Discretization Methods for Stable Initial Value Problems

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1044 Eckart Gekeler

Discretization Methods for Stable Initial Value Problems

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author Eckart Gekeler Mathematisches Institut A der Universit~t Stuttgart Pfaffenwaldring 57, 7000 Stuttgart 80, Federal Republic of Germany

AMS Subject Classifications (1980): 65 L 07, 65 L 20, 65 M 05, 65 M 10, 65M15, 6 5 M 2 0 ISBN 3-540-12880-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12880-8 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

Introduction

In the past twenty years f i n i t e element analysis has reached a high standard and also great progress has been achieved in the development of numerical procedures f o r stiff,

i ° e . , stable and i l l - c o n d i t i o n e d d i f f e r e n t i a l

systems since the communication

of Dahlquist 1963. Both f i e l d s together provide the ingredients f o r a method of l i n e s solution f o r p a r t i a l d i f f e r e n t i a l equations. In t h i s method time and space d i s c r e t i zation are carried out independently of each other, which has the advantage that often available subroutine packages can be applied in one or both d i r e c t i o n s . F i n i t e element or f i n i t e difference methods are used f o r the d i s c r e t i z a t i o n in space d i r e c t i o n and f i n i t e difference methods as multistep or Runge-Kutta methods are used f o r the numerical solution of the r e s u l t i n g semi-discrete system in time, as a r u l e . For example, i f a hyperbolic i n i t i a l

boundary value problem with the d i f f e r e n t i a l

equation

u t t + ut - Uxx = g ( x , t ) and s u i t a b l e i n i t i a l

and boundary conditions is d i s c r e t i z e d by a f i n i t e element method

or more generally by a Galerkin procedure then the semi-discrete system of ordinary d i f f e r e n t i a l equations has the form (*)

My" + Ny' + Ky = c ( t )

where M, N, and K are real symmetric and p o s i t i v e d e f i n i t e matrices. M and N are w e l l conditioned but K is i l l - c o n d i t i o n e d in general, i . e ,

IIKIIIIK-II] >> O. The f i n i t e element

approximation of more general l i n e a r hyperbolic problems leads to s i m i l a r systems. In engineering mechanics the basic p a r t i a l d i f f e r e n t i a l equation is mostly not a v a i l a b l e because the body to be considered is too complex,instead the equations of motion are approximated by matrix s t r u c t u r a l analysis. The r e s u l t i n g ' e q u i l i b r i u m equations of dynamic f i n i t e element analysis' are then alarge d i f f e r e n t i a l

system f o r the displace-

ments y being of the form (*) too. I f also a number of eigenvalues of the associated generalized eigenvalue problem is wanted then methods employing eigenvector expansions may be preferred in the solut i o n of (*) (modal a n a l y s i s ) . In the other case the numerical approximation leads immediately to the study of d i s c r e t i z a t i o n methods for d i f f e r e n t i a l systems y' = f ( t , y ) being stable in the sense that (v - w ) T ( f ( t , v ) - f ( t , w ) ) ~ O. As the system (*) changes dimension and condition heavily with a refinement of the space discretization, methods are of particular interest here whose error propagation

~V

depends as l i t t l e

as possible on these data. Mathematically, the v e r i f i c a t i o n of this

property or in other words of the uniformity of the e r r o r propagation with respect to a class of related problems can be established only by a - p r i o r i e r r o r estimations therefore

p a r t i c u l a r emphasis is placed on them in this volume.

Three d i f f e r e n t classes of methods are at our disposal in the solution of i n i t i a l value problems: multistep methods m u l t i d e r i v a t i v e methods ~ multistage methods Runge-Kutta methods ~ Multistep methods need a m u l t i d e r i v a t i v e or a Runge-Kutta method as start-procedure. S k i l f u l l y mixed procedures can have advantages over t h e i r components without i n h e r i t i n g the bad properties to the same degree. Multistep m u l t i d e r i v a t i v e methods are treated here from a rather general point of view. Runge-Kutta methods are intermediate-step methods actually,and they coincide with m u l t i d e r i v a t i v e methods f o r the l i n e a r d i f f e r e n t i a l system y' = Ay with constant matrix A. Therefore these methods are both denoted

as multistage methods and they have the same properties with respect to the test

equation y' = ~y. Multistep Runge-Kutta methods are not f u l l y investigated to date and besides there are many f u r t h e r combinations which are not treated here. In the d e r i v a t i o n of 'uniform' error bounds we are faced with two p r i n c i p a l problems: the v e r i f i c a t i o n of 'uniform' s t a b i l i t y in multistep methods and a suitable estimation of the d i s c r e t i z a t i o n e r r o r in Runge-Kutta methods. The f i r s t

difficulty

is

overcome by a uniform boundedness theorem being applied here in a version due to Crouzeix and Raviart. The second d i f f i c u l t y

is overcome by the pioneering work in

Crouzeix's thesis 1975. Furthermore, we should name Jeltsch and Nevanlinna whose contributions threw important l i g h t on the shape of the s t a b i l i t y region. In chapter I and I I multistep m u l t i d e r i v a t i v e methods are considered f o r d i f f e r e n t i a l systems of f i r s t

and second order. A - p r i o r i error bounds are derived f o r sys-

tems with constant c o e f f i c i e n t s and a survey is given on modern s t a b i l i t y analysis. Over a long period the t e s t equations y' = ~y and y "

= ~2y have been studied here

only. Nevertheless, many important results have been produced in t h i s way and a large v a r i e t y of numerical schemes been widely used in the meantime. In chapter I I I we leave the constant case and turn to l i n e a r systems with scalar time-dependence. Following a work of LeRoux [79a] and the d i s s e r t a t i o n of Hackmack [81] e r r o r bounds are established f o r l i n e a r multistep methods which show that a bad condition of the d i f f e r e n t i a l system does not a f f e c t seriously the e r r o r propagation here, too. Chapter IV then deals with recent results on the e r r o r propagation in l i n e a r multistep methods and nonlinear d i f f e r e n t i a l systems of f i r s t

order.

For a comparison with multistep m u l t i d e r i v a t i v e methods, Runge-Kutta methods are treated in chapter V but not to the same extent because we must r e f e r here to a f o r t h coming book of Crouzeix and Raviart. These methods haven't l o s t anything of t h e i r fascination and today new variants are known in which the computational e f f o r t is re-

duced considerably. In f i n i t e element analysis of e l l i p t i c boundary value problems a-priori error estimations play a large part and there are celebrated results among them. In Chapter VI some of these error bounds are combined with error bounds established in the f i r s t two chapters. Because of the special form of the l a t t e r results, error estimations are obtained for ' f i n i t e element multistep multiderivative' discretizations of parabolic and hyperbolic i n i t i a l boundary value problems without further computations. The convergence order of the f u l l y discrete schemes with respect to time and space discretization turns out to be the order of their components. My thanks are due to Mrs. E. von Powitz for typing an early draft. I am also grateful to U. Hackmack for reading the manuscript and for some useful comments. Finally, I am indebted to S. Huber, K.-H. Hummel, and U. Ringler for computational examples and the plotting of the figures.

Table o f Contents

I.

Multistep Multiderivative

1,1. Consistence

Methods f o r D i f f e r e n t i a l

Systems o f F i r s t O r d e r .

. . . . . . . . . . . . . . . . . . . . . . . . . .

1.2, Uniform S t a b i l i t y

. . . . . . . . . . . . . . . . . . . . . . . .

1.3, General P r o p e r t i e s o f the Region S o f Absolute S t a b i l i t y 1,4, I n d i r e c t Methods f o r D i f f e r e n t i a l

. . . . . . . .

Systems o f Second Order

1.5. Diagonal Pad~ Approximants o f the Exponential Function

I 1 6 11

. . . . . . .

15

. . . . . . . .

20

1,6, S t a b i l i t y

in the L e f t Half-Plane . . . . . . . . . . . . . . . . . .

23

1,7. S t a b i l i t y

on the Imaginary Axis

30

II.

Direct Multistep Multiderivative

. . . . . . . . . . . . . . . . . . Methods f o r D i f f e r e n t i a l

Systems o f Second

Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2,1. M u l t i s t e p Methods f o r Conservative D i f f e r e n t i a l

Systems

. . . . . . . .

2.2, L i n e a r M u l t i s t e p Methods f o r D i f f e r e n t i a l

Systems w i t h Damping

2.3. L i n e a r M u l t i s t e p Methods f o r D i f f e r e n t i a l

Systems w i t h Orthogonal Damping.

2.4. Nystr~m Type Methods f o r Conservative D i f f e r e n t i a l 2.5. S t a b i l i t y

on the Negative Real Line

55 59 66

. . . . . . . . . . . . . . . .

69

L i n e a r M u l t i s t e p Methods and Problems w i t h Leading M a t r i x A ( t ) = a ( t ) A

72

3,1. D i f f e r e n t i a l Matrix

Systems o f F i r s t Order and Methods w i t h Diagonable Frobenius

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3,2, D i f f e r e n t i a l

Systems of F i r s t Order and Methods w i t h Non-Empty S

3.3. An E r r o r Bound f o r D i f f e r e n t i a l IV.

Systems . . . . . . .

34 45

. . . . . . . . . . . . . . . .

2,6, Examples o f L i n e a r M u l t i s t e p Methods III.

. . . . .

34

Systems o f Second Order

L i n e a r M u l t i s t e p Methods and N o n l i n e a r D i f f e r e n t i a l

4.1. An E r r o r Bound f o r Stable D i f f e r e n t i a l

. . . .

. . . . . . . .

Systems o f F i r s t Order

Systems . . . . . . . . . . . .

72 75 82 88 88

4.2, The M o d i f i e d M i d p o i n t Rule

. . . . . . . . . . . . . . . . . . . .

92

4,3. G - S t a b i l i t y

. . . . . . . . . . . . . . . . . . . .

96

and A - S t a b i l i t y

4.4. Uniform S t a b i l i t y V.

under Stronger Assumptions on the D i f f e r e n t i a l

Runge-Kutta Methods f o r D i f f e r e n t i a l

Systems o f F i r s t Order

5,1, General M u l t i s t a g e Methods and Runge-Kutta Methods 5,2, Consistence

Systems

. . . . . .

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

5.3, E r r o r Bounds f o r Stable D i f f e r e n t i a l

Systems

. . . . . . . . . . . .

5.4, Examples and Remarks . . . . . . . . . . . . . . . . . . . . . . .

106 114 114 117 126 136

VHI

Vl°

Approximation of Initial

6.1.

Initial

Boundary Value Problems . . . . . . . . . . .

Boundary Value Problems and G a l e r k i n Procedures . . . . . . . .

142 142

6 . 2 . E r r o r Estimates f o r G a l e r k i n - M u l t i s t e p

Procedures and P a r a b o l i c Problems .

146

6 . 3 . E r r o r Estimates f o r G a l e r k i n - M u l t i s t e p

Procedures and H y p e r b o l i c P r o b l e m s

150

Appendix A.I.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Auxiliary

A.2. Auxiliary

Results on A l g e b r a i c Functions

Results on Frobenius and Vandermonde M a t r i c e s . . . . . . . .

A.3. A Uniform Boundedness Theorem A . 4 . Examples t o Chapters I and IV A . 5 . Examples o f Nystr~m Methods A°6. The ( 2 , 2 ) - M e t h o d

References

for

157 157 173

. . . . . . . . . . . . . . . . . .

177

. . . . . . . . . . . . . . . . . .

181

. . . . . . . . . . . . . . . . . . .

Systems o f Second Order

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Glossary o f Symbols S u b j e c t Index

. . . . . . . . . . . . . .

188 194

196

. . . . . . . . . . . . . . . . . . . . . . . .

200

. . . . . . . . . . . . . . . . . . . . . . . . . . .

201

I. Multistep Multiderivative Methods for Differential S~stems of First Order

1.1. Consistence

Let us begin with an introduction of numerical approximation schemes for the general i n i t i a l value problem (1.1.1)

y' = f ( t , y ) , t > O, y(O) = YO"

We assume that f : IR+XRm ÷ ~m is s u f f i c i e n t l y smooth and denote by @f/@y the Jacobi matrix of f . Let At be a small fixed time increment and recall that (1.1.2)

f ( J ) ( t , y ) = (@f(J-1)/@t)(t,y) + [ ( B f ( J - 1 ) / B y ) ( t , y ) ] f ( t , y ) ,

j = 1,2,...

,

where f(O) = f . Then a general multistep multiderivative method - below b r i e f l y called multistep method - can be written as

(1.1.3)

k vn+i + ci=O~j=1 ~k ~ a t j mj i f(J-1)(Vn+ i ) = O, Zi=O:Oi n+i

n = 0,1, . . . .

By virtue of (1.1.2) the total derivatives f(J) = dJf/dt j are to be expressed here as far as possible by partial derivatives of f . vn shall be an approximation to the solution Yn = y(nat) of (1.1.1) at the time level t = nat and we always assume in a multistep method that the i n i t i a l values Vo,...,Vk. I are given in some way by an other method, e.g., by a Runge-Kutta method or a single-step multiderivative method. A scheme (1.1.3) can be described in a twofold way by the polynomials k i p~(~)j : _~i:O~ji{ ,

j : O,. . . . ~,

ai(n) : ~j:O~ji nJ ,

i : O,...,k.

or by the polynomials

We introduce the differential operator ® = B/@t and the s h i f t operator T defined by (Ty)(t) = y(t+at). Furthermore, we use the notation f ( - 1 ) ( t , v ( t ) ) = v(t) i f thereby no confusion arises. Then we can write instead of (1.1.3)

(1.1.4)

k ( A t e ) T i f ~ - 1 ) ( V n ) = 0, ~:0Pj(T)AtJoJf~-1)(Vn) ~ Zi=0oi

In p a r t i c u l a r ,

we obtain f o r the d i f f e r e n t i a l

n = 0 , I , . ..

equation y' = xy

~(T,At~)vn ~ Zj=0Pj(T)(At~)Jvn ~ Z~=0~i(AtX)TiVn = 0 and ~(~,q) is c a l l e d the characteristic polynomial of the method ( 1 . 1 . 3 ) . Obviously not a l l polynomials pj(~) as well as not a l l polynomials o i ( n ) must have the same degree but we suppose t h a t (1.1.5)

mOk ~ O, pc(~) ~ O, and ~O(n) ~ O.

This c o n d i t i o n guarantees t h a t the method (1.1.3) is e x a c t l y a k-step method w i t h d e r i v a t i v e s up to order ~. That number is sometimes c a l l e d the stage number and the method s h o r t l y a (k,~)-method. The d i s c r e t i z a t i o n

error

or defect

d ( A t , y ) of a method is obtained i f the exact

s o l u t i o n y is s u b s t i t u t e d i n t o the approximation scheme:

d(At,y)(t) = Z~=0AtJpj(T)y(J)(t).

(1.1.6)

(I.1.7)

Definition.

the method (1.1.3)

is consistent if there exists a positive in-

teger p such that for all u £ cP*IIR;IRm)~ p, = max{p+1,~},

l l d ( A t , u ) ( t ) l l ~ rAt p+I where £ does not depend on At. The maximum p is the order of the method.

The f o l l o w i n g lemma generalizes a r e s u l t due to Dahlquist [59, ch. 4 ] ; see also Lambert [73 , § 3.3] and

Jeltsch [76a]. I t proves the important f a c t t h a t c o n s i s t e n t

m u l t i s t e p methods a l l o w an estimation of the d i s c r e t i z a t i o n pend on the d a t a of the d i f f e r e n t i a l

e r r o r which does not de-

equation.

(1.1.8) Lemma. If the method (1.1.3) is consistent of order p then p + I A t J o j ( T ) , u ( j ) (t)ll l l d ( A t , u ) ( t ) I I ~ r a t p t+kAt f llu [~+~ TM ")(~)lld~ + ~ = t where r does not depend on t ,

v u E cP*(IR;IRm)

At, u, and the dimension m.

Proof. I t s u f f i c e s to prove the assertion f o r p ~ ~. We w r i t e 00 = I and s u b s t i t u t e the Taylor expansions of u,

u(J)(t+iAt) = zp-j ~=0 ~(iAt) . v u(j+u)(t ) + - ~I.

iAt (iAt-T)P-Ju(P+I)(t+T)dT' ~

j = 0 . . . . . p, into ( 1 . 1 . 6 ) . By t h i s way we obtain d(At,u)(t)

~ ~p-j,~k i~)AtJ+~u(j+u)(t ) = Lj=0L~:0~Li=0~ji v-'F

(I.}.9)

k

At j

+ ~J=OZi=O~ji ~ "

iAt

I

(iAt-T)P-Ju(P+I)(L+T)dT"

The assumption that the method is consistent of order p implies p-j

k

i~^+j+u,(j+~)ft~

Zj=oZv=o(Zi=o~ji ~-'F.'. . . . . (1.1.10)

rp vmin{u,~}~k L~=0Lj=0

i ~-j

~Li=0~ji ~ .

)At~u(p)(t) = 0.

Let i i f z > 0, ifz<0

z+=

then we have by (1.1.9) and (1.1.10)

111111 dCtul
ip-j

IId(At,u)(t), ~ (~j=O~i=Olajil - ~ ) A t

~ t+kAt

~~

{~+I

FLu~ ")(~),dT

which proves the assertion. By ( I . 1 . 1 0 ) we immediately obtain the f o l l o w i n g r e s u l t : (1.1.12) Lemma. The method ( I . 1 . 3 ) is consistent of order p iff ~min{~,~}~k i ~-j j=O Li=Omji ~

= O,

u = 0,1,...,p.

In p a r t i c u l a r , the method (1.1.3) is consistent i f f (1.1.13)

P0(1) = p~(1) + o i ( I )

= 0

which are the well-known conditions f o r the consistence of l i n e a r m u l t i s t e p methods,

i.e.,

methods w i t h L = I. A study of the t r i v i a l

always suppose t h a t P i ( I )

equation y' = < shows t h a t we must

m 0 hence we may s t i p u l a t e as customary t h a t

(1.1.14)

P1(1) = - 1.

Sometimes i t is advantageous to use the consistence c r i t e r i a in the subsequent form. (1.1.15) Lemma. the method (1.1.3) is consistent of order p iff the characteristic polynomial ~(~,q) satisfies

×(eAt,At) = ×pAtp+I + ~(AtP+2), At ÷ 0, ×p ~ 0.

Proof. I f the method (1.1.3) has order p then we obtain by a s u b s t i t u t i o n of u: t ÷ e t i n t o the d i s c r e t i z a t i o n

e r r o r d(At,u)

d(At,e t ) : Z~:0AtJpj(T)e t : Z~:0AtJpj(eAt)e t = ~(eAt,At)e t = ×pAt p+I + ~(AtP+2), At ÷ 0, where ×p = 0. For t = 0 t h i s proves the necessity of the c o n d i t i o n . On the other sidej observe t h a t (iAt) v ~(eAt'At) = ~ o = 0 A t J ~ = 0 ~ ' i ~ j =0 ~ v. I f the c o n d i t i o n of the lemma is f u l f i l l e d

~ ~min{~,~},~k - (i-~-j ~.)At = L~=0Zj=0 tLi=0~ji

n.

then we obtain from t h i s the c o n d i t i o n a l

equations of Lemma ( I . 1 . 1 2 ) which proves the s u f f i c i e n c y . In other words, the lemma says that in a method of order p the value n = 0 is exactly a (p+1)-fold root of ~(en,q), or, ~ = I is exactly a (p+1)-fold root of ~(~,log~). The number - ×p/P1(1) is called the error constant of the method. In general, the comparison of the discretization error of two methods with the same order via (1.1.11), i . e . , via their Peano kernels, is too involved. Hence under two otherwise comparable methods that with the smaller modulus of the error constant is usually preferred. Recall now once more that ~(eAt,At) : Z~:0AtJpj(e At) : Z~:0oi(at)e iAt and assume that the polynomials pj(~) have a common divisor,

5

oj(c) = ¢ ( c ) ~ j ( c ) ,

¢ ( I ) = O,

j = O, . . . . m.

I f the method with the c h a r a c t e r i s t i c polynomial ~(~,n) has order p and o i ( I )

m0

then we obtain by Lemma (1.1.15) •

×p

~(eAt,At) = ~=0AtJ~j(e ~t) : ~--(-F)-AtP+I + ~(AtP+2),

At + 0.

Consequently, the method with the c h a r a c t e r i s t i c polynomial ~(~,n) has the same order and the same e r r o r constant as the method with the c h a r a c t e r i s t i c polynomial ~(~,n). I f the polynomials ~i(n) have a common d i v i s o r , ~i(n) = ~ ( n ) ~ i ( n ) , ~(0) ~ 0,

i = 0 . . . . . k,

then • ^ . . i A t = ~--(-~TAtP+I Xp ~(eAt,At) = ~=0AtJ~j(e At) = ~k ~i=0~i~At)e + d(at p+2)

At + 0,

and the method with the c h a r a c t e r i s t i c polynomial ~(~,n) has again the same order as that with the polynomial ~(~,n). Moreover, l e t r

~(n) = ~+=0~n

1
, ~0 = ~(0) = 0,

then o0(+) = ~0£0(+) and Ol(C) = Yl~O(C) + YO~l(¢). Because o0(I) = YO~O(1) = 0 by (1.1.13) we obtain o1(1) = e(0)~1(1) and the e r r o r constants of both methods coincide, too. Thus we assume henceforth without loss of g e n e r a l i t y t h a t the polynomials ~ j ( ~ ) , j = 0 , . . . ~ , and that the polynomials o i ( n ) , i = 0 , . . . , k ,

have no common roots

have no common roots.

1.2. Uniform S t a b i l i t y

Instead of the general problem (1.1.1) we consider in the remaining part of t h i s chapter the l i n e a r i n i t i a l

(1.2.1)

value problem y' = Ay + c ( t ) ,

t > O, y(O) = YO'

w i t h constant leading matrix A, In Section 1.4 however t h i s system arises from a t r a n s formation of a l i n e a r d i f f e r e n t i a l

system of second order. For the problem (1,2.1) the

m u l t i s t e p method (1.1.3) has the form

Z~:0~i(AtA)TiVn = - Z~ IzJ-~AtJA~p.(T)C~j - l - v ) j= ~= J '

n = 0,I,...,

~ oij(AtA)AtJc~J-1) Z~:0qi(atA)TiVn : - zk ~i=0T i Lj=I

n = 0,I,.,.,

or (1.2.2) where (1.2,3)

oij(n)

~g m-j = Lm=jmmin .

In order to estimate the global e r r o r we have to w r i t e every method as a s i n g l e - s t e p procedure. For t h i s we introduce the f o l l o w i n g n o t a t i o n s , Vn = (Vn_k+ I . . . . . vn)T, (1.2.4)

k i ~ = 1 ~ i j ( ~ t A ) ~ t J c ( Jn-k -1))m ' Cn = (0 . . . . . O,~k(^tA)-IZi=oT

and the Frobenius matrix associated w i t h the c h a r a c t e r i s t i c

polynomial ~(~,n) of the

method ( 1 . 1 . 3 ) ,

(1.2.5)

F (n) =

L'°o(n)/~k(n)

. . . . . . . . . . . . . . . .

-Ok-1 (n)/~kCn)J

Let the Euclid norm and the associated matrix lub norm (spectral norm) be denoted by I.I,

l e t lllhIIIn = maxo~tsnAtlh(t)I , and l e t Sp(A) be the set of the eigenvalues, i . e , ,

the spectrum of the matrix A. < and r denote always p o s i t i v e constants which are not n e c e s s a r i l y the same in two d i f f e r e n t

contexts.

Instead of (1.2.2) we now can w r i t e

(1.2.6) For Cn

Vn = F (AtA)Vn. I - Cn, :

0, n = k,k+1,,

o,,

n = k,k+1 . . . . .

the increasing or decreasing of the sequence {II V nIl}n=k de-

pends on the eigenvalues of F (atA) which are the roots of the polynomial ~ ( ~ , n ) . For s i m p l i c i t y we define the region of absolute s t a b i l i t y

- below b r i e f l y

called stability

region - in the f o l l o w i n g way: (1.2.7) D e f i n i t i o n , L e t ~ = C v {~} and let ~(~,~) = pL(~), The stability region S of

(1,1.3) consists of the q £ C with the following properties: (i) ~k(n) ~ O, q £ S ~ C, (ii) all roots ~i(n) of ~(C,n) satisfy ICi(n) I ~ I, (iii) all roots ~i(q) of ~(c,n) with l~i(n) I = I - i.e. the unimodular roots - are simple roots of ~(~,n).

a multistep method

By ( i ) we exclude from S i s o l a t e d points where ~(C,n) degenerates. E.g., by Stoer and B u l i r s c h [80 , Theorem 6.3.4] the Frobenius matrix F (n) is undiagonable i f f teristic

polynomial ~(C,n) has a m u l t i p l e root. Hence the d e f i n i t i o n

in the s t a b i l i t y

region S i f f

i t s charac-

says t h a t n l i e s

the sequence {1IF (n)n,}~=1 remains bounded. The proof of

the uniform boundedness w i t h respect to n £ S is the c r u c i a l step of the otherwise simple e r r o r e s t i m a t i o n . We postpone a study of some properties of S to the next s e c t i o n . But i t should be remarked here t h a t the time step at is u s u a l l y a small number t h e r e f o r e only methods are of p r a c t i c a l

i n t e r e s t in t h i s context whose s t a b i l i t y

region contains the negative

real l i n e i n c l u d i n g zero in a neighborhood of zero. The property 0 £ S is the c l a s s i c a ; D-stability

introduced by Dahlquist [59 ] f o r l i n e a r m u l t i s t e p methods. Together w i t h

consistence i t implies the convergence of m u l t i s t e p methods in w e l l - c o n d i t i o n e d stable or unstable d i f f e r e n t i a l

systems i f the considered time i n t e r v a l

Stoer and B u l i r s c h [80 , Theorem 7.2.10.3]

is f i x e d . See e.g.

and Brown {74 ].

Let now y be the exact s o l u t i o n of the s l i g h t l y

modified i n i t i a l

value problem

(1.2.1), (1.2.8)

y' = Ay + c(t) + h ( t ) , t > O, y(O) = YO'

The function h plays here only the role of a perturbation but i t is needed in the last chapter for the error estimation of f u l l - d i s c r e t e approximation schemes to i n i t i a l

boundary value problems. I f y is substituted into the discretization error d(At,u) then we obtain a f t e r a repeated application of the d i f f e r e n t i a l equation (1.2.8)

(1.2.9)

k i k ~ (AtA)AtJTi(c~J-1) + h~J-1)) + d(At,Y)n" ~i=0oi (AtA)? Yn = - ~ i = 0 ~ j = 1 ° i j

On the other side, l e t Vn, n = k,k+1 . . . . .

be defined by (1.2.2) as above. Subtracting

(1.2.2) from (1.2.9) and writing the result in single-step form we obtain (1.2.10)

En = F (AtA)En_ I + D(at,y) n - Hn,

n = k,k+1 . . . . . .

where (1.2.11)

D(~tw) n : (0 . . . . . O,Ok(atA)-Id(at,Y)n_k )T.

The following theorem gives an error estimation which holds uniformly for all normal matrices A satisfying the spectral condition that Sp(atA) is contained in a fixed closed subset R of the s t a b i l i t y region S. (I.2.12) Theorem. ( i ) Let the (msm)-matrix A in (1.2.8) be diagonable, A = XAX-I" let the solution y of (1.2.8) be (p+l)-times continuously differentiable. (ii) Let the method (1.2.2) be consistent of order p ~ ~ ~ I with the stability region S. (iii) Let Sp(atA) C R C S where R is closed in ~. Then for n = k,k+1 .....

IX-1(y n - Vn) I s


-

Vk_11

+ At p

ly(p+1)(~)ld~ + nat

max O~i~z-1

lllh(i)lll n

.

Proof. A = (~I . . . . . Xm) is the diagonal matrix of the eigenvalues of A. (1.2.10) yields (1.2.13)

IX-IEn I ~ IF (AtA)n-k+111X-IEk_11 + ~=NIF (atA)n-vllX-1(D(at,y) v - H ) I ,

n : ksk+1,...,

and we have IF (AtA)nl = maxlsusmlF ( ~ t X ) n I.

