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Multistep and Runge-Kutta convolution quadrature methods for coupled dynamical systems
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Abstract
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential equation. The linear subsystem can be eliminated in frequency domain and for the numerical solution of the resulting integro-differential algebraic equations, we propose a a combination of Runge-Kutta or multistep time stepping methods with appropriate convolution quadrature to handle the integral terms. The resulting methods are shown to be algebraically equivalent to a Runge-Kutta or multistep solution of the coupled system and thus automatically inherit the corresponding stability and accuracy properties. After a computationally expensive pre-processing step, the online simulation can, however, be performed at essentially the same cost as solving only the small nonlinear subsystem. The proposed method is, therefore, particularly attractive, if repeated simulation of the coupled dynamical system is required.
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In some applications, there arises the need of a spatially distributed description of a physical quantity inside a device coupled to a circuit. Then, the in-space discretised system of partial differential equations is coupled to the system of equations describing the circuit (modified nodal analysis) which yields a system of differential algebraic equations (DAEs). This paper deals with the differential index analysis of such coupled systems. For that, a new generalised inductance–like element is defined. The index of the DAEs obtained from a circuit containing such an element is then related to the topological characteristics of the circuit’s underlying graph. Field/circuit coupling is performed when circuits are simulated containing elements described by Maxwell’s equations. The index of such systems with two different types of magnetoquasistatic formulations (A* and T-Ω) is then deduced by showing that the spatial discretisations in both cases lead to an inductance-like element.
We introduce a new algorithm for approximation by rational functions on a real interval or a set in the complex plane, implementable in 40 lines of Matlab. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points, (2) greedy selection of the support points to avoid exponential instabilities, and (3) least-squares rather than interpolatory formulation of the overall problem. The name AAA stands for "aggressive Antoulas--Anderson" in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a Matlab code and eight applications and describe variants targeted at problems of different kinds.
We describe an efficient algorithm for time-domain simulation of elements described by causal impulse responses. The computational bottleneck in the simulation of such elements is the need to compute convolutions at each time point. Hence, direct approaches for the simulation of such elements require time O(N<sup>2</sup>), where N is the length of the simulation. We apply ideas from approximation theory to reduce this complexity to O(N log N) while maintaining double-precision accuracy. The only restriction imposed by our method is that the impulse response h(t) gets “smoother” as t goes to infinity. Essentially all physically reasonable impulse responses have this characteristic. The ideas presented can also be applied to time-domain simulation of elements described in the frequency domain, including those characterized by measured data. In this paper, we demonstrate the efficiency of the algorithm by applying it to the simulation of lossy transmission lines
We present algorithms for reducing large circuits, described at
SPICE-level detail, to much smaller ones with similar input-output
behavior. A key feature of our method, called time-varying Pade (TVP),
is that it is capable of reducing time-varying linear systems. This
enables it to capture frequency-translation and sampling behavior,
important in communication subsystems such as mixers and
switched-capacitor filters, Krylov-subspace methods are employed in the
model reduction process. The macromodels can be generated in SPICE-like
or AHDL format, and can be used in both time- and frequency-domain
verification tools. We present applications to wireless subsystems,
obtaining size reductions and evaluation speedups of orders of magnitude
with insignificant loss of accuracy. Extensions of TVP to nonlinear
terms and cyclostationary noise are also outlined
We give an algorithm to compute $N$ steps of a convolution quadrature
approximation to a continuous temporal convolution using only $O(N \log N)$
multiplications and $O(\log N)$ active memory. The method does not require
evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of
its Laplace transform, which is assumed sectorial.
The algorithm can be used for the stable numerical solution with
quasi-optimal complexity of linear and nonlinear integral and
integro-differential equations of convolution type. In a numerical example we
apply it to solve a subdiffusion equation with transparent boundary conditions.
When using distributed magnetoquasistatic field models as additional elements in electric circuit simulation, the field equations contribute with large symmetric linear systems that have to be solved. The naive coupling and solving (using direct solvers) is not always efficient, since the electric circuit is coupled only via coils, which are often represented only by a small subset of the unknowns. We revisit the Schur complement approach, give a physical interpretation and show that a heuristics for bypassing Newton iterations allow for efficient multirate time-integration for the field/circuit coupled model.
