。此外,在单自由度和双自由度动力学模型下,由三种方法的稳定性叶瓣图可以看出,HAMM方法预测铣削稳定性的精度均好于1st-SDM方法和2nd-FDM方法,计算效率远远高于1st-SDM方法和2nd-FDM方法。实验结果表明,HAMM方法是一种有效的预测铣削稳定性的方法。

非饱和土渗透系数的一种测量方法和预测公式

总结了非饱和土渗透系数的各种测量方法和预测模型。在此基础上,提出一种直接测量瞬态渗透系数的方法和预测模型。这一测量方法采用多步溢出法测量土水特征曲线,同时实时测量渗透流量和土样的含水率。在试验过程中,把气压强增量分为两个部分:一部分克服渗透阻力,驱赶水分运动,另一部分平衡静水平衡条件下的表面张力,转化为基质势,从而根据达西定律公式求出瞬态渗透系数。另一方面,利用孔隙水的平衡微分方程,引进雷诺层流理论来解释土体渗透产生的土骨架和孔隙水之间的相互作用力,推导出了非饱和土渗透系数的理论公式,该公式只有一个参数,可以通过讨论不同孔隙率、饱和渗透系数与土水作用力系数之间的关系拟合得到,从而预测出非饱和土渗透系数。通过与试验结果的对比,证实了所提出的测量方法和预测模型的正确性。