Stability of Numerical Methods for Ordinary Differential Equations

Variable stepsize stability results are found for three representative multivalue methods. For the second order BDF method, a best possible result is found for a maximum stepsize ratio that will still guarantee A(0)-stability behaviour. It is found that under this same restriction, A()-stability holds for 70°. For a new two stage two value first order method, which is L-stable for constant stepsize, A(0)-stability is maintained for stepsize ratios as high as aproximately 2.94. For the third order BDF method, a best possible result of (1/2)(1+ ) is found for a ratio bound that will still guarantee zero-stability.     

Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations

Stability of numerical methods for nonlinear autonomous ordinarydifferential equations is approached from the point of viewof dynamical systems. It is proved that multistep methods (withnonlinear algebraic equations exactly solved) with bounded trajectoriesalways produce correct asymptotic behaviour, but this is notthe case with Runge-Kutta. Examples are given of Runge-Kuttaschemes converging to wrong solutions in a deceptively ‘smooth’manner and a characterization of such two-stage methods is presented.PE(CE)^m schemes are examined as well, and it is demonstratedthat they, like Runge-Kutta, may lead to false asymptotics.     

一类求解常微分方程数值解的块方法

本文讨论了一类并行计算常微分方程初值问题的带有高阶导数的块隐式混合单步方法，这种方法可以在Ｋ台处理机上并行进行数值计算，本文对方法的一般性质及收敛性进行了讨论，得知该方法的阶数为２ｌ＋１，并且指出当ｌ＝１，２时，方法是Ａ－稳定的，最后给出了一个数值例子。     

Numerical Integration of Systems of Stiff Ordinary Differential Equations with Special Structure

Algorithms for the solution of general systems of stiff differentialequations commonly use implicit integration formulae. The associatednon-linear equations at each step of the integration are efficientlysolved by an iteration such as the parallel chord method, wherethe matrix is an approximation for the Jacobian at a calculatedpoint of the solution. This iteration frequently gives sufficientlyrapid convergence over a number of integration steps beforeupdating and re-inversion of the matrix is required. When thedifferential equations have a special structure, satisfactoryconvergence may be maintained by updating a partition of theJacobian less frequently than the remainder and an efficientcomputational procedure consists in calculating the correspondingupdate of the inverse. Sufficient conditions for local convergencemay be expressed in terms of the difference between the iterationmatrix and the derivative at the solution or in terms of thedifference of the corresponding inverses. Similarly the asymptoticrate of convergence is estimated in terms of the norms of theseperturbations. To assess the effectiveness of updating a partitionof the Jacobian or its inverse we set the corresponding perturbationto zero and evaluate the estimate of the rate of convergence.Variable transformation and "weighting" of equations may beused to obtain more accurate computable estimates of convergencerates and tighter conditions for convergence. This is particularlyrelevant in non-linear stiff systems arising in applicationsfrom physics, chemistry and engineering and associated withfast and slow motions. Such systems exhibit special structurein the Jacobian and higher derivatives related to the sensitivityof the system to the components of the fast motion which makesthem particularly amenable to matrix updating techniques. Anumber of illustrative problems from the literature are cited.A worked example has been solved numerically using an inversewhose partitions are updated irregularly as required for convergenceand a comparison is made of iteration counts and inversion statisticswith updating of the full inverse. Computational savings maysometimes belarge.     

Stability of Adams-Type Formulae for Second-Order Ordinary Differential Equations

The stability of methods for systems of second-order equationsis discussed. Stability regions are obtained for a single equationand the existence of stable step-sizes is shown for systems.An example is used as an illustration of the effect of the usualorder selection strategies on stability and accuracy.     

On Implicit Ordinary Differential Equations

In a recent paper Fox & Mayers discuss the numerical solutionof implicit ordinary differential equations of the form f(x,y(x), y'(x)) = 0. They find that numerical methods can be veryunreliable near the point where f_y' = 0. In this paper we givea theoretical analysis of the problem which enables us to explainwhen to expect numerical difficulties. We suggest a possibleline of approach for the solution of such problems, and discusssome numerical examples. Research supported by the National Science Foundation, the Officeof Naval Research, the Army Research, and the Air Force Officeof Scientific Research. Travel funding provided by the Universityof Toronto and the British Council.     