The Uniform Boundedness Theorem which is proved in the Appendix yields immediately supn£RSUPnEiNIF (n)nl ~i ~R' Poles of ~k(n) -I cannot l i e in ~ hence {~k(~t^)-11 ~ supqESI~k(n)-II ~ ~. Therefore we obtain from (1.2.13) by Lemma (1.1.8) (1.2.14)

IX-IEn I ~
+ ~v:klX n -IHvl ] .

Hn has the same form as Cn in (1.2.4) w i t h c n replaced by hn so we have n -I n X-I k i z j (j-l) i ~:kIX H i : ~=k i Ok(AtA)-1~i=0T ~ j = 1 ~ i j ( A t A ) A t h~_ k k Z i ~ k ( A t A ) - 1 o i j ( A t A ) A t J - l l Ix-lllll <=
h(J-1) lil n •

I f R C S is bounded then AtA is bounded by assumption and we obtain (1.2.15)

i~k(AtA)-laij(AtA)i

~ KR,

i = 0 . . . . . k, j = I . . . . . ~.

I f R C S is unbounded then ok(n) is n e c e s s a r i l y a polynomial of exact degree L by Lemma ( A , I . 3 ) whereas the polynomials ~ i j ( n ) cordingly,

have degree less than ~ by ( 1 . 2 . 3 ) . Ac-

(1.2.15) is v a l i d in t h i s case, too. Therefore we have

Z~=klX-IH I ~ ~RnAtlX-11maxosi~c_iIllh(i)Illn and a s u b s t i t u t i o n of t h i s bound i n t o (1.2.14) proves the theorem, The s t a b i l i t y

region S has not n e c e s s a r i l y a non-empty i n t e r i o r

and i t is not

n e c e s s a r i l y a closed set in ~, For instance the Milne-Simpson method is the l i n e a r 2-step method of order 4 w i t h the c h a r a c t e r i s t i c ~(¢,n) = 2

(1.2.16)

This method has the s t a b i l i t y

polynomial

_ I - ~(¢2 + 4~ + I ) .

region S w i t h

: {n = i ~ , - /3 ~ ~ ~ ~ } but the polynomial (1.2.16) has double unimodular roots f o r n = ± i ~ S = S \{±i~}.

hence we obtain

The conjecture suggested by t h i s example reveals to be g e n e r a l l y t r u e :

By Theorem(A.1.4)the roots ~ i ( n ) of ~(~,n) are continuous f u n c t i o n s in simply connected open domains w i t h exception of the points where ok(n) = O. Therefore S consists of the n E ~ w i t h spr(F (n)) ~ I up to i s o l a t e d p o i n t s . For n E S \ S we thus must have at least one unimodular root of m u l t i p l i c i t y greater than one.° As is proved in the next s e c t i o n , the i n t e r i o r S of S consists of the n E ~ w i t h spr(F (n)) < I where spr(F (n)) denotes the spectral radius of the Frobenius m a t r i x . We now introduce the regions where spr(F (n)) is u n i f o r m l y less than one in a s l i g h t m o d i f i c a t i o n of S t e t t e r {73 , D e f i n i t i o n 2.3.15 and § 4 . 6 ] . (1.2.17) D e f i n i t i o n .

The region of b-exponential stability S C S of a method (1.1.3)

consists of the n E C with the property that all roots

i ~ i ( n ) i ~ I - 2~, 0 < ~ < I / 2 .

~i(n) of ~(~,n) satisfy

10 By a c o n t i n u i t y argument, too, S

is closed in C f o r u > O. Using t h i s concept and

the Uniform Boundedness Theorem we e a s i l y obtain exponential s t a b i l i t y . tial

The exponen-

decreasing f a c t o r u/At is however somewhat vague at present.

(1.2.18) Theorem. Let the assumptions of Theorem (1.2.12) be fulfilled and let Sp(AtA) C S for a fixed ~ lx'l(yn

=

< <~

-

> O.

Then for n = k , k + l , . . . ,

Vn) I

IX-1][e-(W~t)(n-k)~t I

Yk-1 - Vk-11 + atp

i~te-(U/~t)((n-k)At-~)ly(p+l)(~)ld~

+ AtZ~= k e - ( u / A t ) ( n - ~ ) A t m a x ( v _ k ) a t ~ T ~ v & t m a x 0 ~ i ~ L _ l l h ( i ) ( T ) l ] .

Proof. The r e s u l t f o l l o w s in the same way as in Theorem (1.2.12)

i f we s u b s t i t u t e the

bound (1.2.19)

IF (AtA)nl ~ K (1 - ~)n =<

into (1.2.13). (I.2.20)

Hence we have to prove ( 1 . 2 . 1 9 ) .

<

e -~n

But i f Sp(AtA) c S then

SUPn EIN I[(1 - u ) - l F ~ (AtA)]nl ~ SUPnEINSUpnE S

I[(1 - u)-lFjn)]nl

and we have spr([(1

- ~ ) - I F ( n ) ] ) ~ (1 - 2u)/(1 - p) < 1 - u < 1

f o r a l l n C S . Consequently, the Uniform Boundedness Theorem y i e l d s (1.2.21)

SUPnEINsuPnES I[(1 - u ) - l F ~ ( n ) ] n l ~ < .

A s u b s t i t u t i o n of t h i s bound i n t o (1.2.20) proves ( 1 . 2 . 1 9 ) . I f the s t a b i l i t y

region S is closed in ~ then we can obviously choose R = S in

Theorem (1.2.12) and the constant KR becomes a f i x e d constant depending only on the data of the m u l t i s t e p method. This remains also true i f f o r instance Sp(AtA) c IR and IR ~ S is closed in ~. A s i m i l a r remark holds f o r the constant ~

in Theorem (1.2.18)

and f o r a l l subsequent e r r o r estimations which contain a spectral c o n d i t i o n of the form ( I . 2 . 1 2 ) ( i i i ) .

11 1 . 3 . General Properties of the Region S of Absolute S t a b i l i t y

The closed hull T of the s t a b i l i t y region S contains no s i n g u l a r i t y points of k i ~(C,n) = Z i = 0 ~ i ( n ) ( , i . e . , no points n* with ~k(n*) = 0. Let n* £ T and l e t ~* be a r o o t of ~(~,n*) of m u l t i p l i c i t y

r. In a neighborhood of ~* every r o o t ~ i ( n ) of ~(~,q)

then can be w r i t t e n as a Puiseu~ series; cf. Theorem ( A . I . 4 ) .

In a somewhat s i m p l i f i e d

form we w r i t e instead of ( A . I . 5 ) (1.3.1)

~ i ( n ) = ~* + X(n - n*) p/q + ~((n - n * ) s ) , X ~ 0, n + n*,

w i t h p, q EIN having no common f a c t o r and s > p/q. N a t u r a l l y , ( I . 3 . 1 )

represents q

d i f f e r e n t roots f o r n ~ n* in dependence of the chosen branch of (~ - n*) I / q . The general connection between p, q, r, and × is ruled by the Puiseux diagram in Appendix A . I . If

I¢*I = I and n* E T is not an i s o l a t e d s t a b i l i t y

that p E {I,

...,

p o i n t then C o r o l l a r y ( A . I . 2 1 ) says

~} and q E { I , 2}. In general, q can be less t h a t r and there can be

several d i f f e r e n t growth parameters × to the r - f o l d r o o t ~*. By C o r o l l a r y ( A . I . 2 4 ) however t h i s case does not occur i f the method ( I . 1 . 3 )

is l i n e a r . Then we have p = I

and q = r , i . e . , (1.3.2)

¢i(n) : ~* + x(n - n*) I / r + ~((n _ q , ) s ) , × = 0, s > I / r ,

n ÷ n*.

The expansion (1.3.1) shows t h a t the roots of ~(~,n) depend continuously on n. If

I~*I = I then there e x i s t in every neighborhood values of n such t h a t I ~ i ( n ) I > I

f o r some index i . These both statements lead to the f o l l o w i n g r e s u l t : (1.3.3) Lemma. n E S i f f

spr(F (n)) < I.

Let n o w ~ = {n E ~, I ~ i ( n ) I = I f o r some i } be the r o o t says

locus curve then the lemma

t h a t the boundary ~S of S is a subset of ~ . ~ c o n s i s t s

a n a l y t i c curves which i n t e r s e c t

themselves

in f i n i t e l y

The complement o f ~ i n

~ consists of a f i n i t e

number of

many p o i n t s . ~S is not analytic

in the points where ~(~,n) has unimodular roots of m u l t i p l i c i t y in ~ \ S .

of a f i n i t e

greater than one, i . e . ,

number of connected components

~I'

. . . . ~r" I f npE~p then ~p C ~ i f f np E S and i f ~p ¢ ~ then ~pon S = ~. Therefore, i f ~ is known, one can test whether a component ~p belongs to S by testing just one sample point of ~ . See e.g. Jeltsch [76a, § 4.2.3, 78c]. ~ i s p real Zine and can be written geometrically as

symmetric to the

J~= {n E ~', ~ ( e l ~ , n ) = 0, 0 _<- ¢ < 2~}.

Using a r o o t s o l v i n g a l g o r i t h m we obtain by t h i s way a method f o r the computation of

12

where ~ is n a t u r a l l y to be d i s c r e t i z e d . A method with 0 E S is sometimes also c a l l e d zero-stable. A consistent and zeros t a b l e method is convergent in the c l a s s i c a l sense as already mentioned above. For these methods we now study the behavior of spr(F (n)) in a neighborhood of n = 0 more exactly.

(1.3.4) Definition. Let the method defined by ~(~,q) be convergent, i.e., consistent and zero-stable. Then

~1(n) with lim n+O~1(n) = I is the principal root of ~(~,n), (ii) the roots ~i(n) with lim n+Ol~i(n)l = I are the essential roots of ~(~,q) num(i) the root

bered by ~1(n) . . . . . Ck.(n),

I s k. s k.

(iii)

x~ : ~ ( O ) / ~ i ( O )

= - p1(~i(O))/[P~(~i(O))~i(O)],

i = I ..... k.,

are the growth parameters of the essential roots.

( i ) and ( i i )

are w e l l - d e f i n e d only in a neighborhood of zero. Note t h a t x~ = I and

0 E @S holds f o r a convergent method, xT, i = 2 , . . . , k . , method is l i n e a r by the i r r e d u c i b i l i t y

are nonzero at l e a s t i f the

of x ( c , n ) . The f o l l o w i n g r e s u l t shows t h a t in

m u l t i s t e p methods the p r i n c i p a l r o o t ~1(n) r e t a i n s the approximation p r o p e r t i e s of s i n g l e - s t e p methods, ( 1 . 3 . 5 ) Lemma. Let 0 E S then the method defined by ~(~,n) is consistent of order p iff (1.3.6) where

Xp is

~l(n) = e n - Xpnp+I + d(nP+2), Xp = O,

n + O,

the error constant.

Proof. In a s u f f i c i e n t l y

small neighborhood of n = 0 we define

(1.3.7)

~*(~,n) = ~ ( ~ , n ) I ( ~ - e l ( n ) )

and then obtain by (1.1.13) (1.3.8)

~*(I,O)

: p6(I)

= - pi(I)

= o.

I f the method is c o n s i s t e n t of order p then Lemma (1.1.15) y i e l d s by a s u b s t i t u t i o n of e ~ f o r ~ in ~(c,n)

(1.3,9)

~(en,n) : (e n - ~1(n))~*(en,n) : ×pn p+I + ~(nP+2),

This implies (1.3.6) because of ( 1 . 3 . 8 ) . On the other s i d e , i f

n ÷ O.

(1.3.6) holds then a

13 s u b s t i t u t i o n i n t o the f i r s t

equation of (1.3.9) y i e l d s

~(en,n) = ~*(en,n)(×pn p+I + ~(nP+2)),

n ~ O.

Therefore the method has order p by Lemma (1.1.15) and ×p is the e r r o r constant by (1.3.8). The essential roots s a t i s f y Ci(n) = ~ i ( 0 ) ( I (1.3.10)

+ x~n + d(n2)) or

l ~ i ( n ) I = I + Re(x~)Re(n) - Im(x~)Im(n) + ~ ( I n l 2 ) ,

q + O, i = I . . . . . k . .

This equation has some i n t e r e s t i n g consequences because there e x i s t s a ~ > 0 such t h a t (1.3.11)

spr(F (n)) = m a x 1 ~ i ~ k . l ~ i ( n ) I ,

(1.3.12) Lemma. Let ~ ( m , p ) Let the method defined by

Inl ~ ~:

: {n E ~, largn - ~} ~ ~, Inl ~ P} be an angular domain.

~(~,n)

be convergent and let the growth parameters ×~s i =

Is...~k.~ be nonzero. Then (i) ~ ( ~ p ) (ii) ~(~,p)

C S for some ~ > 0 and p > 0 iff Re(×~) > O, i = I ..... k., C C \ S for some ~ > 0 and p > 0 iff Re(x~) < 0 for some i, I ~ i ~ k.,

(iii) there exists a p > 0 such that ~ \ S contains the ball {n £ C~ In - Pl < P}.

Proof. (Cf. also Crouzeix and Raviart [80 ] . ) The f i r s t

two assertions f o l l o w from

(1.3.10) and ( 1 . 3 . 1 1 ) . For the t h i r d assertion observe t h a t

n-~O.

l~1(n) I = I + Re(n) + ~(In12), S u b s t i t u t i n g n = p + p(cos¢ + i s i n ¢ ) ,

0 ~ ¢ < 27, we obtain

I ¢ i ( n ) I : I + p(1 + cos¢) + ~(o2(I + cos¢)),

p ÷

O,

0

< ¢

< 2~s

which proves the r e s u l t . The Milne-Simpson method defined by (1.2.16) has the growth parameters x~ = I and ×~

= -

I / 3 . This example shows t h a t S is not n e c e s s a r i l y bounded away from zero i f

Re(xT) < 0 f o r some i , By ( I . 3 . 1 0 ) ,

I ~ i ~ k,.

there e x i s t s a a t ' 0 < al < ~' such t h a t l ~ i ( n ) } ~ I + (Re(x~)/2)n,

al ~ n ~ O, i = I . . . . , k . .

14

I f X = m i n 1 ~ i ~ k . { R e ( × # ) / 2 } > 0 then a s u b s t i t u t i o n of these bounds i n t o ( 1 . 3 . 1 1 ) y i e l d s spr(F(n))

s I + xn,

- Sl ~ n ~ O.

Observing now t h a t t h e r e is a 0 < ~ ~ 81 such t h a t a l l e s s e n t i a l r o o t s of ~(~,q) are simple f o r

-a ~ q ~ O, the Uniform Boundedness Theorem y i e l d s f o r F ( n ) / s p r ( F ( n ) ) :

( 1 . 3 . 1 3 ) C o r o l l a r y . Let the method defined by ~(~,n) be convergent and let Re(x~) > O, i = 1,...,k,.

Then there are positive constants 8,
IF (q)nl ~
such that

,

- 6 ~ n s O.

I f ~ ~ ~ then thereo e x i s t s by Lemma ( 1 . 3 . 3 ) and the c o n t i n u i t y of spr(F (n)) a ~0 > 0 such t h a t Su c S f o r 0 < ~ ~ U~'u The boundary ~S

of S

is a subset o f the r o o t locus

curve

~

= {~ E ~, ~ ( ( I - 2 u ) e i ~ , n ) = O, 0 ~ ~ < 2~}

as again f o l l o w s from J e l t s c h [76a, § 4 . 2 . 3 ] a n d ~ t

depends c o n t i n u o u s l y on ~.

( 1 . 3 . 1 4 ) Lemma. Let the method defined by ~(~,n) be convergent, let Re(x~) > O, i = I , o

...,k,,

and let [-s, O) C S, 0 < s ~ ~. For n I ~ n O ~ 0 then there exist positive con-

stants ~s and K*s depending only on n I such that K*nn

IF (n)n[ ~ mse s

0

- s ~ n ~ n O.

Proof. Let 8 > 0 be the constant defined in C o r o l l a r y ( 1 . 3 . 1 3 ) then we have [ - s , - ~ ] C S

f o r some ~I > O. Hence we o b t a i n in the same way as in Theorem ( 1 . 2 . 1 8 )

(1.3.15)

JFjn)nJ

s <2 e

-~ln

~ ~2 e

I~l/nl[nno

- s ~ n ~ - ~.

This proves the a s s e r t i o n f o r nO ~ - a. I f - ~ < n O ~ 0 then we o b t a i n by ( 1 . 3 . 1 5 ) and C o r o l l a r y ( 1 . 3 . 1 3 )

IFJn)nl

< max{~2e [ ~ l / q l I n n O :

,

<1 e

~nqO

Let us now r e t u r n once more to the i n i t i a l A is hermitean and n e g a t i v e d e f i n i t e , (iii)

o f Theorem ( 1 . 2 . 1 2 ) reads

}

~ Ks e

<*nn 0 s

,

-

value problem ( I . 2 . 8 ) o

s

s

n

~

no•

I f the m a t r i x

A ~ - ~ I , ~ > O, then the s p e c t r a l c o n d i t i o n

15

Sp(atA) c [ - a t I A l , - a t y ] c [ - s , - a t y ] c S,

S e t t i n g n0 = ~t~ in Lemma (1.3.14) and s u b s t i t u t i n g the r e s u l t i n t o ( I . 2 . 1 3 ) we obtain uniform e r r o r bounds with exponential decreasing m u l t i p l i c a t i o n initial

f a c t o r f o r the class of

value problems (1.2.8) with hermitean and u n i f o r m l y negative d e f i n i t e matrix A:

( I . 3 . 1 6 ) C o r o l l a r y . Let the assumption of Theorem ( 1 . 2 . 1 2 ) b e f u l f i l l e d

but let the

matrix A of the initial value problem (1.2.8) be hermitean and negative definite, A ~ - yl,

0 < ~ ~ x O, o

(ii)

L e t 0 < At S at O, 0 E S, I - s ,

O) C S, 0 < s s =, and Re(×~) > O, i = 1 , . . . , k . .

Then the a~sertion of theorem (1.2.18) holds with (IXl = EX-ll = I,~ ~

<s' and

~/At = <~¥ where <*s depends only on ~t 0 and ~0" Examples of l i n e a r m u l t i s t e p methods with I - ~,0) C S are given in Appendix A.4.

1.4. I n d i r e c t Methods f o r D i f f e r e n t i a l

Systems of Second Order

For an approximate s o l u t i o n of the i n i t i a l (1.4.1)

y"

= Ay + By' + c ( t ) + h ( t ) ,

value problem

t > O, y(O) = YO' y'(O) = y~,

by a m u l t i s t e p method of the type (1.1.3) we have to w r i t e t h i s problem as f i r s t

order

problem, (I .4.2)

z' = A*z + c * ( t )

+ h*(t),

t > O, z(O) = z O,

where z = (y, y , ) T and

However, A was always supposed to be a diagonable m a t r i x in the above e r r o r e s t i mations hence A* must be also a diagonable matrix in order t h a t the r e s u l t s of Section 1.1 to 1.3 apply. In m a t r i x s t r u c t u r a l analysis i t

is f r e q u e n t l y supposed t h a t the

damping m a t r i x B has the same system of eigenvectors as the leading m a t r i x A and t h a t both are diagonable. I f

A = (~I . . . . . Xm) and ~ = (¢I . . . . . Cm) denote the diagonal matrices of the eigenvalues of A and B r e s p e c t i v e l y then the sys-

16 tem (1.4.1) can be decoupled a f t e r a suitable transformation into m scalar d i f f e r e n t i a l equations, (1.4.3)

}''

= ~i ~ + ¢i ~' + ~ i ( t ) ,

i = I . . . . . m,

which can be solved independently of each other, This modal analysis is advantageous i f the decomposition of the matrix A is a v a i l a b l e at a reasonable price or i f the eigenvalues of A must be computed anyhow by other reasons, Every solution of the scalar differential

equation (1.4,3) o s c i l l a t e s f o r Xi < 0 and ¢i s 0 only i f

I¢i[ < 2 ~ i

~

therefore the value 2 / I x i [ is called the critical value for the eigenvalue ~i' The following lemma provides the technical tools for an error estimation in the present case of 'orthogonal' damping. (1.4,4) Lemma, Let the real syn~netric (m,m)-matrices A and B have the same system of eigenvectors, A = XAX T, B = X@X T, xTx = [, let 0 < X[ ~ - A, 0 ~ - @ ~ /~2(-A)1/2t 0 ~ 6 < I, and let P = (I,(-A) -I/2) be a block diagonal matrix. Then the matrix A* is

diagonable,

A* = X*A*X*

-I

,

with

Re(A*) ~ O, I x * - l l s 2[(1 + y- 1)/(1 - a) ]1/2 , ]PX*I ~ 2.

Proof. We have

and the eigenvalues of A* are therefore (t.4.5)

~*p = [¢~ ± (¢~ + 4~u)1/2]/2,

p = 1, . . . . m.

These are by assumption 2m d i f f e r e n t numbers with Re(x*) ~ 0 and nonzero imaginary part therefore A* is diagonable with Re(A*) ~ O. Let now A* = (~1' -=2) be the block diagonal matrix consisting of the diagonal matrices of the eigenvalues (1,4,5) with p o s i t i v e and negative sign r e s p e c t i v e l y . Then we obtain X* = X

III ] "--I

0

X*-1

= (-=2 ° -=I )-I

--2

and I

'

T

I

-=2

-Z I

i][i =

-=2 By d i r e c t computation we f i n d that

I I i]

X

(-A)-I/2-=I

-I

T

X ,

I

(-A)-I/2-=2

17 I ( - A ) ' 1 / 2 - :.1

i = 1,2,

= I,

hence we obtain s (NPX*II fIPX*Tf1 )1/2 = (2.2) 1/2 = 2.

IPX*I

For the e s t i m a t i o n of [ x * - l l x*-Hx * - I = X ~IgI + -2-2 L-(~I + z 2)

we observe t h a t -(:I

+ 5~) (Z 2 - £I)-2X T = 21

-1

X(4A + O2) -I

XT

2I

Because of 0 ~ - { s ~ 2 ( - A ) I / 2 , 0 ~ ~ < I , we thus derive (_A)I/2 21(-4A + 4 a A ) - l ( I

- A) I =

21A-l(I

= 21(4A + O 2 ) - I ( I - A)I/(1

-

- ~) ~ 2(1 + ¥ - 1 ) / ( 1

A) I - ~),

Let now (y, y , ) T = z be the s o l u t i o n of the i n i t i a l value problem (1.4.2) and l e t v*n = (Vn' wn)T , n = k , k + 1 , . . . , be the numerical approximation obtained by the scheme (1.2.2), i.e., k " ~i=0~i(AtA*)T1Vn

(1.4,6)

k T i ~ = l a i j (AtA.)AtJ c n ( j - l ) = - ~i=0

then we have the f o l l o w i n g c o r o l l a r y (1.4.7)

n = 0,1

i

I t * l l

to Theorem ( 1 . 2 . 1 2 ) ,

Theorem. Let the initial value problem ( 1 . 4 . 2 ) and the numerical approximation

(I.4.6) fulfil the assumptions of T;~eorem (1.2.12) and Lemma (I.4.4).

Then for n = k,

k+1 , . . . , [Yn

-

Vnl

+

]AI-I/21 Yn'

-

Wn

+ At p

I <=

KR[

(I

+

y-I)/(I

[y(p+l)(~) I

-

6)]I/2[IYk_ I

ly(p+2)(~)l)dT

-

Vk_l[

+

I Y'k-1

-

Wk-1

i

+ nat max0
0

V* Proof. Let E* = Z - V* where Z = (Zn_k+ I . . . . . zn)T and V*n n n n n = ( n-k+l"" we obtain by Theorem (1.2.12) (I.4.8)

IX*-IEn I _-< ~ R I X * - I I [ I Z k _ I - V ~ _ I I

which up to the p r e - f a c t o r

+ AtpiAtIz(P+I)(z)Idz 0

,V~) T then

+ nat max llih(i)llin] 0<=i<~-I

y i e l d s the r i g h t side of the e r r o r bound and the p r e - f a c t o r

on the r i g h t side of the a s s e r t i o n

is the bound of { X * - I I

from Lemma ( 1 . 4 . 4 ) .

For an

18 a r b i t r a r y regular matrix P we have IPE~I ~ IPX*II(PX*)-IpE~I = IPX*[IX*-IE~I. Choosing f o r P the block diagonal matrix P = ( I , (-A) - I / 2 ) and s u b s t i t u t i n g the l a s t bound of Lemma (1.4.4) we f i n d

IYn - Vnl + IAI-1/21Y'n - Win =< lYn - Vnl + IA-1/2(Yn - Wn)l <= ~'2IX*-IE~ l" This estimation and (1.4.8) prove the assertion. I t should be emphasized that in the error bound of t h i s theorem the i n i t i a l

error

on the r i g h t side is m u l t i p l i e d neither by IAI I/2 nor by At -I unlike the subsequent e r r o r estimations of t h i s and the f o l l o w i n g chapter. On the other side, i f the second or the f i r s t

term on the l e f t side is omitted then we obtain the f o l l o w i n g bounds re-

spectively:

ly n - Vnl ~

] r i g h t side of the i n e q u a l i t y in Theorem ( 1 . 4 . 7 ) .

IY~

Wnl ~ Ial I/2x

This r e s u l t r e f l e c t s a f a c t which is well-known in e l l i p t i c

finite

element analysis

namely that in i l l - c o n d i t i o n e d problems the approximation of the d e r i v a t i v e of the solution is always worse than that of the solution i t s e l f . The next r e s u l t concerns an other special form of the damping matrix B in (1.4.1) which appears e.g. in D i e t r i c h [81 ]. (1.4.8) Theorem. Let the initial value problem (1.4.2) and the numerical approximation fulfil the assumptions of Theorem

(I.4.6)

(1.2.12).

Let the real (m,m)-matrices A

and B be symmetric and negative definite and skew-symmetric respectively, B T = - B.

lYn

Then for

- Vnl + IY~ - Wnl ~
A ~ - yl

nat

f

(I(-A)

I)I/2[

1/2y(p+1)

[(-A)I/2(YN_

(~)1 +

ly(p+2)

I

VN_I)I

+ IYk_ I - Wk_iI

(~)l)d~ + nat maxo~i~_1111h(i)illn

Proof. We consider instead of (1.2.10) the modified error equation (I " 4.9)

< O,

n = k,k+1,...,

QE~ = F (AtQA*Q-I)QE~ = I ÷ QD*(At'Z)n - H*n '

where Q = ((-A) I / 2 , I) is a block diagonal matrix and hence

n = k,k+1,...,

]

.

19

(1.4.10)

A** =- QA*Q-I =

I 0

I/2

-(-A)

I;1,2]

•

By assumption, t h i s matrix is real and skew-symmetric hence normal, A**HA ** = A**A **H, hence u n i t a r y diagonable. Therefore we obtain in the same way as in Theorem (1.2.12) by the Uniform Boundedness Theorem ( c f . Appendix) (1.4.11)

IQE~I ~
n = k,k+1,...

~=kIQD*(At,Z) v - QH~I],

But QD*(At,Z) n = Q(O . . . . . O,ok(AtA*)-'Id(At,Z)n_k )T = (0 . . . . . O,Ok(AtA**)-Id(At,Qz)n_k )T,

k ~ AtA.)AtJh~(~-I))T QH~ = Q(0 . . . . . 0,Ok(Ai;A . ) - I ~i=0~j=1~ij( • ~ "=I k ~ (ALA**)AtJQhn( = (0 . . . . . 0,~k(AtA ** ) -I ~i=0~j=1oij _~ ))T ,

and Qh*(t) = Q(0, h ( t ) ) T = h ( t ) . Consequently, as in the proof of Theorem (1.2.12), nAt ~=kIQD*(At,z)~ I sYAtP~ IQz(P+I)(T)Id~ s AtP~ At 0 0 and

( - A ) I / 2 y ( p + I ) ( T ) I + ly(p+2)(~)l)d~

~:klQH~I ~ ~RnAt max0si~_ I lh(i)llln. Finally, IQE~I

IQE~_II s I(-A)I/2(Yk_ I - Vk_1) I + IY~_I :

Wk_11 and

I((-A)I/2(Y n - Vn) , (Y~ - Wn))i ~ 2-1/2(y1/21y n _ Vnl + IY~ - Wnl)

which proves the desired r e s u l t . Under the assumption of Theorem (1.4.7) and (1.4.8) the spectral condition (I.2.12)(iii)

claims f o r the leading matrices A* and A** r e s p e c t i v e l y t h a t

Sp(AtA*) C {n ~ C, - 2 ~ A t I A i I/2 ~ Re(n) s O,

IZm(n)l

~ AtIAi I / 2 } c R c S

and Sp(AtA**) C {n E £, Re(n) = O, ilm(n)I s At(IB i + I a [ 1 / 2 ) } C R C S. This leads to the demand f o r multistep methods whose s t a b i l i t y

region S contains a large

i n t e r v a l {n = i~, I~i ~ s} on the imaginary axis. The special class of single-step m u l t i d e r i v a t i v e methods studied in the next section has even the property that S contains the e n t i r e imaginary axis. For multistep methods the problem of s t a b i l i t y imaginary axis is discussed more d e t a i l l e d in Section 1.7.

on the

2O 1.5. Dia~onal Pad~ Approximants of the Exponential Function

In view of Theorem (1.4.7) and (1.4.8) i t suggests i t s e l f

to ask whether there are

numerical schemes which do not need the d i a g o n a b i l i t y of the matrix A f o r uniform e r r o r bounds. In order to answer t h i s question we consider s i n g l e - s t e p m u l t i d e r i v a t i v e cedures of order 2~ defined by the c h a r a c t e r i s t i c

~(¢,n) = ~(~,n)

(1.5.1)

= ~(~)(-n)~

pro-

polynomials

- o(~)(n)

where (1.5.2)

°(C)(n)

~FT)(_n)

en + ~(n2~+I),

n ÷ 0,

are the well-known diagonal Pad~ approximants of the exponential f u n c t i o n : 4! s(~)(n) = ~:0 ~ .

