The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods
The term differential-algebraic equation was coined to comprise differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such problems arise in a variety of applications, e.g. constrained
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Ernst Hairer Christian Lubich Michel Roche
The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods
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Authors
Ernst Hairer Michel Roche Universite de Geneve, Departernent de Mathematiques Case Postale 240, 1211 Geneva 24, Switzerland Christian Lubich Universitat Innsbruck, Institut fur Mathematik und Geometrie Technikerstr. 13, 6020 Innsbruck, Austria
Mathematics Subject Classification (1980): 65L05, 65H 10, 34A50
ISBN 3-540-51860-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51860-6 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
Preface The term differential-algebraic equation has been coined to comprise differential equations with constraints (differential equations on manifolds) and singular implicit differential equations. Such problems arise and have to be solved in a variety of applications, e.g., constrained mechanical systems, fluid dynamics, chemical reaction kinetics, simulation of electrical networks, and control engineering. From a more theoretical viewpoint, the study of differential-algebraic problems gives insight into the behaviour of numerical methods for stiff ordinary differential equations. As a consequence, this subject has attracted the interest of many engineers and mathematicians in the last years. The purpose of these lecture notes is to give a self-contained and comprehensive exposition ofthe numerical solution of differential-algebraic systems arising in applications, when treated by Runge-Kutta methods, here included also extrapolation methods. While multistep methods (BDF) have been considered since the early seventies (Gear (1971)), the study of Runge-K utta methods for differentialalgebraic systems has begun only a few years ago. Runge-Kutta methods also have interesting computational and theoretical properties. They combine high order with good stability, allow a simple step size selection, are self-starting and have advantages in parallel computing. The first two sections are introductory and review differential-algebraic problems and Runge-Kutta methods for their numerical solution. In Sections :3 to G we study existence and uniqueness of the numerical solution, influence of perturbations, local error and convergence, and asymptotic expansions. We
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