Numerical Analysis of Ordinary Differential Equations and Its Applications PDF
Preview Numerical Analysis of Ordinary Differential Equations and Its Applications
NUMERICAL ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS This page is intentionally left blank NUMERICAL ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS Editors T Mitsui Nagoya University, Japan Y Shinohara Tokushima University, Japan fe World Scientific WT Singapore * New Jersey * London•Hong Kong Published by World Scientific Publishing Co. Pie. Ltd. P O Box 128. Farrer Road, Singapore 9128 USA office: Suite IB. 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. NUMERICAL ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or pans thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in (his volume ,please pay a copying fee through th eCopyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA. ISBN 981-02-2229-7 This book is printed on acid-free paper. Printed in Singapore by L)to-Print V Preface Numerical solutions of ordinary differential equations (ODEs) are broadly recognized that they are not only interesting in theoretical study but also useful in practical applications. It is the reason why the numerical analysis of ODEs has been attracting many research works in the scientific computation community. One might be aware that this year is the centennial memorial one since the historical article of C. RUNGE "Uber die numerische Auflosung von Differeutialgleichungen" appeared in Mathematiscke Annalen as the pioneering work of more sophisticated and effective numerical solution of ODEs. Hoping that this volume contributes to the progress of numerical analysis of ODEs, we are publishing it as a collection of original research articles. The contributions in this volume are mainly based on those which were submitted in 1994 Kyoto Workshop on Numerical Analysis of ODEs held in November of 1994 at the Research Institute for Mathematical Scicences, Kyoto University. The topics of the articles are widely spreading, although they are touching more or less upon the numerical solutions of ODEs. They reflect the state-of-the-art of the study in numerical analysis. Actually topics treated in the volume are: discrete variable methods, Runge-Kutta meth ods, linear multistep methods, stability analysis, parallel implementation, self-validating nu merical methods, analysis of nonlinear oscillation by numerica lmeans, differential-algeraic and del ay-differential equations, stochastic initial value problems and so on .Readers will be able to recognize the recent development of these topics. Last, but not least, we express our sincere gratitude to the present authors of the volume as well as to the contributors of the Workshop. June 1995 Taketomo Mitsui Nagoya University Yoshitane Shinohara Tokushima University This page is intentionally left blank vii CONTENTS Preface v Limiting Formulas of Eight-Stage Explicit Runge-Kutta Method of Order Seven H. Ono 1 A Series of Collocation Runge-Kutta Methods T. Mitsui and H. Sugiura 15 Fourth Order P-Stable Block Method for Solving the Differential Equation y" = f(x, y) K. Ozawa 29 Two-Point Hermite-Birkhoff Quadratures and Its Applications to Numerical Solution of ODE C. Suzuki 43 Improved SOR-like Method with Orderings for Non-Symmetric Linear Equations Derived from Singular Perturbation Problems E. Ishiwata and Y. Muroya 59 Analysis of the Milne Device for the Finite Correction Mode of the Adams PC Methods I M. Fuji 75 A New Algorithm for Differential-Algebraic Equations Based on HIDM T. Watanabe and G. Gnudi 91 Semi-Explicit Methods for Differential-Algebraic Systems of Index 1 and Index 2 H. Skintani 113 Computational Challenges in the Solution of Nonlinear Oscillatory Multibody Dynamics Systems J. Yen and L. Petzold 127 Existence and Uniquess of Quasi periodic Solutions to Quasiperiodic Nonlinear Differential Equations Y. Shinohara, A. Kohda and H. Imai 147 viii Absolutely Stable Delay Differential Equations and Natural Runge-Kutta Methods T. Koto 165 An Interval Method of Proving Existence of Solutions for Nonlinear Boundary Value Problems S. Oishi 179 Experimental Studies on Guaranteed-A ecu racy Solutions of the Initial-Value Problem of Nonlinear Ordinary Differential Equations M. Iri and J. Amemiya 195 Numerical Validation for Ordinary Differential Equations Using Power Series Arithmetic M. Kaskiwagi 213 Statistical Error Analysis in Numerical Simulation for Stochastic Integral Processes Y, Saito and T. Mitsui 219 1 LIMITING FORMULAS OF EIGHT-STAGE EXPLICIT RUNGE-KUTTA METHOD OF ORDER SEVEN HARUMl0N0 Faculty of Engineering, Chiha University 1-SS Yayoicko, Inage-ka, Chiba, 263, Japan E-mail: aB9600Stansei.cc.u-tokyo.ac. jp ABSTRACT It is well known that eight-stage explicit Runge-Kutta formulas are of order at most six. However, by taking th elimit as the first abscissa approaches zero, the formulas can achieve seventh order. Such formulas are calle dlimiting formulas, which requre the evaluations of the second derivatives of the solution. In this paper, eight-stage seventh order limiting formulas using the second derivatives are derived. And based on these limiting formulas, new eight-stage numerically seventh order methods without derivatives are proposed. 1. Introduction The attainable order of s-stage explicit Runge-Kutta methods is s — 1 for s = 5, 6 and 7. However, they can achieve sth order in the limiting case where the distance between some pairs of abscissas approaches zero. Such formulas are called s-stage sth order limiting formulas. Previously 3, we derived five-stage fifth order and six-stage sixth order limiting formulas. Furthermore, we presented five- and six- stage formulas of orders numerically five and six. They are obtained by replacing the second derivatives involved in the limiting formulas with the simplest numerical differentiation. The reason to be able to do so is that the values of the second derivatives in the limiting formulas do not require full significant figures carried in the computation and we can choose free parameters so as to minimize the error caused by numerical differentiation. In this paper, eight-stage seventh order limiting formulas are presented. And based on these limiting formulas, new eight-stage numerically seventh order formulas without derivatives are derived by the similar way as in the five-stage case. 2. Limiting formulas The problem is an initial value problem ^=Mv), y{t) = yo a where / and y are vectors and / is assumed to be different iable sufficiently often for the definition to be meaningful. The parameters of an s-stage explicit Runge-Kutta 2 method are represented in the following Butcher array 2: 0.21 "31 132 • " ,l-l s W h •• And, yi is used to denote the y ordinate at the abscissa Cj, namely, i—1 Kf = y + ft^a'j/j' n j=i where /l =/(*«,¥»), (i = 2,3, •••,*). fi = f(tn+Cih,yi) Using them, the method can be written as • Many eight-stage sixth order formulas are known5 ,5 and their properties are precisely reported 5. An eight-stage limiting formula that uses the values of the second derivatives at the point (t„,J/„) has the form A = /(*».*,), F = D(f(t,y))-v(f,), 2 n n V* + HaaJi + haF), ¥3 = 3 2 h = f(t + ch,y), n 3 3 Vi = •i—3 j—j /, = flU + Cih^) (i = 4,5,--,8), B S<n+1= Vn + Khfl + Y, bif' + kfaFj), (1) 1=3 where D(f(t, y)) and v(f,) denote the Jacobian matrix of / at the point (t„, y) and n n n the vector (1, f\, • • •, /")r respectively (the superscripts denote the component numbers). The parameters of this limiting formula can be written in the following array analogous to Butcher array: a 31 <*3 0.43 ©S "51 "S3 «S4 "s Cs aS3 OB4 • • "87 ClB Is <*< • °S ft.
The list of books you might like
