Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
GENERAL LINEAR METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
ZDZISLAW JACKIEWICZ
WILEY A JOHN WILEY & SONS, INC., PUBLICATION
dcd-wgc1.jpg
This Page Intentionally Left Blank
GENERAL LINEAR METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
This Page Intentionally Left Blank
GENERAL LINEAR METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
ZDZISLAW JACKIEWICZ
WILEY A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 0 2009 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 1 River Street, Hoboken, NJ 07030, (201) 748-601 1, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information about Wiley products, visit our web site at www.wiley.com.
Libra y of Congress Cataloging-in-Publieation Data:
Jackiewicz, Zdzistaw, 1950- General linear methods for ordinary differential equations / Zdzislaw Jackiewicz.
Includes bibliographical references and index. p. cm.
ISBN 978-0-470-40855-1 (cloth) 1. Differential equations, Linear. I. Title. QA372.5145 2009 5 15'.352-dc22 2009007428
Printed in the United States of America.
1 0 9 8 7 6 5 4 3 2 1
http://www.copyright.comhttp://www.wiley.com/go/permissionhttp://www.wiley.com
To my wqe, Elzbieta my son, Wojciech Tomasz
and my daughtec Hanna Katarzyna
This Page Intentionally Left Blank
CONTENTS
Preface xiii
1 Differential Equations and Systems
1.1 The initial value problem
1.2
1.3 1.4
Examples of differential equations and systems
Existence and uniqueness of solutions
Continuous dependence on initial values and the right-hand
side
Derivatives with respect to parameters and initial values
Stiff differential equations and systems
Examples of stiff differential equations and systems
1.5
1.6 Stability theory
1.7
1.8
2 introduction to General Linear Methods
2.1
2.2 Preconsistency, consistency. stage-consistency, and
Representation of general linear methods
zero-stability
16
22
27 37 50
59
59
69
vii
1
1 3 9
viii CONTENTS
2.3 2.4 2.5 2.6 2.7 2.8
2.9 2.10 2.11 2.12
Convergence Order and stage order conditions Local discretization error of methods of high stage order Linear stability theory of general linear methods Types of general linear methods Illustrative examples of general linear methods 2.8.1 2.8.2 2.8.3 2.8.4 Algebraic stability of general linear methods Underlying one-step method Starting procedures Codes based on general linear methods
Type 1: p = r = s = 2 and q = 1 or 2 Type 2: p = r = s = 2 and q = 1 or 2 Type 3: p = r = s = 2 and q = 1 or 2 Type 4: p = r = s = 2 and q = 1 or 2
3 Diagonally Implicit Multistage Integration Methods
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
3.9 3.10 3.11 3.12 3.13 3.14
Representation of DIMSIhls Representation formulas for the coefficient matrix B A transformation for the analysis of DIMSIMs Construction of DIMSIhk of type 1 Construction of DIMSIMs of type 2 Construction of DINlSIMs of type 3 Construction of DIMSIMs of type 4 Fourier series approach to the construction of DIMSIhls of high order Least-squares minimization Examples of DIILlSIMs of types 1 and 2 Nordsieck representation of DIhlSIhls Representation formulas for coefficient matrices P and G , Examples of DIMSIMs in Nordsieck form Regularity properties of DIMSIMs
4 Implementation of DlMSlMs
4.1 4.2 Local error estimation 4.3 4.4 Construction of continuous interpolants 4.5
Variable step size formulation of DIh/lSIlLIs
Local error estimation for large step sizes
Step size and order changing strategy
74 81 87 90 95
100 100 104 106 109 112 123 125 126
131
131 133 139 143 148 152 157
168 173 179 181 185 189 191
201
201 204 211 215 217
CONTENTS
4.6 4.7 Step-control stability of DIMSIMs
4.8 4.9 4.10
Updating the vector of external approximations
Simplified Newton iterations for implicit methods
Numerical experiments with type 1 DIMSIMs Numerical experiments with type 2 DIMSIMs
5 Two-step Runge-Kutta Methods
5.1 5.2 5.3 5.4
5.5
5.6
5.7 5.8 5.9 5.10 5.11 5.12
5.13
Representation of two-step Runge-Kutta methods
Order conditions for TSRK methods
Derivation of order conditions up to order 6 Analysis of TSRK methods with one stage
5.4.1 5.4.2 Analysis of TSRK methods with two stages
5.5.1 5.5.2 5.5.3 5.5.4 Analysis of TSRK methods with three stages
5.6.1 5.6.2 Two-step collocation methods