(1.5.3) see e.g. G r i g o r i e f f

( 2 z - j ) ! nj ~ T T . '

~ : 1,2 . . . .

;

[72]. For instance we obtain

s ( 1 ) ( n ) = 1 + (n/2) ~(2)(n) = I + (n/2) + (n2/12) q(3)(n) = I + (n/2) + ( n 2 / I 0 ) + (n3/120) o(4)(n) = I + (n/2) + (n2/28) + (n3/84) Below these schemes are b r i e f l y

+ (¢/180).

c a l l e d (~,~)-schemes and the (1,1)-scheme is the trape-

zoidal r u l e . (1.5.4) Lemma. For ~ : 1 , 2 , . . . ,

all roots of the polynomials o(~)(n) defined by (1.5.3)

have negative real parts.

Proof. B i r k h o f f and Varga [65 ] ; see also G r i g o r i e f f

[72 ].

In the next lemma, Re(A)~O means f o r an a r b i t r a r y real or complex (m,m)-matrix A t h a t Re(xHAx) ~ 0 v x E {m, i . e . , t h a t the hermitean part Re(A) = (A + AH)/2 is p o s i t i v e semidefinite. (1.5.5) Lemma. Let s(n) be an arbitrary real polynomial of exact degree ~ such that Re(q)

< 0 holds for all roots n of q(n). Then, for every matrix A with Re(A)

I~(-A)-11

~ K and lo(-A)-lq(A)l

~ I.

~ 0

21

P r o o f . We f o l l o w Gekeler and Johnsen [77] and assume w i t h o u t

loss o f g e n e r a l i t y

that

~(n) i s a normed p o l y n o m i a l . Then we have o(-A) = ~(-A i=I and the components (-A - n i l ) Q satisfies

- nil)

are r e g u l a r m a t r i c e s w i t h Re(-A ~ n i l )

Re(Q) ~ ml > 0 then j Q - I j

Jq(-A)-11 _-<~](-A i=I

~ -I

therefore

> O. I f a m a t r i x

we o b t a i n

- ni)-11 < ~-~(- Re(ni)) -I i=I

where the c o n s t a n t K does not depend on A . In o r d e r t o prove the second a s s e r t i o n we observe t h a t a(-A)-Io(A)

= ~(-A i=I

- nil)-1(A

- nil).

T h e r e f o r e we have t o show t h a t G(A,n) z (-A - n l ) - 1 ( A

- nl)

satisfies (1.5.6)

IG(A,~)I

~ I,

n < O,

and (1.5.7)

JG(A,n)G(A,~) I ~ I

Let f i r s t

Re(n) < O.

n < 0 and l e t y = G(A,n)x then (-A - n l ) y = (A - n l ) x o r

- A(y

+ x)

= n(Y

- x).

Because Re(A) ~ 0 we o b t a i n 0 ~ Re((y + x ) H ( - A ) ( y

+ x))=Re(n(y

Hence we have IYJ < IxJ f o r a r b i t r a r y (1.5.6)

+ x)H(y - x ) ) : n ( l y l

2 - IxJ2),

x E ~m which proves ( I . 5 . 6 ) .

n < O.

In o r d e r t o prove

l e t Re(n) < 0 and l e t y = G ( A , n ) G ( A , ~ ) x then (A 2 + 2Re(n)A + JnJ21)y = (A 2 - 2Re(~)A + JnI21)x

or (A 2 + I n J 2 1 ) ( y - x) = - 2 Re(n)A(y + x ) . I f A i s r e g u l a r then we m u l t i p l y

this

e q u a t i o n by (y - x)H(2Re(n)A) - I and o b t a i n

Re((y - x ) H ( j n J 2 A - I + A ) ( y - x ) / 2 R e ( n ) )

= - (]yj2

ixj2).

22 The l e f t side is nonnegative because Re(A) s 0, Re(A- I ) ~ 0, and Re(n) < 0 hence we have again lYl ~ Ixl

f o r a r b i t r a r y x E ~m . This proves (1.5.7) in the case where A

is regular. I f A is singular then we f i n d in the same way that ( I , 5 . 7 ) holds f o r A - ~I and e > 0. As the r i g h t side of (1.5.7) does not depend on ~ the i n e q u a l i t y must hold f o r ~ = 0, too. Now we return to the i n i t i a l

value problem (1.4.1) and the associated f i r s t

order

problem ( 1 . 4 . 2 ) . An (~,~)-scheme is consistent of oder 24 by (1.5.2) and Lemma ( 1 . 3 . 5 ) . and i t y i e l d s f o r (1.4.2) the computational device s(C)(-AtA*)v~+I - q(~l(AtA*)v~

(I .5.8) L ~ ) (AtA*)c~ ( j - l ) , = - ~ Lj= I ~(~)I j ~- AtA*~c*(J-I) s n+1 + Zj=I~

n = 0,1 . . . .

The associated Frobenius matrix has obviously the form

F~(~) : (s(~)(-n))-lJ~)(~). I f the (m,m)-matrices A and B in the i n i t i a l

value problem (1.4.1) are a r b i t r a r y real

symmetric and negative semidefinite matrices then the matrix A** defined by ( I . 4 . 1 0 ) is no longer unitary diagonable because i t is no longer normal. But A** s a t i s f i e s s t i l l the assumption of Lemma ( 1 . 5 . 5 ) ,

[i ;]0

Accordingly, we deduce from kemma (1.5.4) and (1.5.5) that

io(g)(-AtA**)-11

S < and I F j A t A * * ) i

~ I.

Substituting these bounds i n t o the modified e r r o r equation ( I . 4 . 9 ) we obtain again (1.4.10) without using t h i s time the Uniform Boundedness Theorem. The r e s t of the error estimation is then the same as in Theorem ( I . 4 . 8 ) . We assemble the r e s u l t of t h i s section in the f o l l o w i n g theorem. (1.5.9) Theorem. Let ~ E IN be fixed. In the initial value problem ( I . 4 . 1 )

let A, B

be real symmetric i A ~ - yI < O~ B ~ O, and let the solution y be (2&+2)-times continuously differentiable. Then the numerical approximation v~ =

(Vn,Wn)T defined

(1.5.8) satisfies for n = 1.2 . . . . . lY n - Vnl + [Y~ - Wnl ~ ~(I nat + AtZ~ I ( I ( - A ) I / 2 y ( 2 g + I ) ( T ) I

+ y-1)I/m[i(-A)I/2(y

0 - Vo) i + lY~ - Woi

+ lY(2~+2)(T)i)dT + nat max0~i~g_llllh(i)llln ] ,

by

23 1.6. S t a b i l i t y in the Left Half-Plane

We have seen in Section 1,3 that the s t a b i l i t y

region S can be found geometrically

by a plot of the root locus c u r v e ~ . However, i f i t is demanded that S has a certain shape suggested from practical aspects then general statements on appropriate methods are to be derived in an a n a l y t i c way using tools from algebra and a n a l y t i c function theory. The f i r s t

r e s u l t in t h i s d i r e c t i o n has been provided by Dahlquist [63 ] who

has defined a method to be A-stable i f {n C ~, Ren < O} C S and has shown that l i n e a r multistep methods of order greater than two cannot have t h i s property, With t h i s somewhat depressing r e s u l t a very i n t e r e s t i n g development of numerical analysis has begun which is by no means f i n i s h e d up today. A complete description of the present state of knowledge would go beyound the scope of t h i s volume therefore we r e s t r i c t ourselves to a survey of the most important results of the recent years and omit the proofs of some classical r e s u l t s . In many applications

A - s t a b i l i t y is a too severe requirement therefore Widlund

[67 ] and Cryer [73 ] have introduced the f o l l o w i n g concept. (1.6.1) D e f i n i t i o n . A m e t h o d (1.1.3) i s o

(i)

A(a)-stable

if

0 < a _-< x/2,

{n E { , n ~ O, Ix - argnl < ~} C S,

(ii) A(O)-stable if it is A(~)-stable for some ~ C ( 0 , ~ / 2 ] , o

(iii) Ao-stable if (-~, O) C S.

By Lemma (A.I.17) a convergent method cannot be A(a)-stable with ~ > 7/2 in a neighborhood of n = O. Obviously, an A(O)-stable method is Ao-stable but the converse is not true: The l i n e a r method with the polynomial ~(~,n) = 2

~ _ ~( 2 + 2~ + I)

is convergent and Ao-stable but not A(O)-stable (Cryer [73 ]) because p1(~) = - ~ ( 2+2~ + I ) has a double root ~ = I ; c f . Corollary ( A . 1 . 2 1 ) ( i i ) , (1.6.2) Theorem. There is only one Ao-stable linear k-step method of order p ~ k + I : the trapezoidal rule of order p = 2 with the polynomial

~(~,n)

=

~

-

I -

(n/2)(~

+

I).

Proof. See Cryer [73 , Theorem 3.3]. (1.6.3) Theorem. For each ~ C [0, 7/2) there exists an A(a)-stable linear k-step method of order p : k = 3 and p = k = 4.

Proof. See Widlund [67 ].

24

Some w e l l - k n o w n A ( m ) - s t a b l e methods are r e p r e s e n t e d in Appendix A.4. In o r d e r t o deduce necessary and s u f f i c i e n t the Routh-Hurwitz c r i t e r i o n . k i p ( z ) = ~i=oSi z recall

(1.6.4)

Sk- I

Sk- 3 . . .

sk

Sk_ 2 . . .

Ap =

.

.

.

.

.

.

.

Sk- 3 . . . Sk_ 2 . . .

0 0

Sk_ I . . .

I

.

e

.

I

.

I

we now

] -k+1 ' S-k+ 2

Sk- I

,

for Ao-stability

s

sk

,

conditions

Let Ap denote the Hurwitz matrix o f the p o l y n o m i a l

.

.

I

,

m = 2[k/2],

s i = 0 f o r i < O,

.

s ...s~

.0

oJ

U

then this well-known result reads as follows; cf. e.g. Lambert [73 ]: (1.6.5)

Lemma. All roots Z* of

k i p ( z ) = Zi=oSi z satisfy Rez* < 0 iff all leading prin-

cipal minors of the Hurwitz matrix are positive, i.e., the determinants of all matrices arising from Ap by cancelling the r last rows and columns for I t can be shown t h a t the R o u t h - H u r w i t z c r i t e r i o n (1.6.6)

r : k-1,...,O.

implies

k i Lemma. I f a l l roots z* o f p ( z ) = Zi=oSi z s a t i s f y Rez* < 0 then s i > 0 f o r

i = O,...,k. Both r e s u l t s

cannot be a p p l i e d d i r e c t l y

t o the p o l y n o m i a l ~ ( ¢ , n )

therefore

the M6bius

transformation i s g e n e r a l l y i n t r o d u c e d in t h i s c o n t e x t , z = (¢

+ I)I(~

-

I),

¢ = (z

+ 1)/(z

-

I),

which maps the u n i t d i s k o f the c - p l a n e onto the l e f t

half-plane.

In p a r t i c u l a r ,

~ = 0

i s mapped i n t o z = - I and ~ = I i s mapped i n t o z = ~. Let

~(z,n)

= (z - I j

~k ,z + I , k ~ nj " ~z~_ ,n) = Zi=O~j=O~ji (z + 1 ) l ( z

- I

)k-i

(I . 6 . 7 )

~k r~ a Jz i k s = Li=O£j=O j i n = Zi=O i (n)zi = Zj=orj (z)nj then ~ ~k j sk(n) = L j = o £ i = O ~ j i n ~ 0 if

the method i s c o n v e r g e n t and the method i s A o - s t a b l e i f f

the l e f t

half-plane,

Rez < O, f o r n E ( - ~ , 0 ) .

The f o l l o w i n g

all

roots of ~(z,n)

lemma shows r o u g h l y

lie

in

25 speaking t h a t i t

i s unnecessary f o r the v e r i f i c a t i o n

mediate l e a d i n g p r i n c i p a l

of Ao-stability

t o check the i n t e r -

minors o f the H u r w i t z m a t r i x a ~ ( n ) .

(1.6,8). Lemma. A method (1.1.3) is Ao-stable iff the following three conditions are fulfilled: (i) For some n* E (-~, O) all roots of ~ ( Z , n * )

lie in the left half-plane.

(ii) Sk(n) ~ 0 v n E ( - ~ , 0 ) . (iii) d e t ( a ~ ( n ) )

~ 0 v n E (-~, 0).

P r o o f . See F r i e d l i

and J e l t s c h

[78 ] .

This communication c o n t a i n s a l s o an a l g o r i t h m t o d e t e r m i n e the H u r w i t z d e t e r m i n a n t det(A~(n))

with polynomial entries

by means o f Sturmian sequences. As t h i s

cumbersome we now g i v e some f u r t h e r which are more e a s i l y In l i n e a r

necessary c o n d i t i o n s

for Ao-stability

i s somewhat at least

t o check.

methods one w r i t e s

~(~,n)

= p(~) - n ~ ( ~ ) ,

~(z,n)

= r(z)

customarily

k i k i p(~) = ~i=O~i ~ , o(~) = ~i=OBi ~ ,

and - ns(z)

'

r(z)

=

~k ~i=O

a zi i

'

k i s ( z ) = ~i=obi z

Then we o b t a i n

o(I) hence, i f

= l i m z + p ( z ~ _+ I ) = l i m z + ( z

the method i s convergent and ( 1 . 1 . 1 4 )

a k = O, b k = o ( I ) The c o n d i t i o n a l

= bk,

is s t i p u l a t e d ,

e q u a t i o n s f o r c o n s i s t e n c e o r d e r p ~ I read here ( c f .

+ 2j)],

w i t h the c o n v e n t i o n t h a t a i = 0 f o r Accordingly,

= a k, o ( I )

= I.

ai = 2~jzo[bi+1+2j/(1

(1.6.9)

- 1)-kr(z)

Lemma ( A . I . 3 )

k - p ~ i ~ k,

i < 0 and b4 = 0 f o r j > k; c f .

and Lemma ( 1 . 6 . 6 )

Lemma ( 1 . 1 . 1 2 ) )

Widlund [67 ] .

f o r z ÷ 0 and z ÷ ~ y i e l d :

Lemma. ( C r y e r [73 ] . )

If the linear method with ~ ( I ) = I is Ao-stable then k k a i ~ 0 and b~ ~ 0 f o r j : O, . . . . k. Furthermore, ~ i = o a i > O, ~ i = o b i > O, and 6 k m O. R e t u r n i n g t o the general case and the n o t a t i o n which i s due t o J e l t s c h

[77 ] .

(1.6.7)

we prove an a u x i l a r y

result

26 (1.6.10) Lemma. Let the polynomial ~ ( ~ , n )

be irreducibel.

(i) If q E S ~ ~ then si(q) m O, i - 0 ..... k, and

Re(si+1(~)/si(~))

i = O,...,k-1.

> O,

(ii) I f ~ £ S then the degree of s i ( n ) is ~, i = 0 . . . . .

Re(a~,i+i/a~i)

k, and

i = 0 . . . . ,k-1.

> O,

Proof. I f n E S ~ C then a l l roots of ~(z,q) l i e in the l e f t (a) i = a(a + 1 ) ' " ( a

+ i - I),

h a l f - p l a n e , Rez < O. Let

i E IN, then by a repeated a p p l i c a t i o n of Theorem

( A . I . 4 6 ) we f i n d t h a t a l l roots z* of ~l~Iz ~Z 1 k

~ = (1)isi(q)

~q)

+ ( 2 ) i s i + 1 ( n ) z + . . . + (k - i + 1)iSk(~)z k-i

s a t i s f y Rez* < O, too. Therefore s i ( q ) m O, i = O , . . . , k , and the sum of the reciprocals of a l l roots z* has negative real part. Accordingly, Vieta's root c r i t e r i o n yields

-

< O,

ReL~T~iSi(n) This proves the f i r s t

i

= 0 .....

k-1.

J

a s s e r t i o n . The second assertion f o l l o w s in the same way by con-

sidering n~(z,n-1). I t is now convenient to introduce the f o l l o w i n g n o t a t i o n :

(1.6.11) Definition. A method (1.1.3) is asymptotically A(~)-stable if for all e E (~ - at ~ + a) there exists a P8 > 0 such that {q : pe ie, p > ps} C S.°

Asymptotic A ( O ) - s t a b i l i t y and asymptotic A o - s t a b i l i t y are defined in an analogous way but observe t h a t in these d e f i n i t i o n s the p o i n t ~ i t s e l f

is always excluded. Then the

behavior of a method at the p o i n t ~ is e n t i r e l y ruled by the f o l l o w i n g simple r e s u l t :

(1.6.12) Lemma. A method with the polynomial ~(~,n) is asymptotically A(a)-, A(O)-, or Ao-stable iff the method with the polynomial n~(~,n -I) ~ ~,(~,n) is A(~)-, A(O)-, or Ao-stable in a neighborhood of zero.

Proof. I t s u f f i c e s to prove the a s s e r t i o n f o r A ( a ) - s t a b i l i t y .

We f i r s t

observe t h a t

arg(re ie) E (~ - a, ~ + a) i f f arg(re ie) = arg(re - i e ) E (~ - a, ~ + a) hence we may consider the polynomial x(~, ~). Substituting q = p - l e i e , p ÷ O, we obtain with ~ = ie pe

27

~L~(~,~)

= ~~ n ~j=OPj(~)~ j = ~~ ~ j : x.(~,n) ~ n ~~j : 0 P j ( ~ ) q~-J = ~j=OP~_j(~)~ ~

which proves the a s s e r t i o n . N a t u r a l l y , ~.(~,n)

is not n e c e s s a r i l y the c h a r a c t e r i s t i c polynomial of a consistent

method. But f o r every a l g e b r a i c polynomial ~(~,n) the shape o f the s t a b i l i t y

region S

near n = 0 is determined by the behavior of those roots ~i(q) which become unimodular in n = 0. I f a 'method' with the general polynomial ~(~,n) is A0-stable near q = 0 with a possible exception of the p o i n t n = 0 i t s e l f

then C o r o l l a r y ( A . I . 2 1 )

implies

t h a t a l l roots ~i(n) with I ~ i ( 0 ) I = I have near q = 0 the form ( A . I , 1 8 ) with q E { I , and p E {I . . . .

2}

, ~}. The roots of ~.(~,n) must have t h i s property i f the method is

a s y m p t o t i c a l l y A0-stable. Together with Lemma (1.6.10) f o r n E ( - - ,

0) we thus can

state: (1.6.13) C o r o l l a r y . (Jeltsch [77 ] . ) Let the method (1.1.3) be convergent. Then the following conditions are necessary for Ao-stability: (i)

~k

~ 0, s i ( n ) = 0, q E (-~, 0), i = 0 . . . . . k, and Re(si+1(n)/si(n))

(ii)

~ E ~ and a l l

roots

> 0

~ i ( n ) o f ~.(~,n) = n ~ ( ~ , n

q E (-~, 0), i = 0 . . . . . k-1. -I

) with

I ~ i ( 0 ) I : I have n e a r q =

0 the form

~i(n) = ~i(0) + xn p/q + ~(ns), × ~ 0, p E {I . . . . .

c}, q E { I ,

2}, S > p/q.

~ck m 0 f o l l o w s also d i r e c t l y from Lemma ( A . I . 3 ) . Obviously, an A0-stable method is A0-stable near n = 0, and a convergent method has no m u l t i p l e unimodular roots in n = 0. Hence, as in l i n e a r methods the growth parameters

×7 are nonzero, Lemma (1.3.12) y i e l d s a necessary and s u f f i c i e n t

dition for a linear

a l g e b r a i c con-

convergent method to be A0-stable near n = 0. On the other s i d e , A~

also the growth parameters ×i defined in Section 2.1 are nonzero in l i n e a r methods being not n e c e s s a r i l y c o n s i s t e n t (n 2 replaced by n). Hence, by ( 1 . 6 . 1 3 ) ( i i ) , (2.1.25) with respect to x . ( ~ , n ) y i e l d s a necessary and s u f f i c i e n t for a linear

Lemma

algebraic condition

method to be a s y m p t o t i c a l l y Ao-stable.

The next r e s u l t is also due to Jeltsch [77 ]. (1.6.14) Lemma. A method (1.1.3) is A(O)-stable iff it is Ao-stable , A(O)-stable near q : O, and asymptotically A(O)-stable.

Proof. The necessity of the three c o n d i t i o n s is obvious. For the s u f f i c i e n c y we observe t h a t the a l g e b r a i c f u n c t i o n ~(n) defined by ~(~(~),q) = 0 s a t i s f i e s C(T) = T(n) because

28 a l l c o e f f i c i e n t s of the a l g e b r a i c equation ~(~,n) = 0 are r e a l . Therefore we can restrict

ourselves to the upper h a l f - p l a n e , Imn > O. By assumption, there e x i s t two pairs

of p o s i t i v e numbers, (mO' PO) and (m , p ), such t h a t ~ * ( m o , P o ) : {n E ¢, 0 < In[ S PO' X - mO < argn < ~} C S, o

~*(~

,p~) = {~ E

~, I~I > P~, ~ - ~

< argn

As x(~,n) is i r r e d u c i b l e there are only a f i n i t e larities

<

~} C S.

number of branching points and singu-

hence a l l roots of x(~,~) are continuous in a set {~ E { , - ~ < Ren < O,

0 < Imq < y } , y > O. Accordingly, as (-~, O) c S by assumption, i . e . ,

as a l l roots

are less than one in absolute value on the negative real l i n e there e x i s t s an ml > 0 such t h a t

and the method is A ( ~ ) - s t a b l e with ~ = min{~o,~1,~ }. Lemma ( A . I . 4 0 ) y i e l d s necessary and s u f f i c i e n t

algebraic conditions for A(O)-stability

near n = 0 and, by Lemma ( 1 . 6 . 1 2 ) , f o r asymptotic A ( O ) - s t a b i l i t y , the i r r e d u c i b i l i t y

too. In p a r t i c u l a r ,

of ~(C,n) implies f o r l i n e a r methods (1.1.3) t h a t Pv = I in ( A . I . 3 8 ) .

Therefore we can s t a t e e.g. the f o l l o w i n g c o r o l l a r y to Lemma ( 1 . 6 . 1 4 ) .

(1.6.15) Corollary. (Jeltsch [76b].) ~(~,n)

Let the linear method

(1.1.3)

with the polynomial

= p(~) - no(e) be convergent. Then the following conditions (i) - (iv) are ne-

cessary and sufficient for A(O)-stability. (i) The method is Ao-stable, (ii) The unimodular roots of ~(~) are simple. (iii) If ~ is a unimodular root of p(~) then Re[q(~)/(~p'(~))]

> O.

(iv) If ~ is a unimodular root of

> O.

Further useful s t a b i l i t y

q(~) then Re[p(~)/(~a'(~))]

concepts are those of r e l a t i v e s t a b i l i t y

and of s t i f f

stability: (1.6.16) D e f i n i t i o n . Let ~ be the largest star into which the principal root ~1(n) of the consistent method

~=

(1.1.3)

{n c ~, l ~ i ( ~ ) l

has an analytic continuation. Then

< l ~ 1 ( n ) I , i = 2 . . . . . k}

is the region of relative stability.

2g Notice t h a t l~1(n) I is not n e c e s s a r i l y bounded by one i n ~ , h e n c e r e l a t i v e s t a b i l i t y deals also with unstable d i f f e r e n t i a l

equations. Obviously, a necessary and s u f f i c i e n t

c o n d i t i o n f o r a c o n s i s t e n t method ( 1 . 1 . 3 ) to be r e l a t i v e l y hood o f n = 0 is t h a t i t

stable in a ( f u l l )

neighbor-

is ' s t r o n g l y D-stable' in n = 0 which means t h a t a l l roots of

~ ( ~ , 0 ) / ( ~ - I) = po(~)/(~ - I) are less than one in absolute value. (1.6.17) D e f i n i t i o n .

(Gear [69 ] , Jeltsch [76b, 77 ] . ) Let

RI = {n c ~, Ren < - a } , R2 = {n C ~, Ren ~ -b, llmnl < c}, R3 = {n c C, IRenl < b, llmnl < c}. Then a convergent method is stiffly stable iff there exist positive numbers a, bs c such that (i) RI u R2 C ~ and R3 C ~R,, (ii) the method is Ao-stable. Condition ( i i )

is introduced here in order to deal with the demand of s t i f f

stability

in the o r i g i n a l meaning of Gear [69 ]. (1.6.18) Lemma. ( J e l t s c h [76b].) If a convergent linear method ( 1 . 1 . 3 ) satisfies

(1.6.17)(i) then it is Ao-stable hence stiffly stable. Proof. We have to show t h a t (-b, O) c ~ and reconsider the polynomial ~(z,n) = r ( z ) -

ns(z) introduced above. As the method is convergent and s t i f f l y

stable r ( z ) and s(z)

have only roots with Rez ~ O, and ak_ I = 2b k. For an n* E (-b, 0), ~ ( z , n * ) i s a

poly-

nomial with p o s i t i v e c o e f f i c i e n t s and we have to show that i t s roots l i e in the l e f t h a l f - p l a n e , Rez < O. C l e a r l y ~(z,n*) has no p o s i t i v e roots. Moreover, ~(O,n*) = a 0 - n*b 0 m 0 since otherwise p and s in ~(¢,n) = o(~) - no(t) would have a common f a c t o r . Finally,

l e t z be a r o o t of ~(z,n*) with Rez~ 0 and Imz

> O. Then 7 is a r o o t , too.

But then ¢ = (z + 1 ) / ( z - I) and ~ are two roots of ~(¢,n*) of the same modulus. Hence n* does not belong to the region of r e l a t i v e s t a b i l i t y sequently, the roots of ~(z,n*) l i e in the l e f t

which is a c o n t r a d i c t i o n . Con-

h a l f - p l a n e , Rez < 0 and the roots of

~(¢,n*) are less than one in absolute value f o r n* c (-b, 0). (1.6.19) Lemma. (Jeltsch [77 ] . ) A convergent method (1.1.3) i s s t i f f l y

stable i f f

the

following three conditions are fulfilled: (i) pO(~) has the single unimodular root ~ = I. (ii) The method is Ao-stable. (iii) There exists a p > 0 such that {n C ~, In + Pl < P} C S for the method with the polynomial ~,(~,n) : n ~ ( ~ , n - 1 ) .