Coupled systems of differential-algebraic equations (DAEs) may suffer from instabilities during a dynamic iteration. We extend the existing analysis on recursion estimates, error propagation, and stability to (semiexplicit) index-1 DAEs. In this context, we discuss the influence of certain coupling structures and the computational sequence of the subsystems on the rate of convergence. Furthermore, we investigate in detail convergence and divergence for two coupled problems stemming from refined electric circuit simulation. These are the semiconductor-circuit and field-circuit couplings. We quantify the convergence rate and behavior also using Lipschitz constants and suggest an enhanced modeling of the coupling interface in order to improve convergence.
Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals$$f*g (x) = \int_0^x {f (x - t) g (t) dt} (x \geqq 0)$$ by convolution quadrature rules. Here approximations tof* g (x) on the gridx=0,h, 2h, ..., NhtN h are obtained from a discrete convolution with the values of g on the same grid. The quadrature weights are determined with the help of the Laplace transform off and a linear multistep method. It is proved that the convolution quadrature method is convergent of the order of the underlying multistep method.
In this paper we discuss the discrete Fourier transform and point out some computational problems in (mainly) complex analysis where it can be fruitfully applied. We begin by describing the elementary properties of the transform and its efficient implementation, both in the one-dimensional and in the multi-dimensional case, by the reduction formulas of Cooley, Lewis, and Welch (IBM Res, paper, 1967).The following applications are then discussed: Calculation of Fourier coefficients using attenuation factors; solution of Symm’s integral equation in numerical conformal mapping; trigonometric interpolation; determination of conjugate periodic functions and their application to Theodorsen’s integral equation for the conformal mapping of simply and of doubly connected regions; determination of Laurent coefficients with applications to numerical differentiation, generating functions, and the numerical inversion of Laplace transforms; determination of the “density” of the zeros of high degree polynomials. We then...
In today's technological world, physical and artificial processes are mainly described by mathematical models, which can be used for simulation or control. These processes are dynamical systems, as their future behavior depends on their past evolution. The weather and very large scale integration (VLSI) circuits are examples, the former physical and the latter artificial. In simulation (control) one seeks to predict (modify) the system behavior; however, simulation of the full model is often not feasible, necessitating simplification of it. Due to limited computational, accuracy, and storage capabilities, system approximation—the development of simplified models that capture the main features of the original dynamical systems—evolved. Simplified models are used in place of original complex models and result in simulation (control) with reduced computational complexity. This book deals with what may be called the curse of complexity, by addressing the approximation of dynamical systems described by a finite set of differential or difference equations together with a finite set of algebraic equations. Our goal is to present approximation methods related to the singular value decomposition (SVD), to Krylov or moment matching methods, and to combinations thereof, referred to as SVD-Krylov methods.
Part I addresses the above in more detail. Part II is devoted to a review of the necessary mathematical and system theoretic prerequisites. In particular, norms of vectors and (finite) matrices are introduced in Chapter 3, together with a detailed discussion of the SVD of matrices. The approximation problem in the induced 2-norm and its solution given by the Schmidt—Eckart—Young—Mirsky theorem are tackled next. This result is generalized to linear dynamical systems in Chapter 8, which covers Hankel-norm approximation. Elements of numerical linear algebra are also presented in Chapter 3. Chapter 4 presents some basic concepts from linear system theory. Its first section discusses the external description of linear systems in terms of convolution integrals or convolution sums. The section following treats the internal description of linear systems. This is a representation in terms of first-order ordinary differential or difference equations, depending on whether we are dealing with continuous- or discrete-time systems. The associated structural concepts of reachability and observability are analyzed. Gramians, which are important tools for system approximation, are introduced in this chapter and their properties are explored. The last section of Chapter 4 is concerned with the relationship between internal and external descriptions, which is known as the realization problem. Finally, aspects of the more general problem of rational interpolation are displayed.
We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the confolution kernal via a discrete operational calculus. 23 refs., 2 tabs.
