Numerical Analysis of Ordinary Differential Equations and Its Applications

NUMERICAL ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS

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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 an electronic or mechanical, including photocopying, recording or any information storage and 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 the Copyright 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 methods, linear multistep methods, stability analysis, parallel implementation, self-validating numerical methods, analysis of nonlinear oscillation by numerical means, 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

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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

A New Algorithm for Differential-Algebraic Equations Based on H I D M T. Watanabe and G. Gnudi

75

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 L I M I T I N G F O R M U L A S OF E I G H T - S T A G E E X P L I C I T R U N G E - K U T T A M E T H O D OF O R D E R S E V E N 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 the limit as the first abscissa approaches zero, the formulas can achieve seventh order. Such formulas are called limiting 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. I n t r o d u c t i o n 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 , we derived five-stage fifth order and six-stage sixth order limiting formulas. Furthermore, we presented five- and sixstage 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. 3

2. L i m i t i n g 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 2

method are represented in the following Butcher array : 0.21

"31

132

W

h

•

" ,l-l s

••

And, yi is used to denote the y ordinate at the abscissa Cj, namely, i—1 Kf = y

a

+

n

ft^ 'j/j'

j=i

where / l =/(*«,¥»),

fi

=

(i = 2,3, • • • , * ) .

f(tn+Cih,yi)

Using them, the method can be written as •

5 , 5

Many eight-stage sixth order formulas are known and their properties are precisely reported . An eight-stage limiting formula that uses the values of the second derivatives at the point ( t „ , J / „ ) has the form 5

A = /(*».*,),

F =

D(f(t ,y ))-v(f,),

2

¥3 = V* + HaaJi + h = f(t + c h,y ), n

3

n

n

ha F ), 3

2

3

Vi = •i—3

/, = flU + Cih^)

j—j

(i = 4 , 5 , - - , 8 ) , B

S= *+ !

E ic=l + l

W * = E

I E b Aa^

j = fc+l \ ' = j + l

;ai

=a

k

(4 = 4,5,6),

/

E f E ^#U

(27)

i=4

= 0-6

2 8

)

- 1) are

P7 b»'

(29)

>

ass =

1

«75 =

Pi o

i=4

i=4

and a,j (i = 6,7,8; j = 5, S T

8

(

'

Q

6 = |-£fti& 3

7-1

1

(30)

6 The last equation of condition given by Eq.(23) must be satisfied with the solutions above obtained under the assumption given by Eq.{4). By trivial manipulations, we get the following relation between c and C j : 4

14c cJ - 12c e« + 3cs - c, = 0. s

(32)

s

The parameter a obtained from Eq.(23) and c can be rewritten using the relation given by Eq.(32) as 4l i ~ *l • (33) 54

4

c

= - 3

c

And the other a 's are it

a

M

= —(t, - c a ) , s

-

< >

These a, 's are found to satisfy Eqs.(7), (8), (9) and (10) by a straightforward computation. Finally, from Eqs.(5) and (3) we get 3

«a = C - X > „ and

^

(36)

(. = 4,5,--.,8).

(37)

^

^ = f - £ > ^ *

(. = 4,5,..-,8)

,=3

Now, we have obtained a set of parameters of the eight-stage seventh order limiting formula with four free parameters, c , c , c§ and c . 3

t

T

3. Determination of free parameters In the solutions obtained in the previous section, four parameters c , c«, Ce and cj are free to be chosen in any way. In this section we will consider how to determine these parameters. The stability region depends on only one free parameter c,. It is desirable to determine c, so as to maximize the stability region. At the same time it is preferable that every parameter is the number requiring a small number of digits and small in magnitude. We intend to derive the eight-stage formulas which achieve numerically 3

7

seventh order by replacing derivatives with numerical differentiation. The key point to derive these formulas is that the error caused by numerical differentiation does not dominate over the leading error term of the limiting formula. Here, we will present two sets of free parameters. One of them gives the parameters requiring comparatively small number of digits, and the other gives comparatively large stability region. 3.1. Stability The polynomial r which determines the stability of the eitht-stage seventh order limiting formula given by Eq.(l) is z7+

••••

where gj

T* ,

c (3 - 7c,)

1 7 =

8

I + ; + ^ - + --- + ^

4

- W

t

^

^

o

,

=

1

5

1

2

0

f

l

4

c

. _

1

2

c

i

+

3

)

and z is the complex number. The stability region is the set of points for which |r(«)| < 1. Let the simply connected interval ( — d , Q ) be the intersection of the stability region with the negative part of real axis. This interval is called the stability interval. The boundaries of the stability regions for several values of 7 are shown in Fig. 1 with the values of 7 attached. The graph indicates that 7 e { — — , — — } « (0.143 x 10 ,0.147 x 10 ) 70000 '68000' ' -4

7

1

-4

K

K

(38) '

gives the maximum stability region. In the case where the range of c is restricted to the interval (0,1), we get from Eq.(32) s

3 0< c £i, - • Its Lagrange form is given by Lm=£y(®m)

where

i0$=J[t^k,

Theorem 3 For fixed distinct points fo, • • • , 6 on [0,1], the abscissae Ci, the s-stage collocation method satisfy the equation

=

0

w h e r e

c, of

«*' ^(t) be a polynomial of degree (p + 2q) given by 4£,,(f) = td> (t}, then we have p

fiq

j f *;,,(*) / ( c ) = Eb {q