30 Proof, As already mentioned, the f i r s t rectangle R3 C ~ .

c o n d i t i o n is e q u i v a l e n t to the existence of a

The necessity of c o n d i t i o n ( i i )

is t r i v i a l

and on the other side

t h i s c o n d i t i o n implies the existence of a set R2 c S f o l l o w i n g the pattern o f Lemma ( 1 . 6 . 1 4 ) . Condition ( i i i )

finally

is e q u i v a l e n t to the existence of a set RI c ~ be-

cause I p(e ie

I)

=

-

I ~

-

-

sine i

sine

2p(I - cose)

, lim e

~,

÷ 0 1 - cose

and thus the s t r a i g h t l i n e s Ran = - I/2p < 0 are mapped onto the c i r c l e s n = p(e 18- I) by n ÷ I / n . Notice t h a t the f i r s t

two c o n d i t i o n s o f Lemma ( A . I . 5 3 ) - spoken out f o r ~ , ( ~ , n ) -

are e q u i v a l e n t to asymptotic A ( x / 2 ) - s t a b i l i t y

by C o r o l l a r y ( A . I . 2 1 ) and t h a t the p o l y -

nomial ( A . I . 5 2 ) appearing in the t h i r d c o n d i t i o n is l i n e a r i f the method ( I . 1 . 3 )

is

l i n e a r , Hence t h i s l a t t e r c o n d i t i o n is empty in l i n e a r methods and, a c c o r d i n g l y , the disk c o n d i t i o n ( I . 6 . 1 9 ) ( i i i )

and asymptotic A ( ~ / 2 ) - s t a b i l i t y

are e q u i v a l e n t in t h i s

case. Thus we can s t a t e : (1.6.20) C o r o l l a r y . Let the method (1.1.3) be linear, convergent, strongly D-stable

in n : O, and Ao-stable. Then it is stiffly stable iff it is asymptotically A(7/2)stable.

1.7. S t a b i l i t y

on the Ima~inar~ Axis

Methods with a large s t a b i l i t y

i n t e r v a l on the imaginary axis are of p a r t i c u l a r

i n t e r e s t in the s o l u t i o n of d i f f e r e n t i a l

systems of second order by the class of i n -

d i r e c t methods studied in Section 1.4. The f o l l o w i n g n o t a t i o n has become customary in the meanwhile here.

(I.7.1)

Definition.

A multistep multiderivative method is Ir-stable if

{in,

- r < n

< r} C S, 0 < r ~ ~. Recently, Jeltsch and Nevanlinna [81 , 82a, 82b] have developed an a l g e b r a i c comparison theory f o r numerical methods with respect to t h e i r s t a b i l i t y

regions which allows

the treatment of I r - s t a b l e methods from a r a t h e r general p o i n t of view. This technique uses as a fundamental tool r e s u l t s on the shape of the ' o r d e r s t a r '

having been found

by Wanner, H a i r e r , and Norsett [78a] in the necessary global form. L o c a l l y , i . e . ,

in

a neighborhood of n = O, the order s t a r is described by Lemma ( A . I . 1 7 ) , In t h i s section we give a survey on the present s t a t e of knowledge in I r - s t a b i l -

31 i t y . For the proofs however the reader is referred to the o r i g i n a l c o n t r i b u t i o n s . As the class of single-step m u l t i d e r i v a t i v e methods coincides with the class of Runge-Kutta methods f o r the test equation y' = ~y, the below presented results hold also l i t e r a l l y f o r these l a t t e r methods. Let us f i r s t r e c a l l that a method is A-stable i f f teristic

o

{n C $, Ren < O} ~ S. The charac-

polynomial ~(~,n) of a single-step m u l t i d e r i v a t i v e method is l i n e a r with re-

spect to ~ hence we have S = ~ here. Thus a method of t h i s class is I -stable i f i t is A-stable. The same is true f o r l i n e a r multistep methods, too, by remark ( i ) a f t e r Corollary ( A . I . 2 4 ) . In the case of general multistep m u l t i d e r i v a t i v e methods the implication of I - s t a b i l i t y

by A - s t a b i l i t y is ruled by Lemma ( A . I . 5 3 ) .

As concerns the implication of A - s t a b i l i t y by I - s t a b i l i t y ,

the f o l l o w i n g results

are due to Wanner, Hairer, and Norsett [78b]:

(1.7.2) Theorem. A k-step ~-derivative method (1.1.3) of order p is A-stable if it is I-svable and p ~ 2~- I.

(1.7.3) Theorem. A k-step ~-derivative method (I.1.3) of order p is A-stable if it is p ~ 2~- 3, and the coefficients of the leading polynomial Sk(n) have alter-

I-stable,

nating signs. In p a r t i c u l a r , an I -stable consistent l i n e a r multistep method is A-stable which has been proved in an independent way by Jeltsch [78a]. Now, r e c a l l i n g the r e s u l t of Dahlquist [63 ] namely that an A-stable l i n e a r m u l t i step method has order p ~ 2 and that the trapezoidal rule has the smallest error constant ×p, cf. ( 1 . 3 . 6 ) , among a l l A-stable l i n e a r multistep methods of order two, we obtain immediately the f o l l o w i n g r e s u l t , see also Jeltsch [78a].

(1.7.4) Corollary. (i) An I-stable linear multistep method has order p ~ 2. (ii) Among all I-stable

linear multistep methods of order p = 2 the trapezoidal rule

has the smallest error constant. Example (A.4.7) due to Jeltsch [78a] shows that a nonlinear consistent and I -stable method is not necessarily A-stable. The generalization of Dahlquist's r e s u l t to nonlinear multistep methods is known as the Daniel-Moore conjecture, cf. Daniel and Moore [70 ]. I t was proved by Wanner, Hairer, and Norsett [78a]. For the presentation we recall that the c h a r a c t e r i s t i c polynomial, (1.7.5)

k

~(~,n) : ~i:O~i(n)~

i

: ~j:OPj(~)n j

,

is always assumed to be i r r e d u c i b l e , cf. Section 1.1.

32

(I.7.6) Theorem.

Let the k-step k-derivative method satisfy

Ok(O) ~ 0

and ( ~ / ~ ) (0 ,I )

O. (i) If the method is A-stable then p <= 2~ and

Xp

(ii) The error constant

l×pl > l×;I, ×p*

sgn(xp) : (-I)k

for p : 2k.

of an A-stable method of order p = 2~ satisfies

= ( - 1 ) k ( k ,. ) 2 / [ (. 2 k ) I. ( 2 k +. 1 ) I ]

p = 2k.

(iii) Among all A-stable methods of order p = 2k the diagonal Pad~ approximants, cf. (I.5.1) and

(1.5.3),

have the smallest error constants,

Xp.

Theorem ( 1 . 7 . 2 ) and ( 1 . 7 . 6 ) y i e l d immediately the f o l l o w i n g g e n e r a l i z a t i o n of C o r o l lary (1.7.4):

(1.7.7) Corollary.

(i) An I-stable k-step k-derivative method has order p ~ 2k.

(ii) Among all I-stable k-step k-derivative methods of order p : 26 the diagonal Pad¢ approximants have the smallest error constants. A f t e r having s t a t e d the r e s u l t s concerning i m p l i c i t explicit

methods l e t us now t u r n to

methods. As concerns l i n e a r methods, the only c o n s i s t e n t and e x p l i c i t

step method is the e x p l i c i t

single-

Euler method, ( 4 . 2 . 1 ) w i t h m = O, which is not I r - s t a b l e

f o r any r > O. For ~ = I and k = 2 the l e a p - f r o g

method o f o r d e r p = 2 w i t h the p o l y -

nomial

~(~,n)

= 2

I - 2n~

has the l a r g e s t s t a b i l i t y

interval

I r = {in, -r < n < r} £ S w i t h r = I as the f o l l o w i n g r e s u l t o f J e l t s c h and Nevanlinna [81 ] r e v e a l s : (I.7.8)

Theorem. I f

( 1 . 1 . 3 ) is an explicit convergent k-step k-derivative method then

Ik i ~ or T k = ~ and the characteristic polynomial x.(~,n) : 2

where

~(~,q)

has a factor

_ 2i~TL(_ i n / ~ ) ~ + ( - I ) k

Tk(~) = coskarccos~

is the Tschebyscheffpolynomial of degree ~.

Observe t h a t T2v(O) = ( - I ) v hence these methods are not convergent f o r even k because 0 ¢ S in these cases. The next r e s u l t concerns the case ~ = I and k = 3,4, and is proved by an e x p l i c i t c o n s t r u c t i o n o f the d e s i r e d methods; see J e l t s c h and Nevanlinna [81 ] .

33 (1.7.9) Theorem. For every r E [0, I) and k = 3,4 there exists an explicit linear

Ir-stable k-step method of order p = k.

(1.7.10) Theorem. An explicit linear k-step method of order if

p = k

cannot be I -stable r

k : I mod 4.

Proof. See J e l t s c h and Nevanlinna [81 ] . However, f o r any r < ~ there e x i s t s an i m p l i c i t

l i n e a r I r - s t a b l e 4-step method of

order p = 6; c f . Dougalis [79 ] and Lambert [73 , pp. 38, 39]. On the other side, f o r e x p l i c i t

s i n g l e - s t e p m u l t i d e r i v a t i v e methods van der Houwen

[77 ] has proved: (1.7.11) Theorem. If an explicit single-step ~-derivative method is Ir-stable then r ~ 2 [ ~ / 2 ] . The equality sign is attained for ~ odd. Finally,

the question f o r methods w i t h maximum s t a b i l i t y

i n t e r v a l on the imaginary axis

is answered completely f o r ~ = 1,2 by the f o l l o w i n g r e s u l t of J e l t s c h and Nevanlinna [82a, 82b]. Here, the methods are not n e c e s s a r i l y e x p l i c i t again.

(1.7.12) Theorem. (i) If a k-step ~-derivative method is Ir-stable and p > 2~ then

r ~ r~,op t = (ii) If

p = 2~

{

~,

~ =

I,

I/T5, ~ = 2.

and I r C S with r > r~,op t then the error constant ×p satisfies

I(I IXpl ~

- (3/r2))/12,

~ : I,

(I - ( 1 5 / r 2 ) ) / 7 2 0 ,

c = 2.

(iii) The only method with ~ = I and l ~ C S is the Milne-Simpson method (1.2.16), The only method

with ~ : 2 and

I~

C

~ is the method with the polynomial

x(~,n) : (~ - I) 2 - ~n(~ 2 - I) +~r~n2(~ 2 - 8~ + I ) .

For C : I t h i s r e s u l t was proved by Dekker [81 ] in an independent way.

II.

D i r e c t M u l t i s t e p M u l t i d e r i v a t i v e Methods f o r D i f f e r e n t i a l

2.1, M u l t i s t e p Methods f o r Conservative D i f f e r e n t i a l

In Section

(1.4)

the i n i t i a l

Systems of Second Order

Systems

value problem (1.4.1) was transformed i n t o a f i r s t

order problem of twice as large dimension before the numerical treatment which then has provided an approximation of the s o l u t i o n of the o r i g i n a l problem and of i t s d e r i v a t i v e simultaneously. In t h i s chapter we consider d i r e c t approximation schemes w i t h out a - p r i o r i

transformation.

For the general i n i t i a l

w i t h conservative d i f f e r e n t i a l (2.1.1)

y"

: f(t,y),

value problem of second order

equation,

t > 0, y(0) : Y0' y ' ( 0 ) : y~,

a m u l t i s t e p method can be w r i t t e n f o r m a l l y in the same form as in Section 1.1o However, we p r e f e r a s l i g h t l y tives

modified representation in which the even and odd t o t a l d e r i v a -

f ( J ) of f are summed up separately: Z~:0Pj(T)(At2C)2)Jf~ - 2 ) ( v n ) + AtZ~:0P3(T)(At2e2)Jf~-1)(Vn ) =-

(2.1.2)

Z~=0~i(At2c)2)Tif~-2)(Vn) + z~tZ~=0Ol(Z~t2C)2)JTif~-1)(Vn) : 0,

n = 0,I,...

Here we have to i n s e r t k i ~k , i pj(~) = ~i=O~ji c , p~(c) = Li=O~ji c ,

j = 0,. . . , ~,

(2,1,3)

~i(n)

_j=0~ji

~J

,

o#(n) = ~ ~* ~j Lj= 0 j i ~ ,

and @ = ~/~t is again the d i f f e r e n t i a l f(-1

)(Vn)

i : 0,...,k,

operator. Furthermore, f ( - 2 ) ( v n) m Vns and

~ wn plays the r o l e of an approximation to Yn" , The t o t a l d e r i v a t i v e s of f

are again to be expressed by p a r t i a l

d e r i v a t i v e s of f using the recurrence formula

(1.1,2). tn order to overcome the d e f i c i e n c y t h a t the scheme (2,1,2)

is only one recurrence

formula f o r the two unknown sequences {Vn}n= k and { Wn}n=k ~ we have three p o s s i b i l i t i e s : ( i ) Put p~(~) ~ 0, j = 0 . . . . . ~, i , e . , (ii)

o#(n) z 0, i : 0 . . . . . k.

Introduce a f u r t h e r scheme of the same type such t h a t - besides other conditions

described below - (Vn+k, Wn+k) can be computed from (v v , w ), ~ = n . . . . ,n+k-1. (iii)

Choose a f i n i t e

d i f f e r e n c e approximation & t - 1 ~ i ( T ) y n to Yn+i and replace Wn+i

by a t - I T i ( T ) V n , i = 0 . . . . ,k. Rather few is known on the t h i r d way hence i t shall not be discussed here although i t

35 allows a simple g e n e r a l i z a t i o n to nonconservative d i f f e r e n t i a l

systems. The second way

leads to numerical schemes of NystrQm type which are considered in Section 2.4. In t h i s and the next two sections we study the f i r s t

case, i . e o , numerical schemes of the

form

x(T'at202)f~-2)(Vn ) z v~ Lj:0Pj ,T~&t2@2~Jf(-2) ~ J~ J n (Vn) (2.1.4) k 2 2 i (-2) = Zi=O~i(&t e )T fn (Vn) = O,

n = 0,I , . . .

For ~ = I we obtain l i n e a r m u l t i s t e p methods f o r conservative d i f f e r e n t i a l second order which have been proposed f o r the s o l u t i o n of dynamic f i n i t e

systems of element equa-

t i o n s e.g. by Bathe and Wilson [76 ] , Dougalis [79 ] , and Gekeler [76 , 80 ] . However, i f c > I then the numerical approximation wn to y ' ( n A t ) does not appear in (2.1.4) only i f the i n i t i a l (2.1.5)

y"

value problem is of the form

: Ay + c ( t ) ,

t > O, y(O) = YO' y'(O) = y~,

where A is a constant m a t r i x . Therefore the a p p l i c a b i l i t y

of m u l t i s t e p multiderivative

methods is r e s t r i c t e d to t h i s special case; c f . e.g. Baker et a l .

[79 ].

In analogy to (1.1.5) we assume henceforth t h a t (2.1.6)

~Ok m O, pC(S) { O, and CO(q) ~ O,

and we suppose again without loss of generality that the characteristic polynomial ~(~, 2) defined by (2.1.4) The d i s c r e t i z a t i o n (2.1 7) •

d(at,y)(t)

is i r r e d u c i b l e with respect to ~ and n2. e r r o r of the method (2.1.4)

is now

= ~ Lj=0Pj ( T ) A t 2 JY' ( 2 J ) r t )

and the method is consistent i f there exists a constant r not depending on at such that lld(At,u)(t)ll ~ r~t p+2

v u E cP*(IR;Rm)

f o r a p E fN and p, = max{p+2,2c}, the maximum p being the order of the method. (2.1.8) Lemma. If the method (2.1.4) is consistent of order p then

~ (p+3)/2]At2jpj(T)iiu(2J)(t)llv u E cP*(IR;IRm) lld(At,u)(t)ll ~ rat p+I t+kAt i IIu(P+2)(T)IIdT+Zj=[ where £ does not depend on t , a t , u, and m.

36 Proof. I t suffices to prove the assertion for p ~ 2 ~ - I . In the same way as in Lemma (1.1.8) we substitute into (2.1.7) the Taylor expansions ~p+1-2j ~(iAt) I }At u(2J)(t+iAt) : ~v=0 . v u(2J+v)(t) + (p+1-2j)! b (iAt-~)P+I-2Ju(P+2)(t+T)dT and obtain ~ ~p+1-2j~k iv)At2J+vu(2J+v) d ( A t , u ) ( t ) = Lj=0L~=0 ~Li=0~ji 7 (t) (2.1.9)

k At2J iAt + Zj=0Zi=0mji (p+1-2j)t ~ (iAt-T)P+I-2Ju(P+2)(t+T)d~"

The assumption that the method is consistent of order p implies

(2.1.10)

Z~ ~p+1-2j~k vi~.)At2J+vu(2J+v)(t) j=0Lv=0 kLi=0mji ~p+1~min{[u/2] ~} k i u-2j ~u=0Lj=0 ' Zi=0~ji ~

At~u(U)

(t) = 0

where [u] denotes the largest integer not greater than u. This yields kAt C k d ( A t , u ) ( t ) = I [~j=0~i=0~ji (p+1-2j). At2j '(iAt-T)~ +I-2J]u(p+2)(t+T)dT or

k IId(At,u)(t)II < (Zj=0Zi=01~jil =

+ 2 (P

-'J)"

t+kAt i.+o~ )At p+I f 11u~v "J(T)lld~ t

which proves the assertion. From (2.1.10) we immediately obtain: (2.1.11) Lemma, The method (2.1.4) is consistent of order p iff ~min{[~/2],~}~k i ~-2j j=0 Li=0mji T ~ .

= 0,

= O,...,p+1.

In p a r t i c u l a r , the method (2.1.4) is consistent i f f the following conditions for the consistence of linear multistep methods are f u l f i l l e d , (2,1.12) o0(1) = p~(1) = p~'(1) + 2P1(1) = O, The analogue to Lemma (1,1.15) now reads as follows: (2,1.13) Lemma. The method (2.1.4) is consistent of order p iff the characteristic

37 polynomial ~(~,n 2) satisfies

x(eAt,At 2 ) = x p A t p+2 + C(AtP+3), At ÷ 0, ×p ~ 0.

Again a study of the t r i v i a l

equation y "

= ~ shows that we must suppose t h a t

p~'(1) = - 2 P i ( I ) ~ 0 and we may s t i p u l a t e again that (1.1.14) holds, i . e . , P I ( I ) = - I . The conditions (2.1.12) and (1.1.14) together imply that ~ = I is a root of ~(~,n 2) 2 2 f o r n = 0 which has e x a c t l y m u l t i p l i c i t y two. Therefore n = 0 is no longer contained in the s t a b i l i t y

region S defined by ( 1 . 2 . 7 ) . For t h i s reason we have to weaken the

concept of absolute s t a b i l i t y tial

a p p r o p r i a t e l y in d i r e c t multistep methods f o r d i f f e r e n -

systems of second order.

(2.1.14) D e f i n i t i o n . The stability region S of a method (2.1.4) consists of the n

Z

E ~ with the following properties:

(i)

~k(n 2) ~ O, n2 E S ~ C,

(ii) all roots ~i(n)

of ~(~,n 2) satisfy i ~ i ( n ) i ~ I ,

(iii) all roots ~i(n)

of ~(~,n 2) with l ~ i ( n ) 1 = I have multiplicity not greater

than two,

As in Chapter I a method with 0 E S is called z e r o - s t a b l e . Henrici [62 , Theorem 6.6] has proved that consistent and zero-stable l i n e a r multistep methods are convergent in the c l a s s i c a l sense. A generalization of t h i s r e s u l t to nonlinear methods is not d i f ficult

f o r the r e s t r i c t e d class of applications (2.1.5) therefore consistent and zerostable methods are again called convergent below. Observe that then 0 £ ~S by Lemma (I.3.3). For the roots ~i(n) of the polynomial ~(~,n 2) we obtain by i m p l i c i t d i f f e r e n t i a t i o n and Theorem ( A . I . 4 ) :

Case ( i ) . If ~i(O) is a simple root of ~(~,0) then (p~(~i(O)) ~ 0 and) ~i(~) = ~i(O) + xi n2 + d{n4),

~-* O,

(2.1.15) xi = [- pl/p~](~i(O)). Case ( i i )

•

i Ri I f ~i(O) is a double root of ~(~,0) then (po(~i(O)) = O, PO (~i ( 0) )

Ci,i+1(n ) : ~i(O) + ^Xi(±~ ) + ~.~n2 + ~( n3), (2.1.16) ^2 ×i = [-2Pl/PO'](~i(O))' A

and i f ×i ~ 0 then

~ ÷ O,

m O)

38

(2.1.17)

~i = [ ( 2 0 ~ " ~ I - 600" P i ) / 3 ~ 0 ' 2 ] ( ( i ( 0 ) ) "

Accordingly, a convergent method has e x a c t l y two r o o t s , ~1(n) and 52(n) = 51 (- n), c a l l e d again the p r i n c i p a l roots which have the property (2.1.18)

~1,2(n) = 1 ± n + ~(n2),

(2.1.19) Lemma. Let

n ~ O.

0 E S then the method defined by

~(~,n2) is consistent

of order

p iff

(2.1.20)

Proof.

~ l ( n ) = e n - ×pn p+I + d(nP+2), ×p = O,

In a s u f f i c i e n t l y

n + O.

small neighborhood of n = 0 we define

(2.1.21) ~(~,n 2) = ~(C,n2)/[(C

- ~l(n))(~

- 51(-n))]

and then obtain (2.1.22) ~ ( I , 0 )

= p6'(1)

= - 2P1(1) = 0

because by (2.1.12) ~(~,0)

= PO(1)

#~'(1)

=---E----(~

+ p~(1)(~ - I

)2

p~'(1)

I)

+ ~ ( C

+ ~((~

- I)3),

-

- 1) 2 + ~((C

- 1) 3 ) n + O,

I f the method is c o n s i s t e n t of order p then Lemma (2.1.13) y i e l d s by a s u b s t i t u t i o n of e n f o r ~ in ~ ( ( , n 2) (2.1.23)

~(en,n 2) = (e n - ~ l ( n ) ) ( e n - ~1 ( - n ) ) ~ ( e n , n 2) = Xpn p+2 + ~(qp+3),

n ÷ O.

But, by ( 2 . 1 . 1 8 ) , en - ~ l ( - n ) therefore

(2.1.22)

then a s u b s t i t u t i o n

= (1 + n) + g(n 2) - (1 - n) + ~(n2), and (2.1,23)

prove ( 2 . 1 . 2 0 ) .

i n t o the f i r s t

On the other s i d e , i f

n + O, (2.1.20)

holds

equation of (2.1.23) y i e l d s

~(en,n 2) = ~ ( e n , n 2 ) ( e q - ~ l ( - n ) ) ( x p n P + l + ~ ( n P + 2 ) )

= ~(en,n 2 )(2×pn p+2 + a(nP+3)), q + O,

39 hence the method has o r d e r p b y Lemma (2.1.13). Note t h a t ×p = ×/2 where x is the e r r o r constant introduced by Henrici [62 , p.296]; see also J e l t s c h and Nevanlinna [81 , (2.8) and subsequent remark]. Moreover, Lemma (2.1.19) and Lemma (1.3.5) have e x a c t l y the same form although (or b e t t e r because) S is defined in two d i f f e r e n t

ways. The approximation properties of ;1(n) are thus the

same w i t h respect to n in both times. By (2.1.18) we obtain the same equation as in the proof of Lemma ( I . 3 . 1 2 ) , I ~ I ( ~ ) I = I + Re(n) + d ( I n l 2 ) ,

n ÷ O.

S u b s t i t u t i n g again n = p(1 + e i ¢ ) , 0 ~ ~ < 2~, we f i n d t h a t - w i t h respect to n2 - the domain o~) = {n 2 c C, 2 satisfies k c ~\S

= p2(i + ei¢)2 , 0 ~ p < PO' 0 ~ @ < 2~}

f o r some PO > 0 s u f f i c i e n t l y

domain {n 2 £ C, IB - arg(n2)l a half-line

small. But ~ \ ~

contains no angular

~ ~} f o r a p o s i t i v e ~ and f o r the angle ~ = ~ not even

in a neighborhood of zero:

(2.1.24) Lemma. The stability region S of a convergent method defined by ~(~,n 2) does not contain in a neighborhood of zero a domain {n 2 £ $~ IB - arg(n2)[ ~ ~} with ~ 0 for B ~ ~ and ~ > 0 for B : ~.

This r e s u l t corresponds d i r e c t l y to the t h i r d assertion of Lemma (1.3.12) via the mapping n ÷ n 2 . The f i r s t two assertions of t h i s lemma have here a somewhat more complicated form. Let ×7 : × i / ~ i ( 0 ) ' Xj = × j / ~ j ( O ) ,

Xj = Xj/~j(O)

be the growth parameters of the simple unimodular roots ~i(O) and the double unimodular roots ~j(O) of ~(~,0) r e s p e c t i v e l y where the constants × i ' ~ j ' (2.1.15),

and ~j are defined in

( 2 . 1 . 1 6 ) , and (2.1.17).

(2.1.25) Lemma. Let the method defined by ~(~,n 2) be convergent and let all growth parameters Xi* and Xj be nonzero. Then (i)S contains the negative real

n2-1ine in a neighborhood of zero iff all simple and

all double umimodular roots of ~(~,0) satisfy

Re(x#) (ii)$ \

> O, Im(~])

: O, Re(~])

- (~])2

> O,

S contains the negative real line in a neighborhood of zero iff some simple

40

~(~,0)

or some double unimodular roots of

Re(xT) < 0

or

Im(~)

m0

or

satisfy

R e ( ~ ) - ~×j)

< O.

Proof. With respect to simple roots the statement is the same as in Lemma (I.3.12) with q replaced by n2", cf. (2.1.15). For the double unimodular roots we observe that (1.3.10) holds, i . e , ,

l~j,j+1(n) I : I + Re(~)Re(±n)

Im(~)Im(±n)

+ 5(In12),

n ÷0,

and, furthermore, if Im(~) = 0 and n2<0, lCj,j+1(n)l = 11 +~(±n) +'7×i n2I + ~(lnl 3) = I

From these two relations More e x p l i c i t e l y ,

writing

2-1(Re(#~) - ( ~ ) 2 ) n 2 + G(Ini3),

+

n +0,

the remaining assertions follow immediately, id for the i d e n t i t y mapping then we have f o r I m ( ~ )

Re(~) - (~)2=Re{[((2p~,,p

= 0

1_6p~,pl)i d + 6PlP~,)/(3p6,2(id)2)](~j(O))}.

A~

The growth parameters ×T and ×j are nonzero i f all unimodular roots of po(~) are not roots of p1(~); cf. (2.1.15) and (2.1.16). This condition is f u l l f i l l e d at least in two cases: ( i ) I f the method is l i n e a r , i . e . , i f x(~,n 2) = po(~) + n2p1(~) because ~(~,n 2) is irreducibel. ( i i ) I f ~1,2(0) = I is the unique double unimodular root of ~(~,0) because a convergent method s a t i s f i e s P i ( I ) ~ 0 by (2.1.12) and 0 E S. A solution of the test equation y " = - ~2y, 0 ~ ~2 c IR, neither decreases nor increases with increasing t but o s c i l l a t e s . In the long-range solution of a problem (2.1.15) the numerical scheme should have a similar behavior therefore Lambert and Watson [76 ] have introduced the following notation: (2.1.26)

D e f i n i t i o n . A convergent method defined by ~(~,n 2) has the periodicity inter-

val I-s, 0], 0 < s, if all roots ~i(n) of ~(¢,q2) satisfy I~i(n) I : I for n 2 E [-s, 0].

A method with the polynomial

k i is called syrr~etrio ~(~,n 2) = ~j=oPj ( ~)q2J" : ~i=Ooi(n2)¢

if (2.1.27) pj(~) : c k p j ( - I ) ,

j : 0 . . . . . &,

41 then ~i(n 2) = ~k_i(q2), i = 0 . . . . . k, and ~(C,n 2) = ~ k ~ ( c - l , n 2 ) . Lambert and Watson [76 ] have shown that a linear multistep method (2.1.4) with a p e r i o d i c i t y i n t e r v a l is necessarily symmetric. Moreover, i f the p r i n c i p a l root ~1,2(0) = I is the only double root of ~(~,0) = pO(~) then t h i s necessary condition is also s u f f i c i e n t .

The f o l l o w i n g

lemma generalizes t h i s r e s u l t and s i m p l i f i c a t e s at the same time Lemma (2.1.25) essentially

f o r l i n e a r symmetric methods.