Publisher Summary
This chapter discusses the modeling aspect of differential-algebraic equations (DAEs). In computational engineering, the network modeling approach forms the basis for computer-aided analysis of time-dependent processes in multibody dynamics, process simulation, or circuit design. Its principle is to connect compact elements via ideal nodes, and to apply some kind of conservation rules for setting up equations. The mathematical model, a set of so-called network equations, is generated automatically by combining network topology with characteristic equations describing the physical behavior of network elements under some simplifying assumptions. Interconnects and semiconductor devices (i.e., transistors) are modeled by multi-terminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities. Interconnects and semiconductor devices (i.e., transistors) are modeled by multiterminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities.
The coupling between a 3D modified magnetic vector potential formulation discretized by the finite integration technique and an electrical circuit that includes solid and stranded conductors is presented. This paper describes classical time integration methods and the implicit Runge-Kutta methods, the latter being an appropriate alternative to the first ones to solve effectively index 1 differential-algebraic equations arising from combined simulation of electromagnetic fields and electrical circuits.
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.
We present a systematic study of how to take into account external excitation in the eddy current model. Emphasis is put on mathematically sound variational formulations and on lumped parameter excitation through prescribed currents and voltages. We distinguish between local excitation at known contacts, known generator current distributions and non-local variants that rely on topological concepts. The latter case entails the violation of Faraday's law at so-called cuts and prevents us from reconstructing a meaningful electric field. Copyright © 2004 John Wiley & Sons, Ltd.
A class of linear algorithms for Volterra Integro-Differential Equations is studied. The concept of the associated canonical fraction is extended to this class and leads to an algebraic criterion forA-stability.
The treatment of massive and stranded inductors is studied in the frame of dual magnetodynamic finite element h- and a-formulations. On both sides, nodal and edge finite elements are used and source fields are defined when needed as mathematical tools to be used directly in each formulation to lead to circuit relations for inductors. Simplified expressions of source fields are proposed.
This paper is concerned with the numerical solution of delay integro-differential equations. The adaptation of linear multistep methods is considered. The emphasis is on the linear stability of numerical methods. It is shown that every A-stable, strongly 0-stable linear multistep method of Pouzet type can preserve the delay-independent stability of the underlying linear systems. In addition, some delay-dependent stability conditions for the stability of numerical methods are also given. (c) 2007 Elsevier Ltd. All rights reserved.
Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arena that follow this design methodology providing extensible, reusable, and flexible software while staying competitive to traditionally designed point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such that the level of abstraction is essentially not raised. In this paper we discuss an alternative way to approach the design challenge which we call "concept oriented design". We apply this design methodology to Petrov-Galerkin methods leading to a class library for both, boundary element methods (BEM) and finite element methods ( FEM). We show as a particular example the implementation of hp-FEM using the library with special attention to the handling of inconsistent meshes.
Linear multistep methods for ordinary differential equations generate convolution quadrature rules. We investigate the stability
of such methods when applied to Volterra integral equations of the second kind and Volterra integro-differential equations.
It is shown that the classical stability concepts for ordinary differential equations (strong stability, A-stability, A(α)-stability) carry over to convolution equations with positive definite or completely monotonic kernels.
An approach for transient simulation of lossy interconnects
terminated in arbitrary nonlinear elements based on convolution
simulation is presented. The Pade approximations of each line's
characteristics or each multiconductor line's modal functions are used
to derive a recursive convolution formulation, which greatly reduces the
amount of computation used to perform convolutions. The approach can
handle frequency-varying effects, such as skin effects, and general
coupling situations. The errors introduced by Pade approximations are
analyzed, and a scheme for determining the necessary order for an
approximation is developed. The approach has been incorporated in the
stepwise equivalent conductance timing simulator (SWEC) for digital CMOS
circuits. Comparisons with SPICE3.e indicate that SWEC, which gives
accurate results, can be one to two orders of magnitude faster
The paper describes a general methodology for the fitting of
measured or calculated frequency domain responses with rational function
approximations. This is achieved by replacing a set of starting poles
with an improved set of poles via a scaling procedure. A previous paper
(Gustavsen et al., 1997) described the application of the method to
smooth functions using real starting poles. This paper extends the
method to functions with a high number of resonance peaks by allowing
complex starting poles. Fundamental properties of the method are
discussed and details of its practical implementation are described. The
method is demonstrated to be very suitable for fitting network
equivalents and transformer responses. The computer code is in the
public domain, available from the first author
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