Continuous System Simulation

FRANÇOIS CELLIER,; ERNESTO KOFMAN

Springer

Lugar: New York; Año: 2006 p. 663

978-0-387-26102-7

About This BookThis text introduces concepts of simulating physical systems that are mathematicallydescribed by sets of differential and algebraic equations (DAEs).The book is written for modeling and simulation (M&S) practitioners, whowish to learn more about the “intestines” of their M&S environments. Modernphysical systems M&S environments are designed to relieve the occasionaluser from having to understand in detail, what the environment doesto their models. Simulation results appear magically upon execution of themodel.Magic has its good and its bad sides. On the one hand, it enables usto separate the discussion of the tasks of modeling from those of simula-tion. The occasional user of M&S environments may be perfectly happy toonly learn about modeling, leaving the gruesome details of numerical DAEsolvers to the specialist.Yet, for those among our readers, who are not in the habit of leavingthe railway station through platform 9 3/4, this book may be helpful, asit explains, in lots of detail, how M&S environments operate. Thanks tothis knowledge, our readers will understand what they need to do, whenthe magic fails, i.e., the simulation run is interrupted prematurely with anerror message. They will also be able to understand, why their simulationprogram is consuming an unreasonable amount of execution time. Finally,they will feel more comfortable with the simulation results obtained, asthey understand, how these results have been produced. “Magic” is awfullydifficult to explain to your boss.The text contains 12 chapters that are unfortunately rather heavily dependenton each other. Thus, reading one chapter of the book, because itdiscusses a topic that you are currently interested in, may not get you veryfar. Each chapter assumes the knowledge presented in previous chapters.Chapters 1–4 introduce the concepts of numerical ODEs in a fairly classicalway. After a general introduction to the topics that this book concernsitself with, presented in Chapter 1, Chapter 2 offers an introduction to thebasic properties of numerical ODE solvers: numerical stability and accuracy.These are introduced by means of the two most basic explicit andimplicit ODE solvers to be found: the forward and backward Euler algorithms.Chapter 3 offers a discussion of single–step integration algorithms. Newconcepts introduced include a new stability definition, called F–stability orfaithful stability, denoting algorithms, whose border of numerical stabilitycoincides with the imaginary axis of the complex eigenvalue plane. Anothernew concept introduced is the frequency order star, leading to an attractivenew definition of an accuracy domain. New ODE solvers include theor L–stable.Chapter 4 offers a discussion of linear multi–step integration algorithms.All of these algorithms are derived by means of Newton–Gregory polynomials,which offer a much more elegant way of introducing these algorithms,than those found in most other numerical ODE textbooks. NewODE solvers introduced in this chapter include a set of higher–order stifflystable BDF methods that are based on least squares extrapolation.Chapters 5 and 6 complete the discussion of numerical ODEs. Thesechapters can be skipped without making the subsequent chapters moredifficult to understand.Chapter 5 discusses special–purpose ODE solvers for second–derivativemodels, as they occur naturally in the mathematical description of mechanicalsystems. This topic has been discussed in the past in a few mechanicsbooks, but it is hardly ever covered in the numerical ODE literature.Chapter 6 offers a fairly classical discussion of the method–of–lines approximationto partial differential equations (PDEs). Thereby PDEs areconverted to sets of ODEs. This topic is not usually covered in the numericalODE literature, but has been dealt with in the past in more specializedtextbooks on numerical PDEs. New in this chapter is the derivation of theformulae for computing spatial derivatives by means of Newton–Gregorypolynomials. Also innovative is the use of Richardson extrapolation methods,previously introduced in Chapter 3 in their normal context, for thecomputation of spatial derivatives.Chapters 7 and 8 deal with the issues surrounding numerical DAEs.Chapter 7 concerns itself with the symbolic conversion of sets of DAEs tosets of ODEs that can subsequently be dealt with numerically using thetechniques introduced in earlier chapters of the book. Chapter 8, on theother hand, deals with the numerical solution of DAEs without previousconversion to explicit ODE form.The symbolic tools presented in Chapter 7 are the result of a collaborationbetween one of the authors of this book with Hilding Elmqvist ofDynasim, a Swedish company specialized in the development of modernphysical systems M&S environments, and Martin Otter of the GermanAerospace Center (DLR) in Oberpfaffenhofen, Germany.The numerical tools presented in Chapter 8 are a bit more classical. Someof these concepts can be found in the numerical DAE literature. However,the concepts presented previously in Chapter 7 help in presenting thesealgorithms in a clear and easily understandable fashion, which is not truefor much of the existing numerical DAE literature.We are convinced that the material presented in Chapters 7 and 8 makesa significant contribution to advancing the maturity of understanding of therelatively recent research field of numerical DAEs. New in Chapter 8 arebackinterpolation algorithms, which can be designed to be either F–stablethe discussion of inline integration, and the way, in which we deal with theproblem of overdetermined DAEs. The problem of overdetermined DAEshas only recently been recognized in the numerical DAE literature, andfurthermore, the techniques for tackling them proposed by other authors,such as Ernst Hairer and Gerhard Wanner, are quite different from ours.Chapter 9 discusses the problems surrounding the numerical simulationacross discontinuities. This is a topic that both authors of this book werecentrally concerned with in their respective Ph.D. dissertations. Chapter 9presents the tools and technique developed by the first author, whereasthose used by the second author are postponed to Chapter 12.Chapter 10 introduces the reader to the problems of performing simulationruns in real time, i.e., synchronizing the numerical ODE solvers withthe real–time clock. Interesting in this context may be the discussion ofthe linearly–implicit integration algorithms. More results, and more fundamentalresults concerning real–time simulation are provided in Chapter 12of the book.Up to this point, the book follows fairly classical approaches to numericalODE, PDE, and DAE solutions. All of the techniques presented discretizethe time axis, and perform the numerical simulation by means of extrapolationsthat are approximations of Taylor–series expansions. All of thetechniques presented are synchronous algorithms, as all differential equationsare simulated synchronously, in step with the temporal discretization.Chapters 11 and 12 represent a radical departure from these concepts.In these chapters, the state variables themselves are being discretized. Wecall this a spatial discretization instead of a temporal discretization. In contrast,the time axis is no longer discretized. Furthermore, these algorithmsproceed asynchronously, i.e., each state variable carries its own simulationclock that it updates as needed. The simulation engine ensures that thestate variable that is currently the one most behind always gets updatednext. The different state variables communicate with each other by meansof state interpolation.As these are the last two chapters in the book, they can be skipped withoutany problem. Yet, these are easily the most interesting chapters in theentire book, as they revolutionize the way of looking at numerical ODEs,offer an exciting new theory of numerical stability, and lend themselves toplenty of fascinating open research questions.