Simulating digital circuits numerically – a charge-oriented ROW approach

A ROW type approach is considered for integral form DAEs arising in charge-oriented nodal analysis of digital networks. These network equations define very special index-2 systems that can be solved by Rosenbrock-Wanner (ROW) methods suitable for semi-explicit index-1 systems without order reduction. To obtain charge conservation, the charge variables are projected on the linear charge constraint. In contrast to the semi-explicit index-1 case, all order conditions for the algebraic variables up to order have to be fulfilled for a method of order . CHORAL, an embedded charge-oriented method of order (2)3, is introduced and compared with DASSL and RODAS for two industrial applications, the NAND gate and the two-bit adding unit. Received January 22, 1996 / Revised version received January 28, 1997     

Sensitivity of solutions of linear DAE to perturbations of the system matrices

This paper studies the effect of perturbations in the system matrices of linear Differential Algebraic Equations (DAE) onto the solutions. It turns out that these may result in a more complicated perturbation pattern for higher index problems than in the case for (standard) additive perturbations. We give an analysis here for linear index-1 and index-2 problems, which, however, has clear ramifications in nonlinear problems (e.g., via the Newton process). This analysis is sustained by a number of examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global error estimation for index-1 and -2 DAEs

In this paper, we consider the extension of three classical ODE estimation techniques (Richardson extrapolation, Zadunaisky's technique and solving for the correction) to DAEs. Their convergence analysis is carried out for semi-explicit index-1 DAEs solved by a wide set of Runge-Kutta methods. Experimentation of the estimation techniques with RADAU5 is also presented: their behaviour for index-1 and -2 problems, and for variable step size integration is investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Structural analysis of electrical circuits including magnetoquasistatic devices

Modeling electric circuits that contain magnetoquasistatic (MQS) devices leads to a coupled system of differential-algebraic equations (DAEs). In our case, the MQS device is described by the eddy current problem being already discretized in space (via edge-elements). This yields a DAE with a properly stated leading term, which has to be solved in the time domain. We are interested in structural properties of this system, which are important for numerical integration. Applying a standard projection technique, we are able to deduce topological conditions such that the tractability index of the coupled problem does not exceed two. Although index-2, we can conclude that the numerical difficulties for this problem are not severe due to a linear dependency on index-2 variables.     

Jacobi spectral solution for integral algebraic equations of index-2

This paper is concerned with obtaining the approximate solution of a class of semi-explicit Integral Algebraic Equations (IAEs) of index-2. A Jacobi collocation method including the matrix-vector multiplication representation is proposed for the IAEs of index-2. A rigorous analysis of error bound in weighted L² norm is also provided which theoretically justifies the spectral rate of convergence while the kernels and the source functions are sufficiently smooth. Results of several numerical experiments are presented which support the theoretical results.     

Global asymptotic stability beyond 3/2 type stability for a logistic equation with piecewise constant arguments

In this paper, a logistic equation with multiple piecewise constant arguments is investigated in detail. We generalize the approach in two papers, [K. Uesugi, Y. Muroya, E. Ishiwata, On the global attractivity for a logistic equation with piecewise constant arguments, J. Math. Anal. Appl. 294 (2) (2004) 560-580] and [Y. Muroya, E. Ishiwata, N. Guglielmi, Global stability for nonlinear difference equations with variable coefficients, J. Math. Anal. Appl. 334 (1) (2007) 232-247], and establish a new condition for the global stability of the equation. Their results are given as one of the special cases. Moreover, we improve the 3/2 type stability condition under several dominance assumptions on the coefficients of the equation. Some examples and numerical simulations are also presented. All of these examples show that there are several conditions for the global stability of the equation, depending on the coefficients on the delay terms of the equation, beyond the 3/2 type stability condition.     

Applying carvalho's method to find periodic solutions of difference equations

In this paper we apply the method initially developed in [1] for differential-difference equations, to the case of difference equations, in order to find 2 and 3-periodic solutions of some equations that often appear in the literatures as are for instance the case of Applications 2,5 which are examples of population growth models, and Application 4, which is a standard example of nonlinear higher order scalar difference equation depending on two parameters (see, Kocik and Ladas [3]).     