(2.1.28) Lemma. (Jeltsch [78b].) Let the linear multistep method defined by ~(~,q2) be convergent and syn~netric, Then it has a periodicity interval iffall growth parameters

^. £^.~2 Xj of the double unimodular roots cj(O) satisfy t × j } > O. Proof. We quote the proof because i t gives some i n s i g h t in symmetric methods and proceed in several steps assuming throughout convergence and symmetry. ( i ) Let ~* be a root of pO(~) then ~ , - I is also a root of po(~) because pO(~) = k -I pO(~ ). Accordingly, because 0 c S we obtain that a l l roots of po(~) are unimodular, 2 i o e . , q = 0 belongs to the p e r i o d i c i t y i n t e r v a l . ( i i ) Let ~.(0) be a simple (unimodular) root of p~(~) and consider ~.(~) near n = O, 2 J -2 u 2 J . q < O. Then ~j(n) is a root of ~(~,~ ) because ~(~,n ) has real c o e f f i c i e n t s and consequently ~ ( n ) -I is also a root of ~(~,n 2) because of the symmetry. Let J

Cm(~) = 11Ejj(n) then we obtain from l~j(O)I : I , i . e . , ~j(O).

~j(O) = e i¢andcjj(O) - I : e i@, that ~m(O) :

I f m m j then ~j(O) is a double unimodular root which is a contradiction hence

m = j and, consequently, ~j(q) : I / ~ j ( q ) or I ~ j ( n ) l

= I . Therefore the roots emanating

2

from simple unimodular roots of ~(c,O) s a t i s f y l ~ j ( n ) I = I f o r some ~ ~ ~ O, 2 no < 0. ( i i i ) Let ~j(0) be a double (unimodular) root of ~(~,0) then ~j(n) is analytic in a

neighborhood of zero by Lemma (A.I.3) and we obtain ~j,j+1(n)

: ~j(O)(1 + ~ ( ± n )

with the growth parameter ~

+ ~(n2)),

T1 + O s

m O. By the I m p l i c i t Function Theorem ( c f . e.g. Dieu-

donn~ [60, § 10.2] we f i n d that n ÷ ~j(n) is i n v e r t i b l e and an i m p l i c i t d i f f e r e n t i a t i o n of ~(~,q2(C)) with respect to c reveals that in a neighborhood of ~j(O) ~ ~j

(2.1.29) n2(~) ~ m(~) = K(~ - ~j)2 + ~((~ _ ~j)3),

÷~j,

where K = I / ~ 2 C (n) s a t i s f i e s [ c . ( n ) l = I near n2 = O, n 2 < 0 i f f the function ~ J • j J ' @ ÷ m(~je '~) maps a s u f f i c i e n t l y small i n t e r v a l (-00' ¢0 ) ' ¢0 > O, onto an i n t e r v a l [-n#, 0]. But from po(~) + n2p1(~) = 0 we obtain

42 (2.1.30) 2

= m(~) = _ po(~)/p1(~ )

hence m(~je 1¢) = m(e1@),~j = el@O' @ = @+ ~0' is a real function in ¢ because of the symmetry. Moreover, ¢ = 0 is a double root of m(~je i¢) hence i t is negative in a real neighborhood of ~ = 0 i f f the second derivative with respect to @ is negative in ¢ = O. But by (2.1.29) or (2.1.30) we obtain (2.1.31) ~2m(~jei@)/B¢21¢=O : - 2 < ~ : - 2 / ( ~ ) 2 < 0 by assumption which proves the a s s e r t i o n . In the attempt to g e n e r a l i z e Lemma (2.1.28) to n o n l i n e a r m u l t i s t e p methods (2.1.4) we obtain again a f u n c t i o n m which s a t i s f i e s urally fulfilled

no longer hence i t

(2.1.29) and ( 2 . 1 . 3 1 ) . But (2.1.30) is nat-

is more d i f f i c u l t

to show t h a t w(~je i@) is a real

f u n c t i o n near @ = O. A polynomial ~(~,n 2) can have m u l t i p l e roots only in a f i n i t e

number of values

n2; cf. e.g. Ahlfors [53 , § 6 . 2 ] . Therefore, i f a convergent method has a p e r i o d i c i t y interval

[ - S l , 0 ] , s I > O, then always [ - s 2 , 0 1 C ~S f o r some 0 < s 2 ~ s I . I t would how-

ever be i n t e r e s t i n g to know wether the case [-s I , O] = BS can be c h a r a c t e r i z e d a l g e braically. A f t e r these p r e l i m i n a r y r e s u l t s on the s t a b i l i t y

region S we now turn to i n i t i a l

value problems of second order and postpone a study of p r o p e r t i e s of S in special methods u n t i l Section 2.5. We consider the i n i t i a l (2.1.32) y "

value problem

: A2y + c ( t ) + h ( t ) , t > O, y(O) = YO' y'(O) = y~,

where the n o t a t i o n A2 instead of A is used only f o r convenience. As in the f i r s t t e r , the p e r t u r b a t i o n h ( t )

chap-

is neglected in the numerical approximation (2.1.4) and we

obtain the computational device k = ~k i~L (2,1.33) Zi=OOi(At2A2)?iVn - ~. ~? 7 . . ~ I:U

=J:l

lJ

For the d e f i n i t i o n of the polynomials ~ i j ( n )

.~2^2~,.2Jc(2j-2) .(a~ ~ )aL n

n = 0,1 , . . .

see ( 1 . 2 . 3 ) . The e r r o r estimation c o r r e -

sponding to Theorem (1.2.12) has here the f o l l o w i n g form. (2.1.34) Theorem.

(i) Let the (m,m)-matrix A 2 in (2.1.32) be diagonable, A 2 = XA2X -I ,

and let the solution y be (p+2)-times continuously differentiable. (ii) Let the method (2.1.33) be consistent of order p ~ 24 - I ~ith the stability region S. (iii) Let Sp(At2A 2) C R C S where R is closed in ~. Then for n = k,k+1 .....

43

[X-1(yn-Vn)[ ~KRIX-IInAt[At-IIYk-I-Vk-II

nat ] +AtPl [Y(P+2)(~)Id~+nAt0~i~2~-2max lllh(i)Uln .

Proof. Exactly in the same way as in Theorem (1.2.12) we obtain n = k,k+1 . . . . ,

(2.1.35) En = F (At2A2)En_ I + Dn - Hn, where En = ((Yn-k+1 - Vn-k+1) . . . . . (Yn - Vn))T' (2.1.36) Dn = (0 . . . . . O,~k(At2A2)-Id(At,Y)n_k )T, and now 2 2 -I k iv~ r_~2^2~.~2j~(2j-2)~T (2.1.37) Hn : (0 . . . . . O,Ok(At A ) Zi=O T Lj=1oij ~aL " Ja~ "n-k J "

The Uniform Boundedness Theorem (cf. Appendix) yields by assumption ( i i i ) sent case In-IF~ (At2A2)nl : max1_-
in the pre-

<
therefore we obtain ,X-IEnl < ~RnAt[At-I,x-IEk_I,

+ F,X - 1 , A t p i A t l y ( p + 2 ) ( ~ ) , d r

+ at-1~n=k,X-IH

,] .

Now the assertion follows in the same way as in Theorem (2.1.12) observing that ok(n 2) is a polynomial of exact degree ~ in n 2 i f R C S is unbounded. This e r r o r estimation d i f f e r s from that in Theorem (1.2.12) by the m u l t i p l i c a t i o n f a c t o r nat. Moreover, the i n i t i a l e r r o r IYk_ I - Vk_iI is m u l t i p l i e d by At -I therefore the method f o r the computation of the s t a r t vectors Vo,...,Vk_ I must be of order p+1 in order that global order p is obtained. This e f f e c t is due to the occurence of double unimodular roots in convergent methods. However, i f the principal roots, ~1(n) and ~1(-n), are the only roots coalescing to double unimodular roots in the s t a b i l i t y interval I-s, O] then the e r r o r estimation with respect to the i n i t i a l e r r o r can be replaced by a more d i v e r s i f i e d one which seems somewhat more s a t i s f a c t o r y . This is achieved by a s p l i t t i n g o f f of one of the principal roots from the c h a r a c t e r i s t i c polynomial and observing that the r e s u l t i n g modified e r r o r equation has the same propert i e s as in multistep methods f o r f i r s t order systems. (2.1.38) D e f i n i t i o n .

Let x*(~,n)

: ~(~,n2)/(~

- ~1(n)) then a method (2.1.4) is

strongly D-stable in [-S, O] if I-s, O] C S and if all r#~ts ~i(q ) of ~*(~,n) with

44

lq(n)l

= I are simple roots of ~*(~,n) for n2 c I-s, 0].

The polynomial x*(~,n) can be chosen on the entire real n2-1ine with exception of possible poles (which cannot lie in [-s, 0]) as a fixed polynomial of degree k-1 in ~ with coefficients that are continuous in n independently of possible real branching points

of the algebraic function ~(n) defined by ~(~(n),n 2) = 0. (2.1.39) Theorem. Let the assumption of Theorem (2.1,34) be fulfilled but let

R =

[-S, O] and let the method (2.1.4) be strongly D-stable in R. Then for n : k,k+ll...,

IX-1(yn- vn)l ~ IX-IE~_21 + KsIX-11nAt[IAilE~_21 + At-lIE~_1 - E~_21 IY where En 0 = ( Yn-k+2- Vn-k+2 " " '

(-)I dT + nat

max [ilh(i)llln 0si~2c-2

Yn - Vn )T.

Proof. Using the polynomial~*(~,n) instead of ~(~,n 2) the error equation writes as ~(T,At2m2)en : ~*(T,AtA)(T - ~l(AtA))en = d(At,y) n- ~=0Ti~j=isij (At2A2)At2Jh(2j-2) n therefore we can write the corresponding single-step equation in the following way: (2.1.40)

E0n - ¢I(AtA)E~-I

= Fx*(AtA)(E~-I - ~I(AtA)E~-2 ) + D0n - HOR' n = k,k+1, . . . .

Dn O and Hn 0 are here block vectors of the same form as in the proof of Theorem (2.1.34) but with block dimension k-1. By the Uniform Boundedness Theorem we now obtain

IF .(AtA)i S sUP_s< < =2=~uPREINIF .(n) I :<
o - CI ~I (AtA)X-IE~ -I i =< ~RI iX -1 Ek-1

21

(2.1.41) + r[X-1[AtP+liAtly(p+2)(T)[d T + Ix-llnAt2max0~i~2g_2111h(i)ni~]. But i~1(AtA) i ~ I by assumption ( i i i ) and a further iteration yield (2.1.42) IX-1EOI < IX-1E~ -21 n' =

+

of Theorem (2.1.34) hence the triangle inquality

n~R[

] .

.

.

.

.

.

where [ . . . . . . ] is the square-bracketed term on the right side of (2.1.41). Finally,

45 Lemma ( A . I . 8 ) y i e l d s -

<

IX IIEIE I

E _2E÷F tJAIIE 211

A s u b s t i t u t i o n of t h i s estimation into (2.1.42) proves the desired r e s u l t . In Theorem (2.1.39) the i n i t i a l

e r r o r is m u l t i p l i e d by IAI instead by At -I as in Theorem

(2.1.34). This phenomenon was already observed by Dupont [73 ] f o r two-step methods. The difference quotient of the i n i t i a l

error corresponds to the i n i t i a l

condition

y'(O) = y~ of the a n a l y t i c i n i t i a l value problem (2.1.32). A two-step method is always strongly D-stable in S ~IR. But in methods with step number k > 2 the polynomial v*(~,n) is never computed e x p l i c i t e l y . However, i f a method is strongly D-stable in n2 = 0 and i f a l l

roots (or at least a l l unimodular roots) of

the o r i g i n a l polynomial v(~,n 2) are simple in I - s , O) c S then i t is strongly D-stable in I - s , 0]. So t h i s somewhat stronger condition may be used in the general case.

2.2. Linear Multistep Methods f o r D i f f e r e n t i a l Systems with Damping

In t h i s section we consider the real i n i t i a l (2.2.1)

y"

value problem

= a2y + B ( t ) y ' + c ( t ) , t > O, y(O) = YO' y'(O) = y~,

with a s u f f i c i e n t l y smooth matrix B(t) not necessarily commuting with A2, That case of orthogonal damping is studied in Section 2.3. Here, we cannot expect error estimations of that strong uniform character as derived in Chapter I even i f A2 and B(t) are symmetric and negative d e f i n i t e ; c f . e.g. Gear [78 ]. The utmost we can obtain is a e r r o r bound which is uniform with respect to the leading matrix A2and ' c l a s s i c a l ' with respect to B. This error bound remains unaffected by a possible definiteness of B(t) therefore the 'damping' matrix B(t) is allowed to be an a r b i t r a r y matrix in t h i s section. Let us f i r s t

assume that B(t) is a constant matrix, B. Then the o r i g i n a l nonlinear

multistep method (2.1.33) reads for n = 0,1, . . . . (2.2.2)

k ~k T i ~ rAt2A2~At2J(Bv(2j-I) + c(2j-2) )n" Zi=OOi(At2a2)Tiv n = - ~i=O Lj=1~ij ~ J

By means of the d i f f e r e n t i a l equation v ( 2 j ' I ) I

I

is to be expressed in t h i s scheme f o r

j > I by v n and Yn" A f t e r t h i s operation Yn is to be approximated by an expression in

46

V n , . . . , V n + k . This can be achieved only b a a linear d i f f e r e n c e formula because no d i f ferential

equation

is

available for y'(t).

I t thus seems of few p r a c t i c a l

interest

to employ n o n l i n e a r m u l t i s t e p methods in the approximation (2.2.2) and we r e s t r i c t ourselves to l i n e a r m u l t i s t e p methods in t h i s and the next s e c t i o n . We w r i t e f o r s i m p l i c i t y ~k S i Pl (~) = ~i=0 i ~ and introduce f o r every Bi m 0 a f u r t h e r real polynomial ~k y ( i ) ~ u ~i (c) = ~ = 0 ~ ' Then a l i n e a r m u l t i s t e p m e t h o d (2.2.3)

f o r the problem (2,2.1)

is a scheme of the form

p0(¢)v n + At2A2pI(T)Vn = - A t ~ I = U^B I B~ + I .T.(¢)V n - At2p1(¢)Cn 1

In the l i t e r a t u r e

I

n = 0,I

i,,,

mostly the notations p(~) z p0(~) and ~(~) ~ - p1(~) are used in t h i s

c o n t e x t ; see e°g. Lambert [73 ]. The t r u n c a t i o n errors of the method (2.2.3) are d(At,u)(t)

= p0(¢)u(t) + At2p1(T)u''(t)

and

di(At,u)(t) = Ti(T)U(t) - AtTlu'(t),

i = 0 . . . . . k.

(2.2,4) Definition. The method (2.2.3) is consistent if there exists a positive integer p such that for all u E cP+2(]R;JRm) lld(At,u)(t)ll ~ FAtp+2 and lldi(At,u)(t)ll ~ rat p+I,

i = 0,...,k,

where r does not depend on At. The maximum p is the order of the method.

The f o l l o w i n g lemma is a composition of Lemmas (1.1.8) and (2.1.8) hence we omit the proof,

(2.2.5) Lemma. zf the method (2.2,3) is consistent of order p then the composed disoretization error

d*(At,u)(t) = d(At,u)(t) + AtZ~=0BiB(t+iAt)di(At,u)(t) satisfies for all U E cP+2(IR;IRm)

+it+kAt

Id*(At,u)(t)I ~ FAtp ~ t

(lu(P+2)(r)I + NIBIIIt+kAtiU(P+I)(T)I)d~

47 where £ does not depend on t , &t, u, and the dimension m.

In p a r t i c u l a r , (2.2.6) and (2.2.7)

a method (2.2.3) is consistent by (2.1.12) and (1.1.13) i f f

p0(1) = p~(1) = p&'(1) + 2pl(1) = 0, I

Ti(1) = 0 and ~i(1) :

1

,

i : 0,..,ik.

(2.2.8) Definition. The method (2.2.3) for the problem (2.2.1) with damping has the stability region S and it is strongly D-stable in I-s, 0] C S if the corresponding method (2.1.4) for the undamped problem has these properties.

In the following theorem we derive again an error estimation for the i n i t i a l problem (2.2.1) with an additional perturbation h ( t ) , (2.2.9)

y"

value

= A2y + By' + c(t) + h ( t ) , t > 0, y(0) = Y0' y'(0) = y~.

The damping matrix B is assumed to be constant and the linear multistep method (2.2.3) is supposed to be e x p l i c i t with respect to the f i n i t e difference approximations s i ( T ) v ( t ) of y ' ( t + i a t ) . The l a t t e r condition is however no serious r e s t r i c t i o n because cum grano salis- (k-1)-step difference formulas for the approximation of y ' ( t ) can be found which have the same order as k-step formulas for the approximation of y " ( t ) . (2.2.10) Theorem. ( i ) Let the (m,m)-matrix A2 in (2.2.9) be diagonable, A2 = XA2X-I , and let the solution y be (p+2)-times continuously differentiable. (ii) Let the method (2.2.3) be consistent of order p with the stability region S. (iii) Let Sp(At2A 2) C R C S where R is closed in C.

( i ) = 0, i = 0 . . . . . k, and l e t P0(0) + q2p1(0) - ~0(n2) ~ 0 v n2 E R. (iv) Let Yk Then for n = k,k+1, ....

IX-1(yn - Vn) I __
Vk_ll + At p iz~t( 'Y (p+2) (T)I + IBY(p+I ) (T)l)d~ +

nAtlllhllln].

Proof. The error en = Yn - Vn satisfies (2.2.11) ~(T,At2A2)en = - At~=0BiBTi(T)e n + d*(At,y) n - At2p1(T)hn ,

n = 0,1 . . . .

By assumption (iv) we have

Z~=0BiB~i(T)en = Li=0~i°Lu=0Y~ vk ~ nvk-1 (i )T~en = L~=0 ~k-1(~k ( i ) BiB)T~en ~ $*(B)TEn+k-I &i:0 y ~

48 where @.(B)T

(~k (i) ~k (i)B B) = Li=0Y0 BiB . . . . 'Li=0Yk-1 i

and En = (en_k+1, . . . . en )T is the already above introduced block error vector. (2.2.11) is equivalent to the single-step equation X-IEn = (F (At2A 2) + At@(X-IBx))X-IEn_I + X-ID~ - X-IH

n'

n = k,k+1

""'

with the notations D~ : (0 .....0,Ok(At2A2)-Id*(At,Y)n_k )T, Hn = (0 .....0,ak(At2A2)-IAt2p1(?)hn_k)T. Accordingly, IX-IEn i ~ i(F (At2A2) + At¢(X-IBx))n-k+111X-IEk_ll (2.2.12)

+ Z~=k[(F(At2A2) + Atm(X-IBx))n-~ilX-ID ~ - X-IHvi , (2.2.13) IX-IHn [ ~ rAt21x-ll

lllhllln,

and by Lemma (2.2.5) nat (2.2.14) IX-IDol s r I x - l I A t p+I {n_k)At(ly(p+2)(~)i

+ IBy(p+I)(T)I)dT.

The matrix ¢(X-IBx) in (2.2.12) is a (k,k)-block matrix whose last row is ~k(At2A2)-I@*(X-IBx) T and all other rows are zero therefore we obtain (2.2.15)

i¢(X-IBx)i ~ <]X-IBxl.

By (2.2.12), (2.2.13), and (2.2,14) the Theorem is proved i f we show that (2.2.16) [(F (At2A 2) + AL¢(X-IBx))V i ~ ~RV(1 + ~IX-IBXIAL) v,

v = k,k+1 . . . . .

But the Frobenius matrix F (At2A 2) is regular because o0(n 2) is the coefficient of 0 k 2)~ i of thls • matrix and OO(n2) ~ 0 in the characteristic polynomial x(C,n2) ~ Zi=Ooi(n v n2 E R by assumption ( i v ) . Consequently,

(2.2.17) IF (At2A2)-I i ~ sup 2 IF~ (n2)-11 ~ KR" n £R

49 Now, by (2.1.15) and (2.1.17), I(F (at2A 2) + At¢(X-IBx))V[ s vlv-IF (&t2A2)Vl(1 + atlF (At2A2)-III{(X-IBx))V I) wl~-lFJat2A2)Wl(1 +

<~lX-lBXtat) ~,

~

: k,k+l . . . . .

and the Uniform Boundedness Theorem yields Iv-lF(At2A2)v I' ~ sup 2 sup In-lF~ (n2)nl s
(2.2.16).

I t is an open question whether the technical assumption (iv) is necessary here. Cursori l y spoken , i t is introduced because the Kreiss' Matrix Theorem does not apply to F (n 2) in the present situation. In the next theorem we turn again to strongly D-stable methods. Then assumption (iv) can be omitted and the estimation of the propagation of the i n i t i a l error obtains a form corresponding to Theorem (2.1.39). However, these improvements are only obtained by considerable e f f o r t . In particular, the Kreiss' Matrix Theorem supported by the Uniform Boundedness Theorem plays an important role in the estimation of t o t a l l y implicit methods and systems with time-varying damping. (2.2.18) Theorem. (i) Let the (m,m)-matrix A 2 in (2.2.9) be diagonable, A 2 = XA2X -I, let the damping matrix B vary with time, and let the solution y be (p+2)-times contiuously differentiable. (ii) Let the method (2.2.3) be consistent of order p with the stability region S, 2 let it be strongly D-stable in n = O, and let all unimodular roots of ~(~,n 2) be simple fQr n 2 E [-s, 0) C S. (iii) Let &t2[A21 ~ s. (iv) If the method (2.2.3) is implicit with respect to B then let AtlIl X-1BXIII n

~

where the constant ~ is defined in (2.2.36). Then for n = k,k+l . . . . .

[x-l(yn - Vn) I ~ IX-1E~_I[ + KslX-llnAt exp{~lllx-lBxZIInnnt}× [

-

&tP[ (ly(p+2)(~) I+ HIBIIInly(p+l)(~)l)d~+n&tlllhll[n] 0

where En 0 : ( Yn-k+2 - Vn-k+2 " " '

Yn - Vn )T.

For the proof of this theorem we need several auxiliary results: (2.2.19) Lemma. Let assumption (2.2 18)(ii) be fulfilled, let F .(n) matrix of the polynomial ~*(~,q) defined by (2.1.38), and let

be the Frobenius

50 IF.(n) G(n) =

LF~*(n)

]

0

_ I

c1(n)l

'

~en sup

IG(n)nl 2 sup -s~n ~0 n EIN

~ ms•

Proof. We f o l l o w Gekeler [82b], w r i t e s h o r t l y F(n) = F , ( n ) , Z(n) = ( c 1 ( n ) l

- F(n))-I(I

where ~1(q) is the p r i n c i p a l

- F ( n ) ) , n E [ - s , 0), Z(O) = I ,

root defined in Section 2.1. Then we obtain

F(n) n

G(n) n =

and define

0

I

]

2 n

,

(F(n) n - ~ 1 ( n ) n l ) Z ( q )

E I-S, 0],

~1(n)nl

and the Uniform Boundedness Theorem y i e l d s (2.2.20) sup 2 sup ([F(n)nl + e l ( n ) [ n) < = -s~n ~0 n EIN

ms •

Therefore i t suffices to show that

(2.2.21)

SUPn2E[_s,o]SUPnEiN[(F(n)n

- C1(n)nl)z(n)[

~ ms"

Let [-s,

O] = RI ~ R2, RI n R2 = 9, 0 ¢ RI ,

such that the closed set R1 contains in its i n t e r i o r the f i n i t e number of points n2 in which El(q) is a multiple roots of the characteristic polynomial x(~,n 2) and ]Cl(n)I < I. Then we find for n2 E R1 sup 2 n E RI = l(Z~S&F(n)Jcl(q)n-l-J)(F(n) -

I(F(~) n -

< ~ j = <sZj=O[~1(n)] : Ks(IIn order to prove (2.2.21) (2.2.22)

sup 2

IZ(n)l

i)I

[ ~ 1 ( n ) I ) -I <: m*s.

f o r n 2 E R2 i t s u f f i c e s

to show by (2.2.20)

that

~ Ks •

E R2 For the v e r i f i c a t i o n of this bound we observe that by Kato [66 , Theorem 2.1.5]

the

function n ~ (~1(n)I - F(n)) -I is continuous for q2 E R2 \ {0} since no eigenvalues of F(q) coincide with ~1(n) in this set by definition of R2. Therefore we have only to show that Z(n) is bounded near n = 0. For this l e t U(q) be a unitary matrix such that

51 k-1 U(q)F(n)U(n) H = R(n) = [ r i j ( n ) ] i , j = I is an upper t r i a n g u l a r matrix with r11(n) = C2(n) = C1(-n). Then we have near n = 0 (2.2:.23)

[rij(n)[

:< IR(n){

= IF(n)[

< r.

Let e = (1,0 . . . . ,0) T be a column v e c t o r and l e t Z*(n) = ( ~ 1 ( n ) l - R ( n ) ) ( l

- ~1(n)eeT) - I .

Then, o m i t t i n g the argument n, UZUH = I + (I - ~ i ) ( ~ i I and I ( I - ~ i ) ( I

- cleeT)-1{

- R) -I = I + (I - ~ i ) ( I

is bounded near n = O. In o r d e r to prove t h a t

bounded we w r i t e the upper t r i a n g u l a r (2.2.24)

Z* = d i a g ( Z * )

where d i a g ( Z * )

- ~leeT)-IZ *-I [Z*'11

is

m a t r i x Z* as

- R* = d i a g ( Z * ) ( l

- d i a g ( Z * ) - I R *)

is the diagonal o f Z* and R* = R - d i a g ( R ) . A c c o r d i n g l y ,

JR* I is bounded near n = 0 by ( 2 . 2 . 2 3 ) ,

and I d i a g ( Z * ) l

(R*) k - O,

as well as i d i a g ( Z * ) - 1 1

are

bounded near n = 0 by assumption and as by (2.1.18) ~1(n)

Thus, f i n a l l y ,

- ~1(-n)

= 2~ + C~(n2),

~1(n)

- I = n + Ci(n 2)

a v o n Neumann s e r i e s expansion of Z* - I

via (2.2.24)

n ÷ O.

proves t h a t

IZ*-II

is bounded near n = O, t o o . The next lemma is the M a t r i x Theorem o f Kreiss in a somewhat shortened form adapted to our purposes. ( 2 . 2 . 2 5 ) Lemma. Kreiss [62 ] . ) If the assertion of Lemma ( 2 . 2 . 1 9 )

is true then there

exists to every matrix G(n) a hermitean matrix H(n) and a uniform constant

G(n}HH(n)G(n) < H(n) and

0 <

rsII

_-< H(n) =< FSI

?S such

v n2 E [ - s ,

that

0].

Here P ~ Q means again for two (m,m) matrices P and Q that Re(xH(p - Q)x) z 0 v x £ £m. As a corollary to Lemma (2.2.19) and (2.2.25) we prove

52 (2.2.26) Lemma. Under the assumptions of Theorem (2.2.18) there exists to every matrix G(AtA)

a norm

II.IIG

with

IIG(AtA)II G <_- 1 I

IIGIIG = maXx~011GXllG/llXllG

rsl/21x I

<= llxll G

--<

and

rl/2[xl

v x c 12x(k-1)×m

S

Proof. By assumption and Lemma (2.2.19) we have IG(AtA)nl ~ sup

-s~n2~0

[G(n)nl ~ Ks

hence by the Kreiss' Matrix Theorem (2.2.25) there exists to every matrix G(AtA) a Kreiss' matrix HG(AtA) with the property IHG(AtA)II2G(AtA)HG(AtA)-I/21

~ I.

Therefore, the vector norm llxll G ~ IHG(AtA)I/2xl

has the desired properties.