Linear stability analysis of two-step collocation methods Two-step collocation methods with one stage
Two-step collocation methods with two stages
TSRK methods with quadratic stability functions
Construction of TSRK methods with inherent quadratic
stability
Examples of highly stable quadratic polynomials and TSRK methods
Explicit TSRK methods: s = 1, p = 2 or 3 Implicit TSRK methods: s = 1, p = 2 or 3
Explicit TSRK methods: s = 2, p = 2. q = 1 or 2 Implicit TSRK methods: s = 2, p = 2, q = 1 or 2
Explicit TSRK methods: s = 2, p = 4 or 5 Implicit TSRK methods: s = 2, p = 4 or 5
Explicit TSRK methods: s = 3, p = 3. q = 2 or 3 Implicit TSRK methods: s = 3, p = 3. q = 2 or 3
6 Implementation of TSRK Methods
6.1 6.2 6.3 6.4
Variable step size formulation of TSRK methods
Starting procedures for TSRK methods
Error propagation. order conditions. and error constant
Computation of approximations to the Nordsieck vector and local error estimation
IX
220 221
227 229 233
237
237 239 250 253 253 257 262
262 265 267 272
275 275 279 281 286 288 292 296
303
307
315
315 317 321
324
X CONTENTS
6.5 Computation of approximations to the solution and stage values between grid points
Construction of TSRK methods with a given error constant and assessment of local error estimation
Continuous extensions of TSRK methods
6.6
6.7 6.8 Numerical experiments 6.9 6.10
Local error estimation of two-step collocation methods
Recent work on two-step collocation methods
7 General Linear Methods with Inherent Runge-Kutta Stability
7.1 7.2 7.3 7.4 7.5 7.6
7.7 7.8
7.9
7.10 7.11 7.12
Representation of methods and order conditions
Inherent Runge-Kutta stability
Doubly companion matrices
Transformations between method arrays
Transformations between stability functions
Lower triangular matrices and characterization of matrices
with zero spectral radius
Canonical forms of methods
Construction of explicit methods with IRKS and good
balance between accuracy and stability
Examples of explicit methods with IRKS
7.9.1 7.9.2 7.9.3 7.9.4 7.9.5
Methods with p = q = 1 and s = 2 Methods with p = q = 2 and s = 3 Methods with p = q = 3 and s = 4 Methods with p = q = 4 and s = 5 Methods with large intervals of absolute stability
on imaginary axis
Construction of A- and L-stable methods with IRKS Examples of A- and L-stable methods with IRKS Stiffly accurate methods with IRKS
8 Implementation of GLMs with IRKS
8.1 8.2 Starting procedures 8.3 Error propagation for GLMs
8.4
Variable step size formulation of GLRls
Estimation of local discretization error and estimation of higher order terms
327
328 331
336 340 344
345
345
348 355
364 374
381 384
389 393 393 395 396 398
400 404 406 408
417
417
419 423
432
8.5
8.6 8.7
8.8 8.9 8.10 8.11
References
Index
CONTENTS
Computing the input vector of external approximations for the next step
Zero-stability analysis Testing the reliability of error estimation and estimation of higher order terms Unconditional stability on nonuniform meshes Numerical experiments
Local error estimation for stiffly accurate methods Some remarks on recent work on GLMs
xi
435 438
444 448 452 456 460
461
477
This Page Intentionally Left Blank
P refa ce
This book is concerned with the theory. construction and implementation of general linear methods for ordinary differential equations. This is a very general class of methods which include the classical methods such as Runge- Kutta, linear multistep, and predictor-corrector methods as special cases. Some theoretical and practical aspects related to general linear methods are discussed in Numerical Methods f o r Ordinary Dafferentzal Equataons by J.C. Butcher, in Solvang Ordanary Dafferentzal Equatzons I: Nonstaff Problems by E. Hairer. S.P. Nmrsett, and G. Wanner, and in Solvang Ordanary Dzfferentzal Equataons 11: Staff and Dafferential-Algebraac Problems by E. Hairer and G. Wanner. However, these monographs cover the entire area of numerical so- lution of ordinary differential equations and devote only a limited amount of space to the discussion of general linear methods. This monograph is an at- tempt to present a complete analysis of some classes of general linear methods that have good potential for practical use. These classes include diagonally implicit multistage integration methods, two-step Runge-Kutta methods. and general linear methods with inherent Runge-Kutta stability.