Geometric Integration of Ordinary Differential Equations on Manifolds

This article illustrates how classical integration methods for differential equations on manifolds can be modified in order to preserve certain geometric properties of the exact flow. Projection methods as well as integrators based on local coordinates are considered. The central ideas of this article have been presented at the 40th anniversary meeting of the journal BIT.This revised version was published online in October 2005 with corrections to the Cover Date.     

"常微分方程"课程建设探讨

根据常微分方程课程特点及常微分方程的发展,探讨了常微分方程精品课程建设中师资队伍建设和教材改革与建设的重要性,提出了对改革有关教学内容以及编写具有本课程特色的优秀教材的具体设想.     

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

The modelling of many real life phenomena for which either the parameter estimation is difficult, or which are subject to random noisy perturbations, is often carried out by using stochastic ordinary differential equations (SODEs). For this reason, in recent years much attention has been devoted to deriving numerical methods for approximating their solution. In particular, in this paper we consider the use of linear multistep formulae (LMF). Strong order convergence conditions up to order 1 are stated, for both commutative and non-commutative problems. The case of additive noise is further investigated, in order to obtain order improvements. The implementation of the methods is also considered, leading to a predictor-corrector approach. Some numerical tests on problems taken from the literature are also included.     

分数阶常微分方程的改进精细积分法

基于Mittag-Leffler函数的定义式,构造Mittag-Leffler矩阵函数的精细迭代计算格式.与常规指数函数的迭代格式相比,迭代递推中多了修正项,其表达式与分数阶导数的阶次有关.对于以Caputo分数导数定义的动力学分数阶常微分方程,使用基于Mittag-Leffler函数的精细积分法可计算方程解在各时间段端点对应函数值.算例表明了所提计算方法的有效性,其精度可由所增加修正项的阶次控制.     

Numerical Integration of a Class of Ordinary Differential Equations on the General Linear Group of Matrices

This paper concerns with the numerical solution of a class of ordinary differential equations on G l(n), the set of all n×n nonsingular real matrices. In particular, we consider matrix dynamical systems of the form Y′=Y ^?T F(Y). The presence of the inverse of the solution and of possible escape times make the numerical solution of this kind of differential equations somewhat worrisome. Here, we suggest some numerical techniques to avoid some problems arising in its numerical solution.     

时变线性系统的稳定性

本文把文[1]、[2]的变换技巧推广到变系数线性系统(i=1,…,n),用函数方法,研究了n=2、3时零解的稳定性,得到了零解稳定的若干判据.有关系数a_(ij)(t)不必都限制为有界.     

Stability Criterion for Systems of Two First-Order Linear Ordinary Differential Equations

Mathematical Notes - The method of Riccati’s equation is applied to find a stability criterion for systems of two first-order linear ordinary differential equations. The obtained result is...     

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method. AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20     

Numerical Solution of the Eigenvalue Problem for Discontinuous Linear Ordinary Differential Equations

The eigenvalue problem for linear ordinary differential equationswith discontinuous terms is considered. It is shown that onecan effectively use Atkinson's product integration techniquestogether with a Green's function to solve the problem numerically.     

Asymptotic Integration of Linear Ordinary Differential Equations of Order N

This paper introduces the notion of scalar dichotomies, an analog of the Levinson dichotomic conditions for diagonal systems adapted to differential equations of order n. Using this concept, a general theory of asymptotic integration of the nonautonomous equation x(n) + (a1(t) + b1(t)) x(n-1) + ... + (an(t) + bn(t)) x = 0 is given, provided the solutions of the equation x(n) + a1(t)x(n-1) + ... + an(t)x = 0, up to the p-derivative, 0 p n - 1, are known. We study in details the constant coefficient case ai(t) = constant, where we obtain an extension of the Ghizzetti theorem. All results are obtained without using canonical forms.     

On the Convergence of Sequences of Ordinary Differential Equations

Mathematical Notes -