For the proof of Theorem (2.2.18) we now observe that the e r r o r e n s a t i s f i e s with the polynomial x*(~,n) introduced in (2.1.38)

= Yn

-

Vn

(2.2.27) ~*(m,AtA)(T-c1(AtA))en_ I = -AtZ~=0BiBn+i~i(T)e n + d*(At,y) n - At2p1(m)hn, n = 0,1 . . . . . By (2.2.7) a l l polynomials Ti(C) of a consistent method have the root = I. We w r i t e

-

~k-I

= Yk

'

then we obtain using the block e r r o r vectors E0 of block dimension k-1 n

~ =06iBn+i~i(T)en = ~i=0Si ~k B n + i ~~k-1 = 0 6(i) ~ (en+u+1 - en+u Tk-1(~ k a~ i (2.2.29) = au:0 ~i=0 )6iBn+i)TU(en+1 = ¢(B)n(en+ k

_

**,B,T,-0

en)

en+k_ I) + @ t JntLn+k_1

E0

n+k_2 )

where ¢(B) n = Z~=0 Y~i)BiBn+ i , i . e . , ¢(B) n ~ 0 i f the method (2.2.2) respect to B. Using (2.2.29), (2.2.27) writes as

is e x p l i c i t

v*(T,AtA)(T - c1(Ath))x-len

= - at¢(X-IBX)n(X-len+ k - X-len+k_1 )

- At~**(X-IBx)~(X-IE n+k-1 0

X-IE n+k-2 0 ) + X-Id*(at'Y)n

(2.2.30) - At2pI(T)X-Ihn"

with

53 With the a b b r e v i a t i o n s E* -I 0 n = X En , (2.2.30)

= ~1(AtA), F* = F~.(AtA)

is e q u i v a l e n t to the s i n g l e - s t e p equation E* - ~E~ n -I = F*(E~-I

- ~E~-2) - AtOn(E~

-

* En-1) -

At~

* n ( E *n-!, - En-2)

(2.2.31) + X-IDa. - X IH n,

n = k,k+1,...

where D*n and Hn are the same block vectors as in (2.2.12) k-1. o n and ~n are ( k - l , k - 1 ) - b l o c k

but w i t h block dimension

matrices o f which only the l a s t row in nonzero and

l a s t row o f o n = ( 0 , . . . , 0 , ~ k ( A t 2 A 2 ) - I ¢ ( x - I B X ) n ) , l a s t row o f ~n = ak(At2A2)-1°**(X-1BX)~ ' Substituting

(2.2.32)

En*

-

E *n-1

:

( E *n

-

cE~_ 1)

-

( E *n-1

-

~E~

-2)

+ ~ ( E ~. - , ~

* - En-2)

i n t o ( 2 . 2 . 3 1 ) we o b t a i n

(I + AtOn)(E ~ - cE~_ I)

= (F* + atOn)(E~_ 1

~E~_ 2)

(2.2.33) - At(~ n + ~On)(E*n-1 - En-2) * + X-1D n. _ X-1H n' and a s u b s t i t u t i o n

of ( 2 . 2 . 3 1 )

into (2.2.32) yields

(I + AtOn)(E n - E*n-I ) = (F* - I ) ( E *n-I - tE n - 2 ) (2.2.34) + (~I - At~n)(E* * X-ID * - X-IH n-1 - En-2) + n n" (2.2.33) (2.2.35)

and ( 2 . 2 . 3 4 )

t o g e t h e r p r o v i d e the e r r o r equation

Zn = (I + AtOn)-IG(AtA)Z n_1 - AtLnZn-1 + D**n - Hn**,

n = k,k+1,...

where Zn= (E n - ~En_ I * ,E*-E*nn - 1 ) T ' D)* * = ( l + A t ° n T) - I x - 1 ( D n ' D'n ) T ' Ln : (1 + A t O n ) - 1 [ ; ° n

~n

+=°nl ~n

J

Hn*=(l+AtOn)-Ix-1(Hn'Hn

54 and G(AtA) is the m a t r i x introduced in Lemma ( 2 . 2 . 1 9 ) .

Recall t h a t on z 0 i f the method

(2.2.3) is e x p l i c i t with respect to the damping m a t r i x B. I f on ~ 0 and B is constant then the proof of Theorem (2.2.18) f o l l o w s the l i n e s of Theorem (2.2.10) estimating Zn by means of (2.2.35) and Lemma (2.2.19) but not i n v o l v i n g the Kreiss' Matrix Theorem. However, i f the method (2.2.3) is i m p l i c i t with respect to B then the m a t r i x on does not n e c e s s a r i l y commute with the m a t r i x G(AtA). Hence we are forced in t h i s case to apply Lemma (2.2.26) and s i m i l a r l y in the case where B varies w i t h time. We now observe t h a t llI~lll n ~ ~IIlX-IBxllln , III~III n ~ ~IIIX-IBxIIIn where we set ~ = 0 i f the method is e x p l i c i t

with respect to B. With the constant r s

defined in Lemma (2.2.26) we suppose t h a t

(2.2.36) Atrs~IIIX-IBXhl n = I/2, and we denote f o r the moment by I * the i d e n t i t y m a t r i x of the same dimension as G(AtA). Then we have

%lllx-lBxllln

ltLnll G ~

and, by Lemma ( 2 , 2 , 2 6 ) , II(I + At~n)-IG(AtA)IIG S 1[(I + At~n)-II*IIGIIG(AtA)IIG ~ I1(I + A t ~ n ) - l I * l l G

(1-

~trs~lllX-lBxllln )-1 ~ 1 + At~slllx-lBxllIn .

Using these bounds we obtain from (2,2,35) IIZnllG ~ (I +

~IlIX-IBXNIn~t),Zn_I,G + ,D~*

- H~*]IG,

n = k,k+1 . . . . .

and we thus f i n d by induction a f t e r r e t u r n i n g to the Euclid norm t h a t

IZnl ~ ~ exp{~l]IX-IBX]llnnAt}ElZk_11 ID~*I

and

IH~*i

with d i f f e r e n t

have the same bounds as r

IX-ID~I

+ ~=klD~* - H~*I]. and

IX-IHI

in (2.2.13) and (2.2.14)

hence the rest o f the proof f o l l o w s as in Theorem ( 2 . 2 . 1 0 ) .

As already noted above, The l a s t c o n d i t i o n in assumption ( 2 . 2 . 1 8 ) ( i i ) stronger than strong D - s t a b i l i t y

is s l i g h t l y

in I - s . 0). I t claims t h a t in t h i s i n t e r v a l the method

is s t r o n g l y D-stable and t h a t the p r i n c i p a l r o o t ~1(n) does not coalesce to a double unimodular r o o t . This l a t t e r c o n d i t i o n cannot be released in the proof of Lemma ( 2 . 2 . 1 9 ) .

55 2.3. Linear Multistep Methods f o r D i f f e r e n t i a l Systems with Orthogonal Dampin~

The estimations of the preceding section contain the norm of the damping matrix B as exponential m u l t i p l i c a t i o n factor. The question whether there are problems and methods which provide damped approximations i f the exact solution is damped leads again to the class of i n i t i a l

value problems with orthogonal damping which was already con-

sidered in Theorem (1.4.7). Perhaps i t should be remarked once more that the occupation with these problems is not only a t h e o r e t i c a l pastime. At the present state of matrix s t r u c t u r a l analysis the majority of dynamic f i n i t e element equations ( I ) with damping seems to be of that form. Besides the references given in the Introduction we quote Geradin [74 ] and Jensen [74 ] to which we were pointed by Godlewski and Puech-Rauolt [79 ]. The l a t t e r c o n t r i b u t i o n deserves special emphasis here because i t deals with l i n e a r multistep methods and problems with orthogonal damping in d e t a i l . We reconsider in t h i s section the i n i t i a l

value problem (2,2,9) and l i n e a r m u l t i -

step methods (2.2.3) under the assumption of Theorem (1.4.7) but w i t h o u t that the damping is smaller than the c r i t i c a l (2.3.1) Assumption.

the condition

damping:

(i) Let the solution y of (2.2.9) be (p+2)-times continuously

differentiable. (ii) Let the constant (m,m)-matrices A 2 and B be real symmetric and negative semidefinite with the same system X of eigenvectors, A 2 = XA2X T, B = X~X T

xTx = I

The f o l l o w i n g lemma describes orthogonal damping completely in an algebraic way. (2.3.2) Lemma. Let A and B be two diagonable (m,m)-matrices.

Then the following three

conditions are equivalent: (i)

AB = BA,

(ii) there exists a diagonable matrix C and polynomials p(x) and q(x) such that A =

p(C) and B = q(C), (iii) there exists a nonsingular matrix X such that X-IAX and X-IBX are both diagonal.

Proof. See Householder [64, p. 30]. As the damping matrix B is constant throughout we introduce the polynomial k k 6 ~k ¥ ( i ) ~ T(~) = Zi=06i~i(~) = ~i=0 i ~ = 0 u

rk ~ k ~ ( i ) ~ ~k X ~ = L~=0~Li=0 iYu ) = L~=0

then the linear multistep method (2.2.3) writes as (2.3.3)

P0(T)Vn + At2A2pI(T)vn + AtBT(T)Vn = - At2p1(T)Cn ,

n = 0,1 . . . . .

56 and the characteristic polynomial is now ~(~,n2,~) = x(~,n 2) + ~(~) = p0(~) + n2p1(~) + ~T(~). S u b s t i t u t i n g the exact s o l u t i o n y of the i n i t i a l (2.3.3) we obtain again the d i s c r e t i z a t i o n

value problem (2.2.1) i n t o the scheme

error d*(At,y)(t)

of the method,

d*(At,y) n = P0(T)Yn + At2A2p1(T)Yn + AtBT(T)yn + At2p1(T)Cn = P0(T)Yn + At2p1(T)Y'n' + AtB(T(T)yn - &tP1(T )Yni ), or

(2.3.4)

d*(&t,y)(t) = d(&t,y)(t) + AtBd(at,y)(t)

where d ( A t , y ) is the d i s c r e t i z a t i o n

e r r o r (2.1.7) of the corresponding undamped pro-

blem (2.1.5) and (2.3.5)

d(At,y)(t)

= ~(T)y(t) - AtP1(T)y'(t).

I f the method (2.3.3) u E cP+20R;[Rm) (2.3.6)

is c o n s i s t e n t of order p a f t e r D e f i n i t i o n

(2.2.4) then f o r a l l

TId(&t,u)(t), ~ FAt p+2 and TId(At,u)(t)I1 ~ FAt p+I

where F does not depend on At but the contrary is not true in general because the polynomial ~(~) does not define the polynomials ~ i ( ~ ) , theless Lemma (2.2.5) remains true i f

i = 0, . . . . k, in a unique way. Never-

(2.2.4) is replaced by (2.3.6) because, by

(1.1.6) f o r ~ = I , - d ( A t , u ) is the d i s c r e t i z a t i o n

e r r o r of a l i n e a r m u l t i s t e p method

( I . 1 . 3 ) w i t h p0(~) = - ~(~) f o r a first order d i f f e r e n t i a l is a composition of Lemma (1.1.8) and Lemma ( 2 . 1 . 8 ) . (2.3.7) D e f i n i t i o n . teger p such that

The method (2.3.3)

(2.3.6) holds

system and Lemma (2.2.5)

So we may d e f i n e :

is consistent if there exists a positive in-

for all u E

cP+2(4R;iRm),The

maximum p

is the order

of the method.

In p a r t i c u l a r , (2.3.8)

the method (2.3.3)

is c o n s i s t e n t i f f

Po(1) = p~(1) = o~'(1) + 2o1(1) = ~(1) = ~ ' ( 1 ) - P1(1) = O.

Recall now t h a t by Lemma (1.1.12) a polynomial p1(~) of degree p defines in a unique way a polynomial - T(~) such t h a t ( 2 . 3 . 5 ) f u l f i l s

(2.3.6).

S t a r t i n g from a method

(2.1.4) of order p f o r the undamped problem we so derive e a s i l y a method (2.3.3) of order p i f the polynomial p1(~) has degree p.

57 As a f u r t h e r tool f o r the construction of methods (2.3.3) we quote the f o l l o w i n g r e s u l t of Godlewski and Puech-Raoult [79 ]. (2.3.9) Lemma. The method (2.3.3) has order p iff (i) the corresponding method for the undamped problem has order p, (ii) the method (1.1.3) with ~(~,n) = p0(~) + n~(~) has order p+1.

Proof. Let d(At,u) be defined by (2.1.7) for ~ = I and let ~(At,U) = P0(T)u + At~(T)u' be the discretization error of the linear method (1.1.3) for f i r s t order differential systems with the polynomial p0(c) + n~(C). Then d(At,u) = - Atd(At,U') + d(At,u) and hence (2.3.4) proves the r e s u l t . A f t e r having studied the consistence l e t us now consider the s t a b i l i t y

of the

method ( 2 . 3 . 3 ) . The e r r o r equation (2.2.11) has here the form

(2.3.10) [P0(T) + At2A2pI(T) + AtBz(T)]e n = d*(At,Y)n - At2p1(T)hn ,

n : 0,I . . . .

Under Assumption (2.3.1) t h i s is equivalent to the s i n g l e - s t e p equation (2.3.11) XTEn = F (At2A2,AtR)XTEn_I + xTD~ - XTHn,

n

=

k,k+1

....

,

where now F (n2,u) is now the Frobenius matrix with the c h a r a c t e r i s t i c polynomial x(~,n2,v) and the f o l l o w i n g f u r t h e r notations are used: s~(At2A2,AtQ) = m0kI + BkAt2A2 + XkAtQ, xTD~ : (0 . . . . . 0,s~(At2A2,At~)-IxTd*(At,Y)n_k )T,

XTHn : (0 . . . . . 0,~(At2A2,AtQ)-IAt2pI(T)XThn_k)T. As above, l e t ~ - b e

the complex plane extended in the usual sense, and l e t {-~ =

~2 u {~, ~} be extended likewise in the usual sense; moreover, l e t ~(~,~,u) = p1(~), C C, ~(~, 2

) = z(~), n2 E { , and ~(~,=,~) = p1(~) + T(~). For s i m p l i c i t y we d e f i n e

here the two-dimensional region of s t a b i l i t y Puech-Raoult [79 ]:

S2 as f o l l o w s , cf. also Godlewski and

(2.3.12) D e f i n i t i o n . The two-dimensional stability region S2 of the method (2.3.3) consists of the pairs (n2,~) E ~ C with the following properties: ( i ) o~(n2,v) m 0, ( 2 , ~ ) E S2n {2,

58 I~4(n2,v)i ~ I, (iii) all roots ~i(n2,~) of ~(~,n2,u) with I~i(n~,u)i = I have multiplicity not greater

(ii)

a l l roots ~i(q2,~) o f ~(~,n2,u) s a t i s f y than two.

With t h i s notation, the error estimation corresponding to Theorem (2.1.34) and Theorem (2.2.10) is then (2.3.13) Theorem. ( i ) Let Assumption (2.3.1) be f u l f i l l e d . (ii) Let the method (2.3.3) be consistent of order p with the stability region S2. ( i i i ) Let Sp(At2A 2) x Sp(AtB) C R £ S2 where R is closed in ~ . Then f o r n = k , k + 1 , . . . , lY n - v n I --< KRnAt[ At-IiYk_ I - V k _ l i + A t p i A t ( l Y (p+2)(~)i+IBy (p+1)(~)I)d~+nAtlilhlllnl" If supplementary the method is implicit with respect to B and the discretization error

~(At,u) defined

by

(2.3.5) has

order

p+1

then the assertion holds with

]By(p+I)(%)i

cancelled.

Proof. Under the assumption we find by the Uniform Boundedness Theorem again that supnEINin-IF (At2A2,At~)nl ~ KR hence the f i r s t assertion follows as in Theorem (2.1.34) using Lemma (2.2.5) instead of Lemma (2.1.8). The second assertion follows in the same way because under the present assumption Io~(At2A2,At~)-IAt~i

~

and accordingly la~(At2A2,At~)-IxTd*(At,y)(t)

~
+ [~(At2A2,At~)-IAt~IId(At,y)(t)I

~+kAt t+kAt ( + 2 ) <: KrAtP+I ~ lY(P+2)(~)Idr+~rAtP+1 ~ IY p (~)i dT" t t The principal roots ~1(q,u) and ~2(n,v) = ~1(-n,v) now depend on n and ~ but otherwise the situation is the same as in Section 2.1. The concept of strong D-stabil i t y can be generalized to the present case in a likewise simple way: (2.3.14) Definition. Let ~*(~,n,v) = x(~,n2,v)/(~ - ~1(n,v)) then a method (2.3.3) 0] if I-S, 0] x I - r , O] C S2 and if for all (q2,u) E [-s, 0] × [ - r , 0] the roots ~i(n) of ~*(~,n,v) with Ici(n,u)l = I are simple

is strongly D-stable in I-S, 0] x I - r ,

59 roots of

~*(~,q).

F i n a l l y , the following r e s u l t is the analogue to Theorem (2.1.39) and Theorem (2.2.18). (2.3.15) Theorem. Let the assumptions of Theorem (2.3.13) be fulfilled but let [ - s , O] x [ - r , O] and let the method (2.3.3) be strongly D-stable in n = k,k+1,..., IYn - V n l

R.

R =

Then for

s IE~_21 + KRnAtI(IA] + IB])IE~_21 + At-IIE~_ 1 - E~_21

+

At p

~AL(Iy(p+2)(~)I + IBy(p+1)(~)I)d~ + nAtillhllln] 0

where En 0 = ( Yn-k+2 - Vn-k+2 " " '

Yn - v n )T. If supplementary the method is implicit

with respect to B and the discretization error d(~t,u) defined by (2.3.5) has order

p+1 then

the assertion holds with

IBy(p+I)(T)Icancelled.

Proof. Using Lemma(A.1.13) instead of Lemma (A.I.8) the f i r s t assertion is proved as in Theorem (2.1.39) and the second assertion is proved as in the preceding theorem. Both error bounds of this section contain no longer an exponential increasing m u l t i p l i c a t i o n factor but, on the other side, numerical damping is derived neither. The v e r i f i c a t i o n of exponential damping in the error bounds however follows rather closely the lines of Section 1.3. I t is therefore omitted in order to avoid to many repetitions.

2.4. Nystrbm T~pe Methods for Conservative D i f f e r e n t i a l Systems

As announced at the outset of the chapter we turn in this section to the second way of approximating conservative d i f f e r e n t i a l systems of second order d i r e c t l y by multistep m u l t i d e r i v a t i v e methods. Supplementary to the polynomials p j ( ~ ) , p~(~), j = 0 . . . . . ~, and the polynomials ~ i ( ~ ) , o#(n), i = O , . . . , k , defined by (2.1.3) we introduce the polynomials ~k B i ~(~) ~j(~) = ~i=o j i ~ '

= rk B* i Li=O j i ~ '

and x i ( n ) = ~=OBji nj, x~(n) = Lj=O ~ B* j i nj " Then a general multistep m u l t i d e r i v a t i v e method of Nystr~m type f o r conservative systems

60 y"

: f(t,y)

is a scheme formally consisting of two formulas (2.1.2), i . e . ,

)

k

2 2 +if(-2)

~i=O~i (at @ )

n

(Vn) +

+tvk

.t.t2e2~Tif(-1

Li=o~i ~a

)

n )(Vn) = O,

(2,4.1)

Z~=0Tj(T)(~t2+2)Jf~-2)(Vn) + ~tZ~=0T~(T)(At2B2)Jf~-I)(Vn ) ~k

fAt2D2~Tif(-2) (V

z Li=oXi ~

)

n

~ nj + atZ~=OXT(At282)Tif~-1)(Vn ) = O, n : 0,1 " "

Here we have to write f~-2)(v n) = vn and f(-1)(Vn)n is to be replaced by wn which plays i the role of an approximation to Yn" In order to guarantee that these formulas provide a unique solution (Vn+k, Wn+k) for sufficiently small time steps At and to remove a certain arbitrariness in the choice of the coefficients we stipulate that

(2.4.2)

* o~Ok= 1 and 60k

=

I

and that the matrix

Ok +kill 6~kJ 60k

:+k1

60k

is regular. Moreover, in analogy to (2.1.6) i t is supposed that

• C(~)

T~(~)] ~ 0 and

F

L×0(~)

x~(~)J

~ O.

Finally, let k ~

i = IPj/2 (~)

PJ(~) z Zi=Omji~ k ~ ~J(~) z Zi=°BJi~

i f j even

Lp#j_1)/2(~) i f j odd '

i = ITj/2 (c)

i f j even

L~j_1)/2(~)i f

j odd

Then the discretization error of the method (2.4.1) is D(At,u)(t) : (dp(At,u)(t), dT(At,u)(t))T,

u £ C2~+I(IR;IRm),

61 where

(2.4.3)

dp(At,u)(t) = LJ=0~2~+IAtJ~pJ(T)u(J)(t)' dT(At,u)(t) = ~L~IAtJ~.(T)u(J)(t)j= J .

(2.4.4) D e f i n i t i o n . The method (2.4.1) is consistent if there exists a positive integer p such that for all u E

cP+2(IR;IR m)

IID(At,u)(t)ll ~ rAt p+2 where £ does not depend on At. The maximum p is the order of the method.

By the formulas (2.4.3) we find in the same way as in Lemma (1.1.12) (2.4.5) Lemma. The method (2.4.1) is consistent of order p iff ~min{~,2~+1}~k ~ i n-j j=O ti=O~ji ~ .

= ~min{~,2~+1}~k ~ i n-j ~j=O Li=O j i ~

= O, ~ = 0,1 . . . . . p+1.

In p a r t i c u l a r , the method (2.4.1) is consistent i f f (2.4.6)

~0(1) = ~(1)

+ ~1(1) = ~'(1)

+ ~(1)

+ 2~i(1

+ 2p~2(1) = 0

and (2.4.7)

T0(1) = w0(1) + ~ i ( I )

=~'(I)

+ T0(1) + 2TI(I

+ 2~2(I) : 0.

For s i m p l i c i t y we consider in the sequel only nonllnear single-step methods, i . e . , methods (2.4.1) with k = I . Furthermore, we assume that the method is at most semii m p l i c i t in the sense that

o~(n)

= ×1(n)

~ 0,

and with respect to the computational e f f i c i e n c y we suppose that o1(n) = x~(n) = o(n). With these r e s t r i c t i o n s , the approximation of the i n i t i a l (2.4.8)

y"

value problem (2.1.5),

= A2y + c ( t ) , t > 0, y(0) = Y0' y ' ( 0 ) = y~,

yields the following scheme which is at once written in e x p l i c i t form:

62

Vn+ I : _ ~(At2A2)-I[~0(At2A2)v n + ~(At2A2)AtWn ] - ~(At2A2)-Ic~ (2.4.9) AtWn+ I = - ~(At2A2)-1[×0(at2A2)v n + x~(At2A2)AtWn ] - ~(At2A2)-Ic~ * , n = 0,1 . . . . where .

r

, _2-2. _2j (2j-2)

cn = ~=it~0j~At a )A~

cn

C**n =Zj=I~XOj~ ~ 'At2A2~AtZJc(2j-2)' n

xij(n) and ~ i j ( n ) ,

cn

+ ×~j(At2A2)At2j+Ic~ 2 j - I )

,._2_2, _2j (2j-2)~ + Oljkat A )AE Cn+ I ±,

, t2.2,A_2j+1 ( 2 j - I ) + Xlj~a ~ ) z Cn+ I },

vt $ m-j ~ , m-j = Lm=j mi n , x~j(n) = Lm=jBmin ,

~ l.J( n )

By ( 2 . 4 . 2 ) ,

. , ~2.2, _2j+I ( 2 j - I )

+ o0j~A~ ~ )at

are defined in (1.2 ° 3)

j = 0,...,~,

'

( 2 . 4 . 6 ) , and (2.4.7) the method (2.4.9) is consistent i f f

(2.4.10) mOO = m~0 = - I , m10 + m11 : - I / 2 , BOO : 0, B~0 : BI0 = - I . The method (2.4.9) and the corresponding Runge-Kutta-Nystr~m method d i f f e r only in the treatment of the time-dependent r i g h t side c ( t ) . Accordingly, both methods can be studied together as concerns problems of s t i f f n e s s and absolute s t a b i l i t y

since

these concepts are defined a - p r i o r i only f o r systems (2.4.8) with constant matrix A2. Moreover, the d i s c r e t i z a t i o n error of m u l t i d e r i v a t i v e methods can be estimated more e a s i l y whereas in Runge-Kutta method an i l l - c o n d i t i o n e d matrix A2 can a f f e c t the d i s c r e t i z a t i o n e r r o r in a negative way; see Chapter V. Let now Vn = (v n, Atwn)T and G(n) = - ~ ( q ) - I [ ~0(n) i

L×o(n)

x~(n)J

Lg3(n)

g4(n)

Then we can w r i t e instead of (2.4.9) (2.4.11) Vn+ I = G(nt2A2)V n - Cn, where

Cn =

o(At2A2)-1(c~, c~*) T, Obviously, we have

G0[I i]

n = 0,I,...,

63 and the eigenvalues of G(n) are (2.4.12) ml,2(n) = [g1(n) + g4(n) ± ((g1(n) - g4(n)) 2 + 4 g 2 ( n ) g 3 ( n ) ) I / 2 ] / 2 .

(2.4.13) D e f i n i t i o n . The stability region S of the method (2,4,9) consists of the £ ~ with spr(G(n2)) ~ I. The method (2.4.9) is strongly D-stable in [-S, 0] C S if all eigenvalues of modulus one of G(n 2) are simple eigenvalues for n 2 C I-s, 0). n

Because of (2.4.12) the elements gi(n 2) of G(q 2) must be bounded f o r n2 c S therefore s(n 2) in (2.4.9) must be a polynomial of exact degree ~ i f S is unbounded in ~. We reconsider the i n i t i a l value problem (2.1.32) and quote the following r e s u l t : (2.4.14) Theorem. ( i ) Let the (m,m)-matrix A2 in (2.1.32) be diagonable, A2 = XA2X-I , and let the solution y be (p+2)-times continuously differentiable. (ii) Let the method

(2.4.9) be consistent of order p ~ 2~ - I with the stability

region S. (iii) Let Sp(~t2A 2) C R C S where R is closed in ~. Then for n = 1,2, ....

nat , ~, IX-IEn I ~
] + n~t maxo~is2~_1111h(i)ll[n

J

where En (Yn - Vn' ~t(y~ - Wn)) T.

This theorem is proved in the same way as Theorem (2.1.34) by application of the Unlform Boundedness Theorem. However, we note that in the present case the crucial e s t i mation sup ' 'ln-lGtn2)nl " " sup 2 n E R n £1N

s
can be carried out in a straightforward way by a Jordan canonical decomposition; see Gekeler [77 ], In order to derive the optimum order of convergence with respect to the i n i t i a l e r r o r we need the following a u x i l i a r y r e s u l t : (2.4.15) Lemma. Let the method (2.4.9) be strongly D-stable in [-S, O] ~ R C S and let g1(n2; n)

g2(n2;n)].

g3(~2;n)

g4(n2;n)]

G(n2) n :

Then

64

Igl,4(n2;n)l ~
g2(n2;n) = g2(n2)en(n2),

g4(n2;n) : ~2(n2) n + (m1(n 2) - g1(n2))en(n2),

g3(n2;n) = g3(n2)en(n2).

But

]%(n2)] ~ n because Iml,2(n2)l ~ 1 by assumption hence we obtain from (2.4.16) that the assertion is true for g2(n2;n) and g3(n2;n). Furthermore, ]ml(n 2) - gl(n2)l

s Iml(n2)l +

Igl(n2)l ~ ~

and by (2.4.12) lml(q 2) - gl(n2){

~ <~*(In2t + (In212 + In21) 1/2 ~ K~**(1 + [ n [ ) [ n t ,

because g1(n 2) - g4(n 2) = d(n2), n 2 ÷ O, Accordingly,

]~l(n 2) - gl(n2)l ~ Inlmin{~/Inl, ~**(1 + Inl)} ~ ~Rlnl

n 2 + 0,

65

which proves the assertion f o r g l , 4 ( n 2 ; n ) ,

too.