In Chapter 1 we present a short introduction to ordinary differential equa- tions. including existence and uniqueness theory, continuous dependence on the initial data and right-hand side, stability theory, and discussion of stiff differential equations and systems. Chapter 2 is an introduction to general
xii i
xiv PREFACE
linear methods. In particular, we discuss preconsistency, consistency, stage- consistency, zero-stability, convergence, order and stage order conditions, local discretization error. and linear stability theory, and present examples of meth- ods that are appropriate for nonstiff or stiff differential systems in sequential or parallel computing environments. We also discuss briefly algebraic stabil- ity, the concept of an underlying one-step method. starting procedures, and codes based on general linear methods.
Chapters 3 to 8 constitute the main part of the book. In Chapters 3 and 4 we deal with the construction and implementation of diagonally implicit multistage integration methods. In Chapters 5 and 6 the theory and imple- mentation of two-step Runge-Kutta methods is discussed. In Chapters 7 and 8 we describe the theory and implementation of general linear methods with inherent Runge-Kutta stability. The topics in these chapters related to the theory and construction of these methods include the derivation of order and stage order conditions, representation formulas for the coefficient matrices of these methods, construction of formulas with desirable accuracy and stabil- ity properties, and Nordsieck representation of these methods. The topics in these chapters related to implementation issues include the construction of appropriate starting procedures, local error estimation for small and large step sizes. step size and order changing strategies, construction of continuous interpolants of uniform high order. updating the vector of external approxi- mations, and the solution of nonlinear systems of equations for stiff systems by the modified Newton method. We also present many examples of these meth- ods of all types, mainly of order p and stage order q = p or q = p - 1. Many implementation issues are illustrated by the results of numerical experiments with different classes of general linear methods.
I wish to acknowledge the support and assistance of many people during the last several years. I would like to thank Henryk Woiniakowski. who sug- gested, in somewhat unusual circumstances, that I should write this book. I would also like to express my gratitude to John Butcher, Zbigniew Ciesiel- ski. Maksymilian Dryja, Ernst Hairer, Stanislaw Kwapiefi. Marian Kwapisz, Waclaw Marzantowicz, and Marian Mrozek for their help and understanding. I would like to express my gratitude to John Butcher, who has founded this area of research and from whom I have learned so much in the last Several years. He was always very supportive of my work and provided invaluable insight into many topics. I would also like to thank him for his warm hospi- tality during my frequent visits to the University of Auckland and for creating stimulating research environment during these visits.
