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.

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.

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.

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.

DELAY-DEPENDENT TREATMENT OF LINEAR MULTISTEP METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

This paper deals with a delay-dependent treatment of linear multistep methods for neutral delay differential equations y'(t) = ay(t) + by(t - τ) + cy'(t - τ), t > 0, y(t) = g(t), -τ≤ t ≤ 0, a,b andc ∈ R. The necessary condition for linear multistep methods to be Nτ(0)-stable is given. It is shown that the trapezoidal rule is Nτ(0)-compatible. Figures of stability region for some linear multistep methods are depicted.

THE STABILITY OF LINEAR MULTISTEP METHODS FOR LINEAR SYSTEMS OF NEUTRAL DIFFERENTIAL EQUATIONS

Presents information on a study which focused on the numerical solution of initial value problems for systems of neutral differential equations. Adaptations of linear multistep methods; Linear stability of linear multistep method; Presentation of numerical equations.

VARIABLE STEP-SIZE IMPLICIT-EXPLICIT LINEAR MULTISTEP METHODS FOR TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS

Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.

CONJUGATE-SYMPLECTICITY OF LINEAR MULTISTEP METHODS

For the numerical treatment of Hamiltonian differential equations, symplectic integrators are the most suitable choice, and methods that are conjugate to a symplectic integrator share the same good long-time behavior. This note characterizes linear multistep methods whose underlying one-step method is conjugate to a symplectic integrator. The bounded- hess of parasitic solution components is not addressed.

DELAY-DEPENDENT STABILITY OF LINEAR MULTISTEP METHODS FOR NEUTRAL SYSTEMS WITH DISTRIBUTED DELAYS

This paper considers the asymptotic stability of linear multistep(LM)methods for neutral systems with distributed delays.In particular,several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument principle.Compound quadrature formulae are used to compute the integrals.An algorithm is proposed to examine the delay-dependent stability of numerical solutions.Several numerical examples are performed to verify the theoretical results.

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.

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.

CONVERGENCE OF LINEAR MULTISTEP METHODS FOR TWO-PARAMETER SINGULAR PERTURBATION PROBLEMS

Some convergence results are given for A(a)-stable linear multistep methods applied to two classes of two-parameter singular perturbation problems, which extend the existing relevant results about one-parameter problems by Lubich~[1]. Some numerical examples confirm our results.

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.

SYMPLECTIC MULTISTEP METHODS FOR LINEAR HAMILTONIAN SYSTEMS

Three classes of symplectic multistep methods for linear Hamiltonian systems are constructed and their stabilities are discussed in this paper.

数值求解多延迟中立型系统的渐近稳定性(英文)

给出并证明了多延迟中立型系统渐近稳定的充分条件;分析了用线性多步法求解多延迟中立型系统数值解的稳定性,基于Lagrange插值,证明了数值求解多延迟中立型系统的线性多步法渐近稳定的充分必要条件是它是A-稳定的。

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。

基于全隐式紧耦合算法的气动弹性数值仿真

为进行气动弹性问题的计算,提出了一种全隐式紧耦合算法,在子迭代过程中分别采用LU-SGS隐格式和隐式线性多步法交替求解气动和结构方程,以获得物理时间域的高精度解。一种径向基函数和超限插值结合的方法被用来进行气动网格的快速变形。运用该算法,进行了Isogai wing和AGARD 445.6 wing的颤振分析,颤振边界的计算结果与文献值和实验值较符合,表明该全隐式紧耦合算法能够有效地计算气动弹性问题。

非线性微分代数方程的一种离散波形松弛算法

讨论用迭代方法求解微分代数方程。针对一类非线性微分代数方程连续时间波形松弛迭代格式,应用一般的单支方法和线性多步法,得到离散时间波形松弛迭代格式。在假定分裂函数满足Lipschitz条件的前提下,通过矩阵正则分裂和特殊矩阵相关性质的运用,获得离散波形松弛迭代的收敛性条件,拓展和改进了相关文献中的一些结果。