Passivity-preserving model reduction of differential-algebraic equations in circuit simulation

We present an extension of the positive real balanced truncation model reduction method for differential-algebraic equations that arise in circuit simulation. This method is based on balancing the solutions of the projected generalized algebraic Riccati equations. Important properties of this method are that passivity is preserved in the reduced-order model and that there exists an approximation error bound. Numerical solution of the projected Riccati equations using the special structure of circuit equations is also discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Index for linear systems of differential-algebraic equations with partial derivatives

We investigate time-varying linear differential algebraic equations with partial derivatives. We introduce concept of insolubility index as the least possible order of the differential operator which transforms the initial system into a structural form with separated the “algebraic” and “differential” subsystems. The approach does not assume the existence of differential indexes with respect to independent variables.     

Computing periodic solutions of linear differential-algebraic equations by waveform relaxation

We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1. Numerical examples, where discrete waveforms are computed with a backward-difference formula, further illustrate the correctness of the theoretical work in this paper.

     

An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems

Averaging or gradient recovery techniques, which are a popular tool for improved convergence or superconvergence of finite element methods in elliptic partial differential equations, have not been recommended for nonconvex minimization problems as the energy minimization process enforces finer and finer oscillations and hence at the first glance, a smoothing step appears even counterproductive. For macroscopic quantities such as the stress field, however, this counterargument is no longer true. In fact, this paper advertises an averaging technique for a surprisingly improved convergence behavior for nonconvex minimization problems. Similar to a finite volume scheme, numerical experiments on a double-well benchmark example provide empirical evidence of superconvergence phenomena in macroscopic numerical simulations of oscillating microstructures. AMS subject classification (2000)  65K10,65N30     

On convergence conditions of waveform relaxation methods for linear differential-algebraic equations

For linear constant-coefficient differential-algebraic equations, we study the waveform relaxation methods without demanding the boundedness of the solutions based on infinite time interval. In particular, we derive explicit expression and obtain asymptotic convergence rate of this class of iteration schemes under weaker assumptions, which may have wider and more useful application extent. Numerical simulations demonstrate the validity of the theory.     

A note on topological methods for a class of differential-algebraic equations

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine a straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided.     

计算微分代数系统稳定域的迭代Lyapunov函数方法

用迭代Lyapunov函数方法对微分代数系统稳定域进行了研究,根据所研究的微分代数系统形式,构造一个Lyapunov函数,然后对这个Lyapunov函数进行逐次迭代,给出了微分代数系统稳定域逐次扩大的迭代算法,数值实验表明迭代Lyapunov函数方法应用于微分代数系统稳定域的估计比单个Lyapunov函数具有良好的优越性。     

An existence result for elliptic partial differential-algebraic equations arising in semiconductor modeling

We consider a system of partial differential-algebraic equations which model an electric network containing semiconductor devices. The zero-dimensional differential-algebraic network equations are coupled with multi-dimensional elliptic partial differential equations which model the devices. For this coupled system we prove an existence result.     

Improving the accuracy of BDF methods for index 3 differential-algebraic equations

Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, andO(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct toO(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct theO(1) errors in the algebraic variables appearing after a change of order.This author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolívar (CESMa) for permitting her free use of its research facilities.     

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

Positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with Markovian switching. Euler–Maruyama and implicit theta-method discretisations are shown to capture exponential mean-square stability for all sufficiently small time-steps under appropriate conditions. Moreover, the decay rate, as measured by the second moment Lyapunov exponent, can be reproduced arbitrarily accurately. New finite-time convergence results are derived as an intermediate step in this analysis. We also show, however, that the mean-square A-stability of the theta method does not carry through to this switching scenario. The proof techniques are quite general and hence have the potential to be applied to other numerical methods.     

The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations

Summary Two Rosenbrock-Wanner type methods for the numerical treatment of differential-algebraic equations are presented. Both methods possess a stepsize control and an index-1 monitor. The first method DAE34 is of order (3)4 and uses a full semi-implicit Rosenbrock-Wanner scheme. The second method RKF4DA is derived from the Runge-Kutta-Fehlberg 4(5)-pair, where a semi-implicit Rosenbrock-Wanner method is embedded, in order to solve the nonlinear equations. The performance of both methods is discussed in artificial test problems and in technical applications.     

Application of circuit analysis programs to systems of partial differential equations

A general approach for solving systems of time domain partial differential equations using circuit analysis programs is described. The approach is then used to solve a nonlinear one-dimensional transient fluid flow problem. Using the general purpose circuit analysis program SPICE, the approach is fully implicit and should provide a convenient method for physical simulations in one dimension. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 601–615, 1997     

Stability analysis for a nonlinear coupled system of fractional hybrid delay differential equations

The main objective of this research work is to establish existence results as well as to study qualitative aspects of the proposed coupled system of fractional hybrid delay differential equations (FHDDEs). Using the hybrid fixed point theory, we establish appropriate results for the existence of at least one solution to our problem. The powerful tools of functional analysis and dynamical systems are applied to derive different kinds of stability analysis. These include Ulam-Hyers (UH), generalized Ulam-Hyers (GUH), Ulam-Hyers-Rassias (UHR), and generalized Ulam-Hyers-Rassias (GUHR). In order to provide the authenticity of the our results, an illustrative example is given to wind up the present research work.     

Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.     

3-trees with few vertices of degree 3 in circuit graphs

A circuit graph(G,C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F_χ with Euler characteristic χ≥0 has a 3-tree with at most vertices of degree 3, where c_χ is a constant depending only on F_χ.     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

Review of a fast simulation method for the analysis of queueing networks

This paper presents a review of and refinements to a class of discrete-event models for the analysis of unreliable queueing systems. In contrast to conventional piece-by-piece simulators, these models observe a number of rare events that affect the inflow and outflow rates at each queue. Between events, the evolution of the system is approximated by a linear function. Several experiments confirm the accuracy of this approximation and its computational efficiency over conventional simulation. © 1997 by John Wiley & Sons, Ltd.     

Nonoscillations for odd-dimensional systems of linear retarded functional differential equations

This paper is concerned with the nonoscillatory problems of odd-dimensional systems of linear retarded functional differential equations. Based upon the corresponding characteristic equations, we get some criteria for nonoscillations by utilizing the matrix measures.