I would also like to thank my colleagues who offered comments and sugges- tions on earlier drafts of this manuscript or influenced my work in many ways. In particular. I would like to thank Peter Albrecht, Christopher Baker, Zbig- niew Bartoszewski, Alfred0 Bellen. Michal Bra&, Kevin Burrage, Jeff Cash. Philippe Chartier, Joshua Chollom. Dajana Conte, Raffaele D’Ambrosio, Way- ne Enright, Alan Feldstein, Lucian0 Galeone. Roberto Garrappa, Nicola Gug- lielmi, Laura Hewitt, Adrian Hill, Giuseppe Izzo, Marian Kwapisz, Edisanter
PREFACE XV
Lo, Stefan0 Maset, Hans Mittelmann, Brynjulf Owren. Beatrice Paternoster, Helmut Podhaisky. Rosemary Renaut. Elvira RUSSO, Larry Shampine, Stefa- nia Tracogna. Jack VanWieren, Antonella Vecchio, Rossana Vermiglio. Jim Verner, Jan Verwer, Rudiger Weiner. Bruno Welfert. William Wright. Marino Zennaro. and Barbara Zubik-Kowal. I would like to thank Irina Long for her help with editing of some figures. and the editorial and production group at John Wiley & Sons. Finally, I would like to acknowledge the support of Na- tional Science Foundation for many of my projects on numerical solution of ordinary differential equations.
Z. JACKIEWICZ
Arizona State Chiversity April, 2009
This Page Intentionally Left Blank
CHAPTER 1
DIFFERENTIAL EQUATIONS AND SYSTEMS
1.1 T H E IN IT IAL VALUE PROBLEM
Many problems in science and engineering can be modeled by the initial value problem for systems of ordinary differential equations (ODES), which we write in autonomous form as follows:
(1.1.1)
Here f : R" .+ R" is a given function that usually satisfies some regularity conditions, and uo E R" is a given initial vector. Introducing the notation
General Linear Methods for Ordinary Differential Equations. By Zdzislaw Jackiewicz Copyright @ 2009 John Wiley & Sons, Inc.
1
2 DIFFERENTIAL EQUATIONS AND SYSTEMS
(1.1.1) can be written in the following scalar form:
t E [ t o . TI, where we have suppressed dependence on the independent variable t in ut and u:. i = 1 , 2 . . . . . m. Observe that the nonautonomous equation
(1.1.2)
g : IR x R" -+ R", yo E IR", can always be reduced to a system of the form (1.1.1) of dimension m + 1 if we define
Systems of the form (1.1.1) or (1.1.2) can also arise in practice from the conversion of initial value problem for differential equations of higher order. Consider, for example, an autonomous form of such a problem:
where dence on t . Setting
stands for the derivative of order i and we again suppressed depen-
we obtain
EXAMPLES OF DIFFERENTIAL EQUATIONS AND SYSTEMS 3
which is equivalent to (1.1.1) with
r
In Section 1.3 we discuss the existence and uniqueness of solutions to (1.1.1) and (1.1.2) under various conditions on the functions f and g. In this discus- sion we often assume that these problems are defined not only for t E [to.T] but on the larger interval t E I , where I = { t : It - to1 5 T}.
1.2 EXAMPLES OF DIFFERENTIAL EQUATIONS AND SYSTEMS
We list in this section several examples of differential equations and differ- ential systems. These problems are used later in our numerical experiments with various algorithms for numerical solution of ODES. These algorithms are based on some classes of general linear methods discussed herein. All equa- tions in this section are examples of nonstiff equations and systems. Problem of stiffness and stiff differential equations and systems are discussed in Sec- tions 1.7 and 1.8.
SCALAR - the scalar problem [143. p. 2371:
y’(t) = -sign(t)/l - It11 y2. t E [-2,2], (1.2.1)
y(-2) = 2/3.
The solution to this initial value problem has a discontinuity in the first deriva- tive y‘ at the point t = 0 and discontinuities in the second derivative y” at t = -1 and t = 1.
BUBBLE - a model of cavitating bubble [200, 2571:
= Y2, dyl ds
- - +- (1.2.2) ds Y l Yvp Y ? + l >
Yl(0) = 1; Y2(0) = 0,
dy2 5exp(-s/s*) - 1 - 1.5~: ay2 + D 1 + D - -
t E [0, TI. Here sx, a: D , and y are real parameters. As observed by Shampine [257], this problem places great demands on the precision and step size control strategies of numerical algorithms.