(2.4.17) Theorem. Let the assumption of Theorem (2.4.14) be fulfilled but let the method (2.4.9) be strongly D-stable in R ~ [-s, 0]. Then for n = 1,2,...,

Vn) I ~ <siX-l[[(1

IX-1(yn -

+ n a t i A l ) l y 0 - v01 + nAtly ~ - w01 + nAtz n]

IX-1(y~ - Wn) 1 ~ <slX-11[nAtlA2]ly0

- v01 + (I + n A t [ A l ) [ y ~ - w0] + nz n]

where sn = at p

Iy(p+2)(T)IdT + nat max0~is2~_1111h(i)llln . 0

Proof. The e r r o r En

=

(Yn

-

v n , At (Yn' - Wn) ) T s a t i s f i e s

(2.4.18) X-IE n : G(~t2A2)nX-IE0 + ~?n-IG(At2A2)n-u-I(x-ID =0 ' ' ~* - X-IH ) . . n. =. 0,1 . . where Hn is the same vector as Cn in (2.4.11) with c n replaced by hn and D~ : ~(At2A2)-1(dp(at,Y)n , d (At,Y)n)T, Applying Lemma 2.1.8 to the components d p (At,y) and dr(At,y) obtain

defined by (2.4.3) we

(n+1

IX-ID~I ~ rlX-11At p+I f )Atly(p+2)(T)Id~, nat and

IX-1Hn I ~ %lx-l[nt2maxo~i~2~_llllh(i)rlln follows in the same way as in Theorem (1.2.12) using again the fact that ~(n) must be a polynomial of exact degree ~ i f R is unbounded in ~. The Uniform Boundedness Theorem or a d i r e c t v e r i f i c a t i o n

yields again

In-IG(At2A2)n I ~ sup

In-IG(n2)n I s Ks -s~n2~0

and so we deduce from (2.4.18)

IX-1(yn - Vn) ] ~ Ig1(at2A2;n)llX-1(y 0 - v0) I +

Ig2(At2A2;n)IAt]X-1(yG - W0) I +

KslX-1[nAtz n

IAtX-1(y~-Wn) I s Ig3(At2A2;n)IIX-1(y 0-v0) I + Ig4(At2A2;n)IAtlX-1(y ~-wo) I +<slX-11nAtSn , By assumption we have Sp(At2A 2) C [-s, 0] and At2A 2 is a diagonal matrix therefore

66 we can s u b s t i t u t e At2A 2 f o r n 2 in Lemma (2,4.15) which proves the Theorem. By and large, the e r r o r bound of X-1(yn - v n) in t h i s Theorem is the same as t h a t of Theorem (2.1.25) f o r the o r i g i n a l m u l t i s t e p m u l t i d e r i v a t i v e methods ( 2 . 1 . 3 3 ) . Additionally,

Nystr~m type procedures provide an approximation of YnI in a s i m i l a r way as

i n d i r e c t methods do, Roughly spoken we can say t h a t AtX-1(y~ - wn) has the same e r r o r bound as X-1(yn - v n) but w i t h the i n i t i a l e r r o r m u l t i p l i e d by AtlA I. In t h i s s e c t i o n , we have adopted the d e f i n i t i o n of the consistence order p of Section 2.1 f o r both formula (2.4.1) representing the Nystr~m type method. Then, in Theorem (2,4.14) and ( 2 . 4 . 1 7 ) , the order of convergence is p w i t h respect to the solut i o n y of ( 2 , 4 , 8 ) and p-1 w i t h respect to i t s d e r i v a t i v e y ' ,

Hence we cannot speak of

convergence w i t h respect to the approximation of y' i f p = I t h e r e f o r e the consistence order is defined sometimes f o r Nystr~m methods as p* = p-1 so t h a t p* ~ I implies convergence of at least order p* f o r bot~ sequences {v n} and {Wn}. For examples of Nystr~m type methods and Runge-Kutta-Nystr6m methods we r e f e r to Appendix A.5,

2.5. S t a b i l i t y

on the Negative Real Line

In Section 1.4 the l i n e a r problem (2.5.1)

y"

= A2y + c ( t ) ,

t > 0, y(0) = Y0' y'(0) = y~,

order problem and then approximated by a k-step ~-deri= ~2y and define the method (1.1,3) by its characteristic polynomial,

was transformed i n t o a f i r s t v a t i v e method ( 1 . 1 . 3 ) .

I f we restrict ourselves to the test equation y "

k

i nj

x(~,n) = Zj=0Zi=0~ji ~

L

(

- Zj=0pj ~)n j ,

then t h i s i n d i r e c t procedure y i e l d s the computational device

Zj=0P2j(T)At2j~2JVn + Zj=oP2j+1(T)At2j+1~2JWn = O, (2.5.2)

Z. 0P2.(T)At2j~2JWn + Zj=oP2j+I(T)At2j+I~2j+2Vn = O, J: J

n = 0,I,,,.,

where mji = 0 f o r j > ~. Here we can e l i m i n a t e the terms c o n t a i n i n g the approximation wn to Ynt and obtain the f o l l o w i n g scheme (2.5.3)

[ ( Z j : O P 2 j ( T ) A t 2 j ~ 2 j ) 2 - A t 2 ~ 2 ( Z j : o P 2 j + I ( T ) A t 2 j ~ 2 j ) 2 ] V n = O,

n : 0,1, . . . .

67 or ~ ( T , A t X ) ~ ( T , - AtX)v n = O,

n = 0,1 . . . .

Accordingly, the transformation (2.5.4)

s: ~(~,n) + x ( ~ , n ) ~ ( ~ , - n ) = ~(~,n 2)

defines a 2k-step method (2.1.4) f o r the problem (2.5.1) i n v o l v i n g the f i r s t

L - I

even d e r i v a t i v e s of the r i g h t side of (2.5.1). See Jeltsch and Nevanlinna [82b]. For ~ = I the transformation (2.5.4) y i e l d s a l i n e a r multistep method f o r the nonlinear problem (2.1.1). Below a method (2.1.4) given by the polynomial k in2J ~(~,n 2) = Zj=oZi=oaji ~ is also b r i e f l y called a (k,c)-method as the k-step ~-derivative methods (1.1.3) f o r first

order systems. I f the s t a r t i n g values are

chosen s u i t a b l y then the schemes (2.5.2) and (2.5.3)

produce the same sequence { vn}n=O i f rounding errors are not regarded. Therefore the method (1.1.3) and i t s s-transformation defined by (2.5.4) have the same order p and the same error constant. For the sequel we recall that the s t a b i l i t y fined in d i f f e r e n t ways f o r f i r s t

regions S are de-

order and second order problems, cf. D e f i n i t i o n s

(1.2.7) and (2.1.14). I f the method (1.1.3) has the s t a b i l i t y

interval I r = {in,

-r

< n < r} on the imaginary axis then a l l unimodular roots of ~(~,~) are simple for n E I r by D e f i n i t i o n (1.2.7). But i f ~j(q) is a unimodular root and n E I r then ~ j ( - n ) is a unimodular roott too, because ~(~,n) is a real polynomial. Hence the method (2.1.4) given by the s-transformation has at most double unimodular roots for n c I r and therefore has the real s t a b i l i t y

i n t e r v a l ( - r 2, O] c S. I f I r is maximum f o r the

method (1.1.3) then ( - r 2, O] is maximum f o r the z-transformation.

Using t h i s f a c t

some resultsfrom Section 1.7 on the s t a b i l i t y on the imaginary axis of methods (1.1.3) can be transformed into results on the s t a b i l i t y on the negative real l i n e of methods (2.1.4). In these l a t t e r methods only the s t a b i l i t y numerical i n t e r e s t because the test equation y "

interval (-s,

0], s > O, is of

= ~2y has bounded solutions only f o r

negative real ~2. The f o l l o w i n g r e s u l t corresponds to Corollary (1.7.7) and has been proved by Hairer [79 ] and Jeltsch and Nevanlinna [82b]. (2.5.5) Theorem. ( i ) A (k,~)-method (2.1.4) with ( - ~, O] C S has order p ~ 2~. (ii) Let ~(~,n) be the polynomial (1.5.1) of the diagonal Pad~ approximant. Among all (k,~)-methods of order p : 2~ with (- ~, O] C S the method with the polynomial 2~(~,n)

has the smallest error constant.

68

As concerns explicit methods, the result corresponding to Theorem (1.7.8) reads: (2.5.6) Theorem. (Jeltsch and Nevanlinna [82b].) If (2.1.4) is an explicit convergent (k,~)-method then either [ - 442, O] ¢ ~ or [ - 4~2, O] = ~ a n d

~.(~,n 2) = 2

~(~,n 2) has the factor

_ 2i2~T2~(_ i~/2c)~ + ( - i ) 2~

where T~(~) = cos~arccos~ is the Tschebyscheffpolynomial of degree ~.

Recall that Tschebyscheff polynomials T~(~) are even f o r ~ even and odd f o r ~ odd. For = I we have ~.(~,n 2) = 2

_ 2~ + 2 - q2

which defines St~rmer's method of order two with [ - 4 , O] = S. Methods (2.1.4) with ( - ~ , O] c S are necessarily i m p l i c i t .

The r e s u l t correspon-

ding to C o r o l l a r y (1.7.4) has been proved by Dahlquist [78a]:

(2.5.7) Theorem. (i) A linear multistep method (2.1.4) with ( - ~, O] C S has order p ~ 2.

(ii) Let ~t(~,n)

= ( ~ - I) - ( q / 2 ) ( ~ + I) be the polynomial of the trapezoidal rule.

Among all linear methods (2.1.4) of order p = 2 with (- ~, O] C S the method given by s~t(~,n)

has the smallest error constant.

F i n a l l y , the r e s u l t corresponding to Theorem (1.7.12) has been proved again by Jeltsch and Nevanlinna [82b] f o r l i n e a r methods: (2.5.8) Theorem. ( i ) I f

(2.1.4) i s a l i n e a r method o f o r d e r p > 2 w i t h ( - s, O] C S

then S ~ 6. (ii) If p = 2 and s > 6 then the error constant ×p satisfies

Ixpl

~ (1 - ( 6 / s ) ) / 1 2 .

(iii) The only linear method (2.1.4) with s = 6 is the method of aowell (or flumerov) given by

(2.5.9)

~(~,n2) = s~t(c,n) + 4 ( ~ - I ) 2= ( 2 _ 2 ~ + i ) - ~ ( 2 + I 0 c + I ) .

Cowell's method has order four. For every s < 6 there exists a linear 4-step method (2.1.4) with order six and S = I - s , 0], cf. Lambert and Watson [76] , Jeltsch [78b], and Dougalis [79 ].

69

Baker, Dougalis, and Serbin [79 , 80]

have developed high order single-step methods

(2.1.4) with ( - ~ , O] c S for the homogeneous linear problem y"

= A2y, t > O, y(O) = YO' y'(O) = y~

which are based on rational approximations to the cosinus function.

2.6. Examples of Linear. Multistep Methods

Examples of Nystr~m methods are given in Appendix A.5 because of t h e i r strong relationship to Runge-Kutta methods. The general 2-step method (2.1.4) with z = 2 is considered in Appendix A.6. In concluding this chapter we give in this section some examples of linear mutistep methods for d i f f e r e n t i a l systems of second order without and with damping: The general linear 2-step method (2.1.4) of order p ~ 2 has the polynomial

(2.6.1)

x(~,n 2) : 2 _ 2 ~ + I

- n2(m~2+ (1-2m)~+m),

EIR.

For m = 1/12 this method has order 4, cf. (2.5.9), and order 2 else. The s t a b i l i t y region for 2 is S = [ - s , O] where s = 4 / ( I - 4 ~ ) for 0 ~ m < I/4 and s = ~ for m ~ I/4. ~(~,n2 ) has double unimodular roots in n2 = 0 and only for 0 s m ~ I/4 in n2 = -s. Accordingly the method is strongly D-stable in [ - ~ ,

O] for m > I/4.

The 3-step method of order 2 with the polynomial (2.6.2)

~(~,n 2) = 2~3-5~ 2+4~- I - n2~3

is strongly D-stable in [ - ~ ,

0], too, but here we have ( - ~ , O) c ~ hence [ - ~ ,

O]

is not a periodicity interval. The 4-step methods given by

~(~,n 2) = (~-I)2(~ 2-2~cos¢+I) - n2[(9+cos¢)(~ 4+I)+8(13-3cos¢)(~ 3+~)+2(7-97cos¢)~2]/120 have optimum order p = 6 for ¢ E [0, x] and 0 E S holds for ¢ E (0, x]. The s t a b i l i t y region is S = I-s(@), 0], s(@) = 60(I + cos¢)/(11 + 9cos@),

~ £ (0, x)

70 hence there exists for every 0 < s < 6 a @E (0, ~) with s = s(¢); cf. e.g. Jeltsch [78b]. The procedure given by (2.6.2) is a backward d i f f e r e n t i a t i o n method as only the leading coefficent m13 of the polynomial p1(~) is nonzero. Choosing for ~3(~) in (2.2.3) the backward d i f f e r e n t i a t i o n approximation of Table (A.4.3) for k = p = 3,

(2.6.3)

T3(~) = (11~ 3- 18~ 2 + 9 ~ - 2 ) / 6 ,

we obtain d i ( A t , u ) ( t )

z O, i : 0,1,2, and

d3(&t,u)(t) = T3(T)u(t) - ~tT3u'(t) = ~(AL4),

At ÷ O,

for the discretization errors with respect to y ' . A substitution of (2.6.2) and (2.6.3) into (2.2.3) yields Houbolt's method of order 2 for the problem (2.2.1), (12 - 6At2A2 - 11&tBn+3)Vn+3 = (30 - 18&tBn+3)Vn+2 - (24 - 9AtBn+3)Vn+I + (6 - 2AtBn+3)vn + 6at2Cn+3, cf. e.g. Bathe and Wilson [76]. With respect to d(At,u) the order is 3 and the method is strongly D-stable in [ - ~ ,

O ] x [ - ~ , O] C S2.

Recall now that a l i n e a r method for the problem (2.6.4)

y"

= A2y + By' + c ( t ) , t > O, y(O) = YO' y'(O) = y~,

with constant matrices A2 and B is given by the polynomial ~(~,n2,~) = pO(~) + n2p1(~) + ~ ( ~ ) . I f (1.1.3) is a l i n e a r method of order p for systems of f i r s t order with the polynomial ~(~,q)

(2.6.5)

=

p(~) - qO(~) then

~(~,n2,u) = p(~)2 _ n2 (~)2 _ ua(~)p(~)

defines a method of order p for (2.6.4). This is shown in the same way as in the previous section. Moreover, i f the method (1.1.3) is A-stable then the method given by (2.6.5) is strongly D-stable in [ - s ,

0 ] × [ - r , O] for every 0 < s < ~, 0 < r < - .

The general 2-step method of order 2 has the polynomial

~(~,nZ,u) = (~_ i)2 _ 2(m 2+ ( I - 2 ~ ) ~ + m ) For m > I/4 the method is strongly D-stable in [ - ~ ,

- ~[( 2_ I ) / 2 ] . O ] x [ - - , 0].

71 The general 3-step method of order 3 has the polynomial x(~,q2,u ) : (~ + ½)(~ - i ) 3 + (~ - I) 2

- ~ I ~ , ~ ,~II~- ~I~, I~ ,~II~- ,I~+ ~+ olI~-~I, ~ - u[(~

+ ~)(~

- I ) 3 + (m + I ) ( ~

- I ) 2 + (~ -

I)],

m,O C IR.

For o = - m/12 we obtain ~(~,n2,~) : [ ( ~ + ~)~ + ( ½ - m)][(~ 2 - 2 ~ + I )

- n2((~ 2 + I 0 ~ + I ) / 1 2 ) ]

~I~ ~IEc~+~l~2+~, I~ ~I~ Hence the associated method f o r conservative systems w i t h the polynomial ~(~,n2,0)

is

a trivial

m o d i f i c a t i o n of Cowell's method in t h i s case and thus has order 4 with the

stability

region S = [ - 6 ,

stability

region S= ~ IR2 of the general method is the closed t r i a n g l e w i t h the v e r t i c e s

(0,0),

(-6,

O], c f . Theorem ( 2 . 5 . 8 ) .

For o = - m/12 and m > 0 the real

0), and (0, -12m); c f . Godlewski and Puech-Raoult [791.

I II.

Linear M u l t i s t e p Methods and Problems with Leading Matrix A(t) = a ( t ) A

3.1. D i f f e r e n t i a l

Systems of F i r s t Order and Methods with Diagonable Frobenius Matrix

During the last years great progress has been made in the understanding of the behavior of numerical integration schemes for s t i f f d i f f e r e n t i a l equations in the case where the leading matrix A varies with time and even some interesting results were obtained for general nonlinear systems. In this and the next chapter we consider some of these a-priori error bounds for d i f f e r e n t i a l systems of f i r s t and second order as far as they are of that special uniform character which is desired in the context with dynamic f i n i t e element equations. The generalization of the results of this chapter to nonlinear multistep methods is somewhat involved as concerns the notation but can be derived otherwise in a straightforward way. Over a long period the majority of contributions to numerical s t a b i l i t y dealed only with the test equation y' = ~y. Nevertheless the study of this t r i v i a l equation has revealed to be very successful and a large variety of interesting results and useful new methods has been derived by this way. Similarly, i t seems advantageous in the study of time-dependent problems to consider at f i r s t the case where the leading matrix A depends in a scalar way on time in order to derive the optimum results. So we study in this chapter the i n i t i a l value problem (3.1.1)

y' : a(t)Ay + c ( t ) + h ( t ) ,

t > O, y(O) = YO'

where a is a scalar-valued function, and the corresponding second-order problem. This special time-dependent form of the leading matrix is necessary in this and the third section by technical reasons but not in Section 3.2. Using the conventional notations k i ~k S i P(C) = Zi:O~i c = PO( c ) ' ~k > O, q(~) = ~i=O i c = - Pl ( c ) '

Bk ~ O,

l i n e a r m u l t i s t e p methods have f o r (3.1.1) the form (3.1.2)

p(T)vn - AtA~(T)(aV)n = Ato(T)Cn,

n : 0,1 . . . . .

where ~(T)(av) n = Z~=0Bian+iVn+i_ and the defect h(t) is again omitted in the computational device. In this section we follow Hackmack [81 ] and deduce a generalization of Theorem

73 (1.2.12) to the problem (3.1.1) under the r e s t r i c t i o n matrix is diagonable in the considered s t a b i l i t y

t h a t the associated Frobenius

(sub-)region

a mean value theorem is applied to the eigenvalues ~ i ( A t a ( t ) ~ )

R c S. As in the proof of F ( A t a ( t ) X ) w i t h

respect to t , t h i s assumption ensures the necessary smoothness of ~ i ( n ) .

(3.1.3) Theorem, (i) Let the (mlm)-matrix A in (3.1.1) be diagonable, A = XAX -I, let a(t) m 0, t > 0 i and let the solution y of (3,1.1) be (p+l)-times continuously differentiable. (ii) Let the method (3.1.2) be consistent of order p with stability region S. (iii) Let Sp(Ata(t)A) c R C S, t > O, where R is closed in ~ and convex.

(iv) Let the Frobenius matrix F (~) of the method (3.1.2) be diagonable in R. Then for n : k,k+1 .....

IX-1(yn -

Vn)I

[

nat

< R I X - I I e x p { ~ O n nat} IYk_ I - Vk_iI + At p ~

exp{-<~OnT}IY(P+I)(~)IdT +

nAtlllhllln

]

where On = maxk~u~nmaX(~_k)At~t~vAtla'(t)/a(~At)l. Proof. In the present case the e r r o r equation (1.2.10) changes to (3.1.4)

En = (F n + Rn)En_ I + D(At,y) n - Hn,

n = k,k+1, . . . .

where Fn = F (AtanA) and Rn is a ( k , k ) - b l o c k m a t r i x of which only the l a s t block row is nonzero: l a s t block row of R n = AtA(mkl - AtBkanA)-1(Bo(an_ k - an)l . . . . .

Bk_1(an_ I - an)l ) .

Under the above assumptions we f i n d e a s i l y t h a t (3.1.5)

IX-IRnXI s ~10nAt

and the Frobenius matrix F (n) is diagonable to Z(n) in the closed set R, F (n) = W(n)Z(n)W(n) - 1 . Using t h e a b b r e v i a t i o n s Wn = W(AtanA) and Zn = Z(AtanA) we thus can w r i t e instead of (3.1.4)

W~Ix-IEn : (Z n + WnIX-IRnXWn)WnIX-IEn_ I + WnIX-1(D(At,Y)n - Hn) (3.1.6)

: (Z n + WnlX-IRnXWn)(WnIWn_1)Wn! IX- I En_ I + WnIX-1(D(At,Y)n - Hn).

74 By assumption we have IZnl ~ I and observing IW(A)I : maxls~mlW(X )i we obtain from Lemma (A.2.4) that (3.1,7)

IWnl ~ <2'

IWnII ~ ~R"

Hence i f (3.1.8)

•IWn_I i ~ I + ~*KROnAt

1W

then we find from (3.1.6) that IWnIX-IEn i ~ (I + (<~/2)OnAt)21Wn!iX-Imn_iI

+ ~RIX-1(D(At,Y)n - Hn)i, n : k,k+1 . . . . .

writing (I +
- ~i(Atan_iX)i.

In this inequality ~1(n) . . . . ,~k(n) are the analytic roots of the c h a r a c t e r i s t i c polynomial ~(~,n) of the method (3.1.2) in the closed and convex domain R. I f ~k = 0 then R is bounded by Lemma (A,I.3) therefore we obtain by the Mean Value Theorem and assumption ( i i i ) i~i(AtanX ~) - ~i(Atan_iX )i ~ maXncRl~(n)IOnSpr(AtanA)At

~ ~OnAt

which proves together with (3.1,9) the desired i n e q u a l i t y (3.1,8). I f Bk ~ 0 then observe that ~ i ( n ) , i = 1 , . . , , k , are the roots of the normed polynomial ~=0Yi(n)~ i , Yi(n) = (a i

~in)/(a k - Bkn).

Accordingly we have omitting the argument n = Ata(t)X

~Ci~ k ~i( ~Y' (3.1.10) ~-E-tY0 . . . . . Yk ) = ~j=0 ~ j Y0 . . . . . Yk)~-~ ' But

75

max1siskSUPnER I /T~(Ci ( n ) ' n ) l

~ ~R

because ~(~,n) has no double roots ~ i ( n ) in R, and moreover B~

(

B~

)B~i

J

0

TT~jq ) :TT(~i Byj + ~i ~ .

.

.

i = I

.

.,k.

Consequently,

(3.1.11)

l~i

j B~

Ici/~(~i)l

Byj

~ KR.

S u b s t i t u t i n g t h i s bound i n t o (3.1.10) and observing t h a t (3.1.12)

By. l~--~-(Ata(t)~ )I s ~o n,

u = I , . . . . m, kAt ~ t s nat,

we can estimate the r i g h t side of (3.1.9) by I + ~ e n A t which proves the i n e q u a l i t y (3.1.8) f o r i m p l i c i t

methods.

Theorem (3.1.3) does not need the assumption t h a t the domain R is contained in the open i n t e r i o r

~ of the s t a b i l i t y

here t h a t S has a non-empty i n t e r i o r . 1.2 s a t i s f i e s

region S thus, in p a r t i c u l a r ,

i t is not necessary

So~ f o r instance, the Simpson r u l e of Section

the assumption with

R = {n = i ~ , - s < ~ ~ s}, 0 < s < v~~. On the other side, assumption ( i v ) is d i f f i c u l t

to delete in the above way of proof,

and a g e n e r a l i z a t i o n of the r e s u l t to d i f f e r e n t i a l

systems y' = A ( t ) y ÷ c ( t ) w i t h gen-

eral time-dependent matrix A is obstructed by the u n a v a i l a b i l i t y

of a s u i t a b l e dimen-

s i o n f r e e mean value theorem f o r f u n c t i o n s of m a t r i x - v a l u e d f u n c t i o n s f ( A ( t ) ) ;

see e.g.

Gekeler [81 ].

3.2. D i f f e r e n t i a l

Systems of F i r s t Order and Methods w i t h Non-Empty

I t was the merit of LeRoux [79a] to have shown a way of error estimation for the problem studied in Section 3.1 which circumvents a Jordan canonical decomposition of the Frobenius matrix F (n) and avoids so the d i f f i c u l t i e s arising in this way of proceeding i f more general results are desired. We dedicate this section to the work of LeRoux in which linear d i f f e r e n t i a l systems with a rather general operator A(t) are considered. However,for the sake of simplicity, we shall confine ourselves to the

76

special non-autonomous d i f f e r e n t i a l

system (3.1.1) and adopt moreover the assumption

of Corollary (1.3,15), The o r i g i n a l

results of LeRoux concern " s t r o n g l y A(m)-stable"

l i n e a r m u l t i s t e p methods of which the best-known representatives are the backward d i f f e r e n t i a t i o n methods given in AppendixA.4. The subsequent generalization to l i n e a r multistep methods with non-empty ~ ~ IR in a neighborhood of zero suggests i t s e l f under the stipulated r e s t r i c t i o n . Before we formulate the result of this section we have to modify the exponential multiplication factor of Theorem (3.1.3) s l i g h t l y : Let

On = m a x k ~ n m a X ( ~ - 1 ) A t ~ t ~ A t l a ' ( t ) / ( a v - l a v )I/21 and e*

n

:

t max{On'~n} Lgn i f

B0 = BI = , . . = Bk-1 : 0 ( c f .

(3.2,1) Theorem. (LeRoux {79a].)

(3.1.2)).

( i ) Let the (m,m)-matrix a(t)A in (3.1.1) be hez~ni-

team and negative definite, a(t)A ~ - ~I, ~ > O, t > O, and let the solution y of (3.1.1)

be (p+l)-times continuously differentiable.

(ii) Let the method (3.1.2) be consistent of order p with stability region S; let o

0 E S, [ - $ , O) C S, 0 < s ~ ~, and Re(x~) > 0, i = I . . . . . k,. (iii)

Let Atla(t)A I s s, t > 0.

Then f o r n = k,k+1, . . . .

](Yn-Vn)I~K~exp{~e~nat}[IYk_1-Vk_11+~tpiAtexp{-~o~}lY(P+1)(T)IdT+nAtlllhHn].

Proof. Let ~n = Fn + Rn = ~(At'AtanA) be the leading matrix in the e r r o r equation (3,1.4) then we obtain instead of (1.2.13) n-1 IEnl ~ l~n.-.~kIIEk_11 + ~v=kl~n...~ +11[D(At,y)v - Hvl + ]B(At,y) n - Hnl. Consequently, i f l~n'"~l

~ KseXp{K~O~(n-~)At},

v = k . . . . . n,

then the assertion follows in the same way as in Chapter I. But, since A is now unitary diagonable we have l~n...~ I = I~(At,AtanA)...~(At,Ata A) I = max1~mI~(At,AtanXu)...~(At,AtavX ) ] •

77 Writing shortly

(3.2.2)

@n = ~ ( A t ' A t a n ~ )

we thus have to prove

maxo~t~nAtSUP_s~Ata(t)~O[~n...¢

[ ~ ~seXp{~e~(n-~)~t},

~ = k . . . . . n.

The proof of this crucial inequality is derived in several steps. The f i r s t one is of rather general character: (3.2.3) Lemma. Let IN0 =IN ~ {0} and l e t Pi and Qi' i = 0,1 . . . . , be two sequences o f matrices such that

(i)

suPiE]No,JEINoIQ~I

(ii)

maxosi~nIP i - Qi [ s ~iCn ,

n = 0,1 . . . . .

(iii) Then

max15i~n,1~j~n[Q ~ - Q~_I [ ~ ~27n ,

n = 1,2 . . . . .

IPn...P]

~0 (2 + I P o [ ) '

""

the f o l l o w i n g

~ = 0 . . . . ,n, n = 0,1 . . . . .

~0~I + ~2 }"

I t s u f f i c e s to prove the a s s e r t i o n

for n = 0,1,..., Pn

n = O,1

~ ~3exp{<3rn(n-~)},

where ~3 = m a x { [ P o i '

Proof.

~ <0'

f o r v = O. By LeRoux t79a, Prop. 3] we have

decomposition w r i t i n g

"Po = nn+1 ~n+1(nn+1-ip ""Po ~0 + Li=1 " i i-I

Pi.o.Pk = I for i < k

nn+2-i - "i-I Pi-2""Po

)

= nn+1 n n+1-i _ nn+1-i + ~n+IQn+1-i(Pi - Qi 1)Pi 2 " " P o " ~0 + ~i=1(Qi ~i-I )Pi-I""Po ci=1 i-I -I Writing z i : IPi_1...P01 we obtain z I = IPo[ ~ ~3 and

< + r n ~ ~ r ~n+1 Zn+1 = CO IQ~IIPo - Qo [ + ~2 n ~ i = I Z i + 0 1 n L i : 2 z i - 1 or Zn+ I ~ ~3(I + rn~=izi) ,

n : 1,2 ....

By induction, finally, we derive from this inequality that

z i ~ <3(I + ~3rn ) i - I ,

i : I,...,n+I,

Ipi...po[

i = O,,.,,n.

or ~ ~3exp{<3rni},

Under the assumption of Theorem ( 3 . 2 . 1 )

the Uniform Boundedness Theorem y i e l d s

78

sUP-s~n~0SupjEINJFx ( n ) j j

~ <s

and (3.1.5) says that i

]Ri] = IR(At,AtaiX) I = I@i - F (AtaiX) I ~ KOnAt,

= 1,...in.