Stability and bifurcation properties of index-1 DAEs

It is well known that an equilibrium of a semi-explicit, index-1 differential-algebraic equation under a parameter variation may encounter the singularity manifold. It is a generic property of this encounter that one eigenvalue of the linear stability mapping associated with the equilibrium will pass from one half of the complex plane to the other without passing through the imaginary axis. This is known as singularity-induced bifurcation and an equivalent result is proven in this paper. While this property is generic, it is shown how more than one eigenvalue can diverge in an analogous manner, with applications in electrical power systems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical Integration of Differential Algebraic Systems and Invariant Manifolds

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.     

Criteria for the Trivial Solution of Differential Algebraic Equations with Small Nonlinearities to be Asymptotically Stable

Differential algebraic equations consisting of a constant coefficient linear part and a small nonlinearity are considered. Conditions that enable linearizations to work well are discussed. In particular, for index-2 differential algebraic equations, there results a kind of Perron Theorem that sounds as clear as its classical model.     

IRK Methods for Index 2 and 3 DAEs: Starting Algorithms

When semi-explicit differential-algebraic equations are solved with implicit Runge-Kutta methods, the computational effort is dominated by the cost of solving the non-linear systems. That is why it is important to have good starting values to begin the iterations. In this paper we study a type of starting algorithms, without additional computational cost, in the case of index-2 and index-3 DAEs. The order of the starting values is defined, and by using DA-series and rooted trees we obtain their general order conditions. If the RK method satisfies some simplifying assumptions, then the maximum order can be obtained.     

Floquet theorem for linear implicit nonautonomous difference systems

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed.     

On Coxeter recurrences

An interesting family of recurrences of order n ≥ 2, which are globally (n+3)-periodic was introduced by Coxeter in 1971. We prove a surprising property of this family: ‘all’ the possible geometrical behaviours that linear real (n+3)-periodic recurrences can have are present inside the Coxeter recurrences.     

指数—2非线性广义系统的解及其稳定性

本文主要给出指数－2的非线性广义系统的可解性和零解的稳定性分析．     

Relations de dépendance entre les équations fonctionnelles de Cauchy

Résumé Afin d'examiner les relations entre les différentes équations de Cauchy, nous résolvons, sans aucune hypothèse de régularité, l'équation fonctionnellea f(xy) + b f(x)f(y) + c f(x + y) + d (f(x) + f(y)) = 0, pour des fonctionsf, définies sur un anneau unifère divisible par deux et prenant leurs valeurs dans un corps, Les coefficientsa, b, c, etd appartiennent au centre de ce corps. Entre autres applications, nous en déduisons qu'une seule équation, à savoirf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), caractérise les endomorphismes des corps dont la caractéristique est différente de 2. En introduisant la notion d'équations fonctionnelles étrangères et d'équations fonctionnelles fortement étrangères, nous concluons à l'indépendance, au sens de cette notion, des équations classiques de Cauchy.

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Summary In order to study the inter-relations between the four Cauchy functional equations, we solve the functional equationa f(xy) + b f(x) f(y) + c f(x + y) + d(f(x) + f(y)) = 0. The functionf is defined over a ring which is divisible by 2 and which possesses a unit, while the values off are in a(skew)-field. The constantsa, b, c andd belong to this field and commute with all elements of thes-field. No regularity assumption is made onf. Among other applications, we show that the single equationf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), is enough to characterize field endormophisms in fields of characteristic different from 2. We introduce the notion of alien functional equations and that of strongly alien functional equations, to conclude that for such notions, Cauchy equations are indeed largely independent.

Dédié avec nos meilleurs voeux à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire     

Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise

The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index-1 differential-algebraic equations (DAEs). Based on the idea of Defect Correction we developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index-1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper. The first author acknowledges support by the BMBF-project 03RONAVN and the second author support by the Austrian Science Fund Project P17253.     

Index-2 Differential-Algebraic Equations

A class of general nontransferable differential-algebraic equations which contains all linear differential-algebraic equations having the global index 2 in the definition of Gear and Petzold or being tractable with index 2 in the sense of Griepentrog and März as well as nonlinear index-2 equations in the understanding of Brenan, Gear, Petzold and further authors is characterized by a uniform matrix criterion. Existence and uniqueness statements are proved.     

A composition law for Runge-Kutta methods applied to index-2 differential-algebraic equations

This paper presents a new composition law for Runge-Kutta methods when applied to index-2 differential-algebraic systems. Applications of this result to the study of the order of composite methods and of symmetric methods are given.     

Reversible methods of Runge-Kutta type for Index-2 DAEs

Summary. A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.Mathematics Subject Classification (1991): 65L05, 65L06