4 DIFFERENTIAL EQUATIONS AND SYSTEMS
AREN - Arenstorf orbit for the restricted three body problem [12, 13, 1431. This is an example from astronomy which describes the movement of two bodies of scaled masses 1 - p and p in a circular rotation in a plane and the movement of a third body of negligible mass (e.g., satellite or spacecraft) in the same plane. The equations of motion are
t E [0, TI, where
Di = ((yi + p)’ + Y;)~”, Dz = ((yi - 1 + p)’ + Y;)~”. This problem with p = 0.012277471 corresponds to the earth-moon system. The periodic orbits of a satellite or spacecraft moving in such a system were discovered by Arenstorf [12, 131 by theoretical analysis of periodicity condi- tions and numerical calculations. Such periodic orbits may facilitate low-cost space exploration and are of interest to NASA. The initial conditions for which the solution to (1.2.3) is periodic are, for example,
yi(0) = 0.994, yi(0) = 0, yz(0) = 0, yh(0) = -2.001585106379,
with the period of motion Ti given by Ti = 17.06522, or
yi(0) = 0.994, yi(0) = 0, yz(0) = 0, yh(0) = -2.031732629557,
with the period of motion TZ given by TZ = 11.1234. Such orbits are plotted in, for example, [52, Fig. 102(i) and (ii)], and [143, Fig. 0.11.
LRNZ - the Lorenz model [209]. This is a system of three differential equa- tions of the form
Y: = -0Y1 + 0Y2,
Yi = Y1 Yz - by37
Y: = -Y1 Y3 + ry1 - Yz, (1.2.4)
t E [O,T]. Here b, 0, and T are positive constants. For example, for b = 8/3, 0 = 10, and T = 28, this system has aperiodic solutions.
EULR - Euler’s equations of rotation of a rigid body [143]. This is a system of three differential equations given by
EXAMPLES OF DIFFERENTIAL EQUATIONS AND SYSTEMS 5
t E [O,T]. Here y1, y2, and y3 are the coordinates of the rotation vector; 11, 12, and 13 are the principal moments of inertia; and the third coordinate has an additional exterior force f ( t ) .
PLEI - a celestial mechanics problem “the Pleiades” from [143, p. 2451. The equations of motion are
(1.2.6)
t E [0,3]; where
The initial conditions are
= y/l(O) = 0, for all i with the exception of
I C ~ ( O ) = 1.75. z/?(O) = -1.5. yk(0) = -1.25, &(O) = 1.
This problem describes the movement of seven stars in the plane with coor- dinates IC,, y, and masses m, = z, i = 1 . 2 , . . . -7 . The trajectories of these stars are plotted by Hairer at al. [143. Fig. 10.2a], and speeds x: and yi, z = 1 , 2 , . . . , 7 , [143, Fig. 10.2bI. ROPE - the movement of a hanging rope of length 1 under gravitation and the influence of horizontal F1/(t) and vertical F,(t) forces [143]. As explained by Hairer at al. [143], the discretization of this problem leads to a system of differential equations of second order for the angles 6‘1 = &(t) between the tangents to the rope and the vertical axis at a discrete arc length sl. This system takes the form
6 DIFFERENTIAL EQUATIONS AND SYSTEMS
t E [O. TI. 1 = 1 . 2 . . . . . n, where
1 2 u l k = glrc cos(01 - 0 k ) . blk = glk sin(0l - &),
glk = n + - - max{l, I c } . The horizontal force Fy( t ) acting at the point s = 0.75 is
FY(t) =
and the vertical force F,(t) acting at the point s = 1 is
F,(t) = 0.4.
This system will be solved for n = 40 with initial conditions
& ( O ) = & ( O ) = 0. 1 = 1.2. . . . .n,
on the interval [O. 3.7231. Setting
system (1.2.7) can be written in vector form as
(1.2.8)
where d2 denotes componentwise exponentiation and g ( t , 0) is an appropri- ately defined vector function. The solution of (1.2.8) requires computation of the inverse matrix A-l. As explained by Hairer a t al. [143. 1461 this can be done very efficiently in O ( n ) operations, due to the special structure of the matrix A. It can be verified that
A + i B = diag(etel ~ eZez. . . . , ezen) G diag(e-"I. e-'O2 , . . . , ePen) . where G = [ g k l ] . This matrix has the inverse
EXAMPLES OF DIFFERENTIAL EQUATIONS AND SYSTEMS 7
and it follows that
( A + i B ) - ' = C + i D
- - diag(eio1, eio2, . . . , ,ion) G-1 diag(e-201, e-ioz . .
where C and D are tridiagonal matrices of the form
and
C =
D =
1 c12
c2 1 2 c23
en-1.n-2 2 Cn-l,n
cn.n-1 3
0 s12
$21 0 s23
S n - 1 . n - 2 0 S n - 1 , n
S n , n - l 0
. . e-i*n) ,
with ckl = - cos(6'k - el) . skl = - sin(6'k - e l ) .
Since (A + i B ) ( C + i D ) = I , we have A C - B D = I . A D + B C = O
and it follows that
or A-l = C + DC-lD. We also have A-lB = -DC-' and system (1.2.8) can be written as
6 = DC-l(b2 + Dg(t , 6')) + Cg( t , Q), As observed by Hairer at al. [143] this suggests the following efficient algo- rithm for computation of the acceleration vector e .
t E [O. TI.
1. Compute w = b2 + Dg(t. 6'). 2. Solve the tridiagonal system Cu = w.
3. Compute 6 = Du + Cg(t , 13).
8 DIFFERENTIAL EQUATIONS AND SYSTEMS
BRUS - a reaction-diffusion equation (the Brusselator with diffusion) [143]. This is the system of partial differential equations of the form
0 5 x 5 1, 0 5 y 5 1, t 2 0, a = 2 x boundary conditions
together with the Neumann
d U d V
dn dn = 0, - = 0, -
where n is the normal vector to the boundary of the region [0,1] x [ O , 11 and the initial conditions
u(x. y. 0) = 0.5 + y. V(Z, y, 0) = 1 + 52. Let N > 1 be an integer and define the grid in space variables x and y by
Z, = (Z - l ) A x , y, = (j - l )Ay, Z , J = 1 , 2 , . . . , N.
where Ax = Ay = 1/(N - 1). Define also the functions
U,, (t) = ~ ( 2 % . y, . t). V,, (t) = w (x'. y3, t ) . i, J = 1,2. . . . , N.
Discretizing (1.2.9) by the method of lines. where the space derivatives are ap- proximated by finite differences of second order leads to the system of ordinary differential equations
t E [O.T], Z , J = 1 . 2 . . . . , N, of dimension 2N2. The boundary conditions imply that
UO 3 = U2,3, U N + l j = U N - 1 , ~ - u,,O = uz,Z, Uz,N+1 = Uz,N-1.
%,, = v2,j. vN+1 2 = V N - l , , , &,O = 2 , &,N+1 = v, N-1, and the initial conditions are
Uij(O)=O.5+yj, V, j (O)=1+5xi , i , j = l , 2 , . . . ,N .
EXISTENCE AND UNIQUENESS OF SOLUTIONS 9
1.3 EXISTENCE AND UNIQUENESS OF SOLUTIONS
In this section we formulate results regarding the existence and uniqueness of solutions to (1.1.2). We begin with the classical Peano existence theorem for the system (1.1.2), which assumes only that the function g is continuous in some domain. The proof of this result is based on the Arzela-Ascoli theo- rem about a family of vector-valued functions that is uniformly bounded and equicontinuous. Consider a family 3 of vector-valued functions y = y ( t ) de- fined on an interval I = { t : It - to1 5 T } . Define llyll := sup{lly(t)ll : t E I } ; where / I . 1 1 is any norm on R". We introduce the following definitions. Definition 1.3.1 A family 3 of vector-valued functions y = y ( t ) is said to be uniformly bounded if there exists a constant M such that ( ( y ( ( 5 M for every y E 3.
Definition 1.3.2 A family 3 of vector-valued functions y = y ( t ) is said t o be equicontinuous if for every E > 0 there exists 6 > 0 such that the condition It - s / < 6, t , s E I , implies that I l y ( t ) - y(s)( / < E for all functions y E 3.
Theorem 1.3.3 (Arzela-Ascoli; see [212]) Let ~ ~ ( t ) ~ n = 1.2 ; . . ., be a uniformly bounded and equicontinuous sequence of vector functions defined on the interval I . Then there exists a subsequence yn, ( t ) , j = 1; 2 , . . . 1 which is uniformly convergent on I .
We are now ready to formulate and prove the classical existence result for system (1.1.2).
Theorem 1.3.4 (Peano [235]) Assume that the function g ( t , y) is contin- uous in the domain
(1.3.1)
and that there exists a constant M such that 11g(t, y)II 5 M for ( t , y) E D . Then system (1.1.2) has at least one solutzon y = y ( t ) defined for
It - to( 5 TI := min{T. K / M }
and passing through the point ( t o , yo)
Proof: It is generally agreed that the original proof of Peano [235] was inadequate. and a satisfactory proof was found many years later (see e.g., Perron [236]). Here, we follow the presentation given by Birkhoff and Rota [all. We will prove the theorem for the interval [ t o . t o +TI]; the proof for the interval [to - TI, to] is analogous. Consider the integral equation
r t
(1.3.2)
10 DIFFERENTIAL EQUATIONS AND SYSTEMS
t E [ to , t o +TI] , which is equivalent to (1.1.2). Define the sequence of functions ;Yn = Y n ( t ) , n = 1 . 2 , . . .. by the formulas
t E [ t o . t o + T1/n] 3 t -T l /n 1 ::+lo g ( s . ~ n ( s ) ) d s , t E ( t o + Tl/n,to +TI ] . y n ( t ) =
Observe that the right-hand side of the second formula above defines y,(t) for t E ( t o + Tl /n . t o + TI] in terms of y,(t) already defined for t E [ to . t o + Tl/n]. This sequence is well defined since
rt-Tq ln
5 M t - t o - - 5 M ( t - t o ) 5 MT1 5 K ( 3 and y,(t) are clearly continuous on [ t o . to+T1]. We have II1Jn(t)il 5 //yoI/+MTl, which shows that the sequence y,(t) is uniformly bounded. We also have
t2 -TI / n
livn(t2) -gn(tl)il 5 J’ //g(s.yn(S))/lds 5 ~ 1 t 2 - t11 tl--Tl/n
which shows that yn(t) is also equicontinuous. Hence, it follows from the Arzela-Ascoli theorem. Theorem 1.3.3. that there exists a subsequence yn, ( t ) , j = 1 , 2 , . . ., which is uniformly convergent to a continuous function g ( t ) : that is,
lim yn, = g. j+m
This subsequence satisfies the integral equation. which we write in the form
t t
Yn, ( t ) = J’ (s, Yn, (s)) ds - J’ (s, Yn, (s)) t o t--Tl/n,
t t We have
lim lo g ( s . Yn, ( s ) ) d s = g ( s , g ( s ) ) d s 3-m lo
since the function g ( t . y) is uniformly continuous. We also have
Hence, passing to the limit as j + cc in the integral equation for yn,, we obtain
EXISTENCE AND UNIQUENESS OF SOLUTIONS 11
which proves that y ( t ) satisfies the integral equation (1.3.2); hence it also
A solution whose existence is guaranteed by the Peano theorem, Theo- rem 1.3.4, is not necessarily unique. A simple example that illustrates this is given by a scalar initial value problem
satisfies (1.1.2) for t E [to, t o +TI].
y' = 3y2'3, y(0) = 0,
where D = { ( t , y ) : It1 5 1, IyI 5 1). Here the function g ( t , y ) = y2I3 is continuous on D , but the problem has solutions y l ( t ) = 0 and y2(t) = t3.
Assume that a function g ( t , y ) is defined in some region R c R x R". To formulate uniqueness results for (1.1.2), we usually assume that the function g( t l y) is not only continuous but satisfies some additional regularity proper- ties. We introduce the following definitions.
Definition 1.3.5 A function g ( t , y) satisfies a Lipschitz condition in R with a Lipschitz constant L if
I l S ( t , Y l ) - S ( 4 Y 2 ) I I I L l l Y l - Y2lI (1.3.3)
for all ( t , yl), ( t , y2) E R, where / / . 1 1 is any norm in R". Definition 1.3.6 A function g ( t , y) satisfies a one-sided Lipschitz condition in R with a one-sided Lipschitz constant z/ if
(1.3.4)
for all ( t , y 1 ) , ( t , y2) E R. Here / I . / I is the Euclidean norm in R"; that is,
T ( d t , Y1) - S ( t , Y2)) (Yl - Y2) I V l I Y l - Y21I2
l/uI/ := a for 21 E R". One-sided Lipschitz condition (1.3.4) plays an important role in the analysis
of numerical methods for stiff systems of ODES (compare [log. 1461). Assume that the function g ( t , y) satisfies Lipschitz condition (1.3.3) in the Euclidean norm 1 1 . 1 1 with a constant L. Then using the Schwartz inequality and (1.3.3), we obtain
T ( S ( 4 Y1) - g ( t , Y2)) ( Y l - Y2) I jIg(4 Y l ) - S ( t > 1J2) l l I lY l - Yzl l I L I l Y 1 - ?/2Il2
and it follows that g ( t , y) also satisfies one-sided Lipschitz condition (1.3.4) with the same constant L. However, as observed, for example, by Dekker and Verwer [log], the reverse is not true. A counterexample is provided by any monotonically nonincreasing function g : R -+ R which has, for some value of g E R, an infinite slope. We then have
( d Y 1 ) - S(Y2)) (Y1 - Y2) 5 0
for all y1, y2 E R and it follows that g(y) satisfies (1.3.4) with v = 0. However, this function does not satisfy (1.3.3) in any neighborhood of 3, where the slope is infinite.
12 DIFFERENTIAL EQUATIONS AND SYSTEMS
Next we formulate a local existence and uniqueness theorem. We also show that the solution to (1.1.2) can be obtained as a limit of a uniformly convergent sequence of continuous functions starting with an arbitrary initial function that satisfies the appropriate initial condition.
Theorem 1.3.7 Assume that the function g( t , y) is continuous and satisfies a Lipschitz condition (1.3.3) in the domain D defined by (1.3.1). Set
M = max { / / g ( t , Y ) ( l : ( t , Y) E D } . Then (1.1.2) has a unique solution defined on the interval
( t - to1 5 TI := min{T, K I M }
passing through (to, yo).
Proof: [ t o - 7'1, to] is analogous. Define the integral operator
First consider the interval [to,to + T I ] ; the proof for the interval
(1.3.5)
t E [to,to + T I ] . Put Y = {y E Rm : lly - yo(( 5 K } and denote by C([to, t o + 7'11, Y ) the space of continuous functions from [ to , t o + T I ] into Y with a uniform norm. Observe that if y E C([to, to + T I ] , Y ) , then
t
I ( Z ( t ) - Yo11 I 1 I19(s,y(s))/Ids I MTl I M K / M = K , t o
and it follows that the operator q5 takes the functions from C( [ to , t o + T I ] , Y ) into C([to, to + T l ] , Y ) :
4 : C([tO,t" + T l ] , Y ) -+ C([to, to + T l ] : Y ) . Define the sequence of functions y,(t) E C([to,to + T11,Y) by the formula
t
Y " + l ( i ) = O(Y"(t)) = Yo + 1 g(s ,yn(s ) )ds , (1.3.6) t o
n = 0 , 1 , . . ., where yo@) = yo, t E [to, t o + 7'11. Then we have the bound
We prove (1.3.7) by induction with respect to n. Since
( 1.3.7)