Writing Pi = ¢ i ' Qi = F ( A t a i x ) , and rn = e*At,n these are the f i r s t two assumptions of Lemma ( 3 . 2 . 3 ) . Accordingly, t h i s lemma y i e l d s the desired i n e q u a l i t y (3.2.2) i f the t h i r d assumption is f u l f i l l e d (3.2.4)

which now reads as

max1~i~nmax1~j~nSUP_s~Ata(t)x~0JF (AtaiX) j - F (Atai_IX)Jj

~ Ks~nAt.

In order to prove t h i s i n e q u a l i t y we need some a u x i l i a r y r e s u l t s . (3.2.5) Lemma. n-1 . j))-I/2 ~j=1(J(n -

~ ~.

Proof. By a standard argument we f i n d that I/2=argmin0<x<1(x(1 - x)) - I / 2 therefore we obtain f o r j = I . . . . , n - l , n-1 [ ( j / n ) ( 1 -1 n [(j/n)(1

j/n - [4 ~ j / n j~j ] -1/2 ~ [ (x(1 - x ) ) - l / 2 d x , j-1)/n (j+1)/n - ( J / n ) ) ] - I / 2 :< S (x(1 - x ) ) - I / 2 d x , j/n

j / n s 1/2, j / n => I / 2 ,

which leads to the desired r e s u l t , n-1 . j))-I/2 Zj_1(J(n _

< =

I f(x(1

-

x))

-I/2dx

~

~.

o

(3.2.6) Lemma. I f 0 E S, I-S, 0) C S, 0 < s < - , and Re(×~) > 0, i =I . . . . . k., then

there e x i s t p o s i t i v e constants ~s and ~*s such that JF (n) n] ~ <seXp{~nn},

-

S <

n =< O,

n =

Proof. By Corollary (1.3.13) there e x i s t p o s i t i v e constants ~, KI , and ~ (3.2.7)

IF (n)nJ ~ ~lexp{K~nn},

1,2,...

such that

- 6 ~ n ~ 0.

7g

Because [ - s ,

-a] c S there exists

i n Theorem ( 1 . 2 . 1 8 )

IF ( n ) n l

a u > 0 such t h a t

~ ~seXp{-nu},

- s ~ n ~ - 6. from t h i s

(3.2.8)

- s ~ n ~ - 6.

From ( 3 . 2 . 7 )

- a ] c S . In t h e same way as

we then o b t a i n

As an immediate consequence we f i n d

IF ( n ) n l

[-s,

~ ~seXp{(~/s)nn},

and ( 3 . 2 . 8 )

the assertion

inequality

now f o l l o w s

that

w i t h <s = max{K1' Es } and ~*s =

min{K~, u/s}. o

Lemma. I f

(3.2.9)

0 E S, [-~, O) c S, and Re(x~) > 0, i = I . . . . . k . , then there e x i s t

positive constants S0~ K~ and ~* such that

(i)

IF ( n ) n l

~ ~exp{K*nn},

-s0
(ii)

IF ( n ) n l

~ ~exp{-<*n},

-~


s0~

(iii)

IF (n) n - F (~)n I ~ ~ e x p { - K * n } / I n I

-~


s0.

Proof.

By assumption we have s p r ( F ( - ) )

< I hence t h e r e i s a n e i g h b o r h o o d {n E ~ ,

Inl m s O} o f - such t h a t

spr(F (n))

second a s s e r t i o n

as in Theorem ( 1 . 2 . 1 8 ) ,

The f i r s t

w i t h s = s 0. In o r d e r t o p r o v e t h e l a s t

assertion

(3.2.6)

Lemma ( A . I . 3 ) (3.2.10)

follows

and t h a t

IFJn)

- F(~)I

<-

< I - ~0 < I by c o n t i n u i t y

for

Inl

assertion

m s o • Now t h e

is that

observe that

o f Lemma

Bk ~ 0 by

by a s i m p l e e s t i m a t i o n ~


For n > 1 we have F~(n) n - F~(~) n = Tn-I Li=0 F( n )

therefore

we o b t a i n

by ( 3 . 2 . 1 0 )

i (F (n) - F ( ~ ) ) F

and the second a s s e r t i o n

IF (n) n - F (~)n I < ~ -I~n-I -~*i -~*(n-l-i) : < n Li=0 e e

(3.2.11)

(n) n - l - i

Lemma. Let the assumption of Theorem ( 3 . 2 . 1 )

Then, for every fixed ~ > O, and 0 < x < s,

<
< n < s O• - = = : -

be fulfilled and let X = - aX=> O.

8O

(i)

Ix~F (Atx)Jl

~ ms(jAt) -~,

(ii)

Ix~(F (Atx) j - F (~)J)i

S < ~,

S < ( j A t ) -~

Proof. For s < co we have by Lemma (3.2.6) [x~F (Atx)Jl

, =< KsX ~exp{-~sDAtx}

S = ~, j = 1,2,

with K*s> 0 • ~ exp{-KsJAtx}(JAt) ,. . -~ =< ~s(j A t) -~. = ~s(JAtx)

For s = ~ we have spr(F (~)) < I. Consequently there are by Lemma (3.2.9) constants

positive

So, ~, and K* such t h a t f o r 0 < x < S0/At iF (Atx)Jl

=< < e x p { - K * j A t x } ,

and so we obtain by the f i r s t

< Kexp{-K*j},

assertion

Ix~(F (Atx) j - F (~)J)l < <1((jAt)-~

IF (~)Jl

_-< < s 0 ( j A t ) - ~

+ (s0J)• Eexp{-~ , .j } ( A t j )

+ <xCexp{-~*j} -~) _< <2(Atj) -~,

For x > S0/At we have by Lemma (3.2.9) IxC(F (Atx) j - F (~)J)i

< ~exp{-~*j}XE/AtX

< ~exp{-~*j}/At~S~ -E :< ~I J.c exp{-< , j } / ( j A t )

< ~exp{-~*j}/[(AtX)I-EAt

c]

~ < ~2 (j A t) -~,

(3.2.12) Lemma. Let the assumption of Theorem (3,2.1) be fulfilled. Then for i = I,... .,olnl ¢i)

IF (AtaiX) - F ( A t a i _ i x ) I ~ K~nAt,

(ii)

I(F (AtaiX) - F ( A t a i _ i X ) ) ( a i a i _ I X 2 ) - I / 2

(iii)

I(F (AtaiX)

- F (Atai_iX))(azx)-I/21

Proof.

For b r e v i t y

l e t x i = aix and F i = F (AtX i) then we have

IFi-Fi-ll

~ kl/2max0~k-li(~k-

Extending by ( A t x i ) I / 2

i s K~nAt 2, ~ K~nAt 3/2,

8kAtXi)-1(B~k-

Sk~v)At(xi-xi-1)(~k-

and (Atxi_1) I/2 we thus obtain

= i-I ,i.

8kAtXi-1 ) - I I "

81

IF i - F i _ 1 1

s
which y i e l d s the f i r s t

- BkAtxi_1)-1(&txi_1)I/21

assertion observing that ..l&txil is bounded i f Bu9 = 0, The second

and the t h i r d assertion f o l l o w in a s i m i l a r way extending with (Atx~) I~2 in the l a s t case, The f i r s t

assertion of Lemma (3.2.12) is the i n e q u a l i t y (3.2.4) f o r j = I . We

now turn to the v e r i f i c a t i o n

of that i n e q u a l i t y f o r j = 2 , . . . , n

w r i t i n g again x i = a i x

Fi = F ( A t x i ) , and F = F (~): ( i ) For 0 < s < = we choose the p a r t i t i o n

I FJ" - FJ

j-1 ~ rJ-l-v i-ii --
_< (IF~I

+ IF ji - I I)IF i - Fi-1 ] + mj-2 L,~=I xi112-~ rill(Fi

- Fi-1)(xixi-1)

-112 ,112 j-1-~ ~i_iFi_1 I.

Then the Uniform Boundedness Theorem and Lemma (3.2.12) y i e l d IF~ - FJi - I I < <s~nAt(1 + At~-_# xil/2~Vri xi-1ri-11/2"j-l-v ) the f i r s t

assertion of Lemma (3.2.11) y i e l d s f o r c = I/2

IF j,

Fjiii

< ~s~n~t( I + Z~-~(~(J-I-~))-112),

and thus Lemma (3.2.5) proves the desired r e s u l t f o r the case of f i n i t e (ii)

s.

For s = ~ we choose the p a r t i t i o n

IF~

FJi_11<- #-2,,:I~, - '
IF i

- Fi_I I -i-IFJ-l-u _ FJ-l-u I .

(3.2.13)

+ #-I ,,=iIF~.-

o F~I IFi Fi_11IFJ-I-~I

÷ zj-I ~=0 F~I IFi - F i-I IIFJ-I-~ I. Now, Lemma ( 3 . 2 . 1 1 ) ( i i )

f o r ~ = I / 2 , Lemma ( 3 . 2 . 1 2 ) ( i i ) ,

J-l-v ~-~IF~"- FZIIFi- Fi-111Fi-I

_ FJ-l-v

~o

and Lemma (3.2.5) y i e l d

I

j - 2 xII2{F u . ,-I12, I 1 2 ( r j - I - ~
= s n

Lv=1 ~ ( j - 1 - ~ )

)-I12

< ~sen At"

_

FJ-I-~)I

82 Lemma ( 3 . 2 . 9 ) ( i i ) , (3.2.5) y i e l d

Lemma ( 3 . 2 . 1 1 ) ( i i i )

j-l-u Z~-~IF~IIFi Fi-ll ~"i-1 -

-

FJ-I-~ I

< <senatZJ ~ ~ '_ ~e _ < * ~ ( j - l - v ) I / 2_

f o r ~ = I / 2 , Lemma ( 3 . 2 . 1 2 ) ( i i i ) ,

j-2 ,-1/2 xl/2cFj-l-v_ < ~=01F~VIl(Fi-Fi-lJXi 1 I i-1"i-1

__<^<sOnAt(I ~ + Z ~_- _ # ( v ( j _ 1 _ v ) ) - I / 2 )

and Lemma

F£-I-~)I

< = Ks~nAt

and a n a l o g e o u s l y

zj-1 Fu. _ FVIIF i

~=11 1

=

- Fi_ll

iFJ-l-vl

< ~ ~ ,~j-l^-<*(j-l-v) = <sUna~Lv=l =

v

-1/2

< ^ ~ j-2 -1/2 ~ t = <senAt(1 + Z v = l ( ~ ( j - l - ~ ) ) ) < <senZ~ . Lemma ( 3 . 2 . 9 ) ( i i )

and Lemma ( 3 . 2 . 1 2 ) ( i ) y i e l d

zJ-IIF~IIFiv:O - F i - I I I F J - I - ~ I

< ~nZ~tZ~-~ e-<*(w+(j-l-w)):

A s u b s t i t u t i o n of these bounds into (3.2.13), f i n a l l y ,

< ~OnA t j e - K * ( j - I )

"< K~)nZ~t"

proves that

I F~ - FJi-II < : <s~nz~t holds independently of i , j , and x i E [-~, 0]. This is the desired i n e q u a l i t y (3.2.4) for s :~.

3.3. An Error Bound f o r D i f f e r e n t i a l

Systems of Second Order

The exact solution of the l i n e a r i n i t i a l (3.3.1)

y"

value problem

: a(t)2A2y + c ( t ) + h ( t ) , t > O, y(O) = YO' y'(O) : y~,

o s c i l l a t e s without decreasing e x p o n e n t i a l l y i f the d i f f e r e n t i a l

equation is homogene-

ous and the leading matrix a(t)2A 2 is hermitean and negative d e f i n i t e . This behavior should n a t u r a l l y be carried over to the numerical approximation scheme and, indeed, convergent l i n e a r multistep methods with a p e r i o d i c i t y i n t e r v a l [ - s , 0], c f . D e f i n i t i o n ( 2 . 1 . 2 6 ) , i n h e r i t t h i s property from the a n a l y t i c problem under the spectral condition la(t)2A21 ~ s. But the e r r o r estimation of the preceding section u t i l i z e s

in an essen-

t i a l way that the spectrum of the leading matrix is contained in the i n t e r i o r of S, c f . Lemma (3.2.6) and Lemma ( 3 . 2 . 9 ) , and the p e r i o d i c i t y i n t e r v a l obviously is a subset of the boundary of S. Therefore the way of proceeding proposed by LeRoux [79a]

83 f a i l s here. I t is an open question whether the above ingenious technique can be gener a l i z e d to l i n e a r multistep methods for second order problems i f the s t a b i l i t y

region

has a non-empty i n t e r i o r of a suitable form. The main obstacle f o r a straightforward adaption seems to be the fact that now ~(~,n 2) has two p r i n c i p a l roots coalescing to one f o r n2 ÷ 0 therefore the Uniform Boundedness Theorem y i e l d s c o r r e c t l y

- S
IF (n2)n I ~ <s n, which cannot play longer the role of the f i r s t

assumption in the basic Lemma (3.2.3)

without f u r t h e r ado. Because of the two coalescing p r i n c i p a l roots the error estimation of Section 3.1 does not hold longer, too. The bound (3.1.7) f o r the inverse of the Vandermonde matrix W holds only i f a l l roots of ~(~,n 2) are bounded away from each other. In the present s i t u a t i o n t h i s implies that At must be bounded away from zero in the case where the leading matrix is negative d e f i n i t e and the p r i n c i p a l roots are the sole coalescing pair coinciding only in n2 = 0. However, t h i s drawback is only of technical character. Subsequently a more thorough study of the Vandermonde matrix leads to error bounds which correspond to those of the f i r s t

section in a s a t i s f a c t o r y way.

Using the notations introduced in Section 3.1, a l i n e a r multistep method f o r the problem (3.3.1) with general time-dependent matrix A(t) 2 has the form

(3.3.2)

p(T)Vn - At2o(T)(A2V)n = At2o(T)Cn,

n = 0,1 . . . .

Let the exponential m u l t i p l i c a t i o n f a c t o r be once more modified s l i g h t l y ,

e** = max n

Ion : max k~vsnmax(v-k)At~t~vAt la Len

(t)/al 2 ~ = maxk~vsnmax(v_k)At~tsvAtl(a(t) ) ' / a I

a = a(vAt), v

and r e c a l l that the essential roots ~i(n) of x(~,n2), i . e . the roots with I~i(O) ] = 1, are numbered by i = 1 , . . . , k . ,

without loss of g e n e r a l i t y . Then the f o l l o w i n g r e s u l t is

due to Hackmack [81, Theorem 4 . 8 ] . (3.3.3) Theorem.

(i) Let the (m,m)-matrix a(t)2A 2 in (3.3.1) be hermitean and negative

definite, a(t)ZA 2 =< - ml, a > O, t > O, and let the solution y of (3.3.1) be (p+l)times continuously differentiable. (ii) Let the method (3.3.2) be consistent of order p with stability region S, and let [ - s , 0] C S, 0 < s ~

~.

(iii) Let the Frobenius matrix F(~ 2) be diagonable for n2 E I-s, 0). Then for n = k , k + l , . . . , ly n - Vnl < ~s(1 +

0 I _ ~i(0)E ~ 21 + -1)exp{K~C)n*nAt} [ IEk_11 + At-lmax1
84

+ ,

where E0n = (Yn-k+2 - Vn-k+2 . . . .

Yn

Atp ~ -

exp{-<sOn*~}IY(P+2)(T)Id~ =

Vn )T and En

(Yn-k+l

_

Vn-k+l

+

nAtltlhllln 1•

,

E0) T

n- "

Proof. As in Section 3.1 l e t F (n) = W(n)Z(q)W(n) -I be the Jordan canonical decompo2 s i t i o n of F (n). Let A2 = (X ,X m) be the diagonal m a t r i x of the eigenvalues of A2 and w r i t e b r i e f l y Wn = W(AtanA) and Zn = Z(AtanA). Then the e r r o r equation ( 3 . 1 . 6 ) the same as in Section 3.1,

(3.3.4) WnIx-tEn = (Zn + WnIX-IRnXWn)WnIWn-Ix-tEn-1 + WnlX-I(D(At'Y)n - Hn)' whereR n i s again a ( k , k ) - b l o c k

m a t r i x of which only the l a s t block row in nonzero:

l a s t block row of R n (3.3.5) =

(At2A2(mk I

_ At 2 2A2,-I 2 2 - ~k an ) (B0(an-k - a n ) l " " ' B k - 1

D(At.y) n and Hn are the same vectors as in (2.1.37) the same way as in the proof of Theorem (2.1.34) IX-1(a(At,Y)n

tZnl

s

1,

IWnl

and (2.1.38)

2 I) - an) " hence we obtain in

for ~ = I that

- Hn) I s <s(At p+l nat {n-k)At ly(p+2)(~)ld~

For the e s t i m a t i o n of the m u l t i p l i c a t i o n

(a~_ I

+ ~ t 2 l l l h l l l n ),

f a c t o r we observe t h a t by assumption ( i i i )

~ ~,

and consequently (3.3.6)

- I X-IRnXl)IW~IWn_II. [(Z n + WnIX-IRnXWn)(WnIWn_I) I =< (I + mlWn

In order to estimate (3.3.7)

IWnIWn_1 I -

we observe t h a t

IWnIWn_11 = IW(AtanA)-Iw(ALan_IA) I = max IW(ALanX )-Iw(ALan_IX ) I . 1~u~m

Now we consider two cases: ( i ) I f s < ~ then Lemma (A.2,6) y i e l d s f o r - s ~ At2a(t)2~ 2 ~ 0 (3.3.8) (ii)

IW(Atan~)-Iw(Atan_1~)I ~ I + ~ s l A t a n ~ l - I I A t a n ~ - Atan_1~l S I + ~sOnAt.

I f s = ~ then Lemma (A.2.6) y i e l d s f o r - s s At2a(t)2~ 2 ~ 0 and IAtan~ I m I

is

85

IW(AtanX)-Iw(Atan_IX)i

< I + ~smaX1
But now we have (~k ~ 0 by Lemma ( A . I . 3 ) hence the same estimation ding part in the proof of (3.1.8) shows that i~i(AtanX)

- ~i(Atan_iX)i

as in the correspon-

< <s~nAt.

For s = ~ and IAtanX i < I Lemma (A.2.6) yields for - s < At2a(t)2X 2 < 0 (3.3.9)

IW(AtanX)-Iw(Atan_iX) i < I + ~sIAtanXi -I max i~i(AtanX) 1_-
- ~i(Atan_iX) i.

In this case, too, the estimation follows the lines of the proof of (3.1.8) (3.1.12) is to be modified in the following way: ~(At2a(t)2X

2)

= ~-~-~-J- BjAt2a(t)2x2 ~k 6k At2a(t)2X2

K~

(At2a(t)2X 2)

<

but

j '

0, =

,k-1 ....

Therefore we obtain from (3.1.10) Ici(AtanX) and accordingly

- ~i(Atan_IX)i

by ( 3 . 3 . 9 ) ,

222 s ~sIAt anX i~]nAt

because IAtanX i < I in the present case,

IW(AtanX)-Iw(Atan_iX)i

< I + ~sIAtanxI-lIAt2a2nX21~)nAt

< I + Ks~nAt,

NOW we have shown that IW(AtanX)-Iw(Atan_iX)i and a s u b s t i t u t i o n

:< I + Ks~nAt ,

- S < At2a(t)2X 2 < 0, 0 < S < ~,

of t h i s bound into (3.3.7) yields

(3.3.10) IWn~Wn_11 _<-I + ~s~nAt. In order to estimate (3.3.11)

IWnIX-IRnXi we observe that

IWnIX-IRnXi : IW(AtanA)-IR(AtanA)i

= max1~u~mIW(AtanX )-IR(AtanX

We omit the argument AtanX u and w r i t e k

W-I = [ w ~ j ] i , j = 1 , then

R = {rij]

k ,j=1' Z = [zij]i,j=

I - W-IR,

)i.

86

(3.3.12)

zij

:

w ~IK . r . KJ

because the f i r s t

k-1 rows of R are zero. Now we have by (3.3.5)

i r k j i = iAt2X2(~ k

_ At2a2X2,-1 2 - ~k n ) Bj-1(an-k+j-1

2 - an)I

(3.3.13) ^ At 2an2X2 ) -I lenAt' ~IAt2a~X211(~k _ ~k and by (A.2.5) we have (s O = I) (3.3.14)

lW*ikl = i ~ ( ~ v ( A t a n X )

- ¢i(AtanX))-11

=< <si¢i(AtanX)

- ~i+1(AtanX)] -I

~I

because a l l roots ~ (~) with u ~ i+I are bounded away from ~i(n) f o r n 2 E I - s , 0] by assumption (iv) and because 0 E S. ( i ) I f s < ~ then we obtain from (3.3.14) and Lemma ( A . 1 . 1 0 ) ( i i ) (3.3.15)

[W*ikl <: ~smax{ I,

IAtanX] - I }

and la k - 6kn21 -I is uniformly bounded f o r n 2 E [-s, 0] independently of f i n i t e i n f i n i t e s. Thus (3.3.12), (3.3.13), and (3.3.15) y i e l d together

or

^ t <: msOnAt, A IZij i ~ ~smaX{1, IAtanXi -I }IAtanX i 2~nAt ~ ~smaX{IS, S}enA 0 < S < -. ( i i ) I f S = ~ and IAtanX i ~ I then we obtain by Lemma(A.1.10)(ii) lWTk i ~ K hence, because now Bk m 0, (3.3.12) and (3.3.13) y i e l d IZijl

~ ~i At 2anX 2 2 (~k - ~k ^ At 2an2X2'-I ^ t ~ ~OnAt" ^ ) lena

I f S = - and IAtanX I < I then we obtain by Lemma(A.1.10)(ii) fore, by (3.3.12), Izijl

and (3.3.14) that

again (3.3.15) and there-

^ ~ ~IAtanXi-1 IAtanX I 2^enAt ~ ~enAt.

These bounds together imply that under the assumption of the Theorem IW(AtanX)-IR(AtanX)i and a s u b s t i t u t i o n

~ <sGnAt

into (3.3.11) y i e l d s

^ IWn IX-IRn X [ ~ ~sOnAt.

87 The e r r o r equation ( 3 . 3 . 4 ) ,

IW~IEnl s But

(I +

(3.3.6),

the bound (3.3.10) and this bound now y i e l d

<~e~*At)2(n-k+1)IW~ZiEk_11+ ~=k(1 + ~e~*~t)2(n-~)IW-1(D(At,y) ~ - H~)I.

IW~IEnl~ IWnl-11EnI ~ <-11Enland IW~1(D(At,Y)

-

Hv)l

~ At-1~-II(D(At,Y)

-

H)I

follows in the same way as the estimation of (3.3.11) because the f i r s t k-1 block elements of the vectors D and H are zero. The remaining part of the proof follows the repeatedly described way estimating

IWkZIEk_II by Lemma (A.2.10).

In concluding this section i t should be remarked that obviously the following bounds f o r the i n i t i a l error can be inserted in Theorem (3.3.3): 0 IEk_11 + ~t-lmaxlsi~k, IE~_ I - ~i(O)Ek_21 s ~at-11Ek_11 and i f the method (3.3.2)

is strongly D-stable in 0 E S then

max1~i~k. IE~_ I - ~i(O)E~_21 = IE~_ I - E~_21. Thus, with exception of the technical assumption ( i v ) , Theorem (2.1.34) and Theorem (2.1.39) respectively.

Theorem (3,3.3) agrees well with

IV. Linear M u l t i s t e p Methods and Nonlinear D i f f e r e n t i a l

4.1. An Error Bound f o r Stable D i f f e r e n t i a l

Systems of. F i r s t Order

Systems

As e a r l y as 1959 Dahlquist [59] has derived a - p r i o r i e r r o r bounds f o r l i n e a r m u l t i s t e p methods and n o n l i n e a r d i f f e r e n t i a l absolute s t a b i l i t y

systems but he introduced the concept of

f o r numerical methods not before 1963 in a very celebrated communi-

c a t i o n . The c o n t r i b u t i o n s of Dahlquist and the book of Henrici [62] c o n s t i t u t e the foundations of modern numerical a n a l y s i s of i n i t i a l

value problems. In t h i s section

we shall combine one of D a h l q u i s t ' s former ideas with a recent r e s u l t of LeRoux mentioned already above and concerning l i n e a r m u l t i s t e p methods and l i n e a r i n i t i a l

value

problems, (4.1.1)

y' : A ( t ) y + c ( t ) ,

t > O, y(O) = YO'

(4.1.2) Theorem. (LeRoux [79a].)

( i ) For t > 0 l e t the (m,m)-matrix A(t) in (4.1.1)

be hermitean and negative definite, A(t) ~ - ~ I , ~ > O, and let the solution y of

(4.1.1) be (p+l)-times continuously differentiable. (ii) Let there exist two positive constants, g and ~, 0 < ~ < I, such that for t > 0

IA(t + A t ) - ~ ( A ( t + At) - A ( t ) ) A ( t ) - 1 + ~ I ~ e a t , and if o(~) ~ Sk~k in (3.1.2) then

IA(t)-I(A(t)

- A(~))I ~ Oft - ~ I ,

t - kAt < T < t .

(iii) Let the method (3.1.2) be consistent of order p with the stability region S, let o

l e t 0 E SD [- ~, O) C S, and Re(×~) > O, i = I . . . . . k.. Then there exist two positive constantsj ~ and ~*p depending only on ~ the data of the method (3.1.2) such that for n : ktk+1~...s

lYn -

Vnl ~ <eK*OnAt[IYk-1 -

~

AtoJ and the

and 0 < At ~ At 0

nat .~ , , Vk-11 + AtP I e-~ UTlY~P+l)(~)Id~]"

The proof of t h i s theorem f o l l o w s the l i n e s of Theorem (3.2.1) but the necessary a u x i l i a r y r e s u l t s are more d i f f i c u l t

to d e r i v e t h e r e f o r e we r e f e r here to the o r i g i n a l

c o n t r i b u t i o n of LeRoux [79a]. We now f o l l o w Hackmack [81] and turn to n o n l i n e a r i n i t i a l

value problems,

89

(4.1.3)

y' : f ( t , y ) , 0 < t < T, y(0) = Y0'

f : [0, T] × IRm ÷IRm, and l i n e a r multistep methods, (4.1.4)

n = 0,I,...,

p(T)V n - Ato(T)fn(V n) = 0,

or k-1 k-1 ~kVn+k - BkAtfn+k(Vn+ k) = - Zi=0~iVn+i + A t Z i = 0 B i f ( ( n + i ) A t , Vn+i ) , n = 0,1, . . . .

Naturally, if

6k ~ 0 then At must be chosen s u f f i c i e n t l y

small here

such t h a t Vn+k e x i s t s f o r a l l n. With respect to the problem (4.1.3) the f o l l o w i n g n o t a t i o n s are used in t h i s section: (4.1.5)

A(t) = ( ~ f / B y ) ( t , y ( t ) ) , Kf(~) = suP0
where y is the exact solution of (4.1.3). ~ denotes a s u f f i c i e n t l y large convex domain such that y and the numerical approximation v are contained in R.

An interdependence

of this form appears always in the error estimation of genuine nonlinear problems. I t is however not a serious r e s t r i c t i o n because i t can be assumed under s u f f i c i e n t smoothness of f that this function is replaced by a function ~ having bounded second derivatives with respect to y such that f and ~ coincide inside a tube in [0, T] x IRm containing the exact solution y. For the following a u x i l i a r y result we refer to Ortega and Rheinboldt [70, § 3.3}. ^2,rE; iRm~ (4.1.6) Lemma. Let ~ C IRm be convex and let f E C j then

If(v)

where f '

- f(w)

- f'(w)(v

- w) I ~ s u P 0 ~ 1 1 f " ( W

denotes the Jacobi matrix and f "

+ ~(v - W))llV

- Wl 2

the (m,m2)-matrix of the second derivatives

of f.

The next lemma is the announced result proposed by Dahlquist [59 ] for the error e s t i mation of nonlinear problems. (4.1.7) Lemma. Let F > 0 be a fixed constant. Let @ and ~ be two nonnegative continuous functions such that

¢(0) = ~(0) = 0, ~(At) s ¢(At) + rC(At) 2, 0 < At < At 0, 4r~(At 0) = I ,

90

and let t be strictly monotonically increasing for 0 ~ At ~ At 0. Then 0 < At < At 0.

~(At) ~ 2@(At),

Proof. We c o n s i d e r At E (0, At 0) and o b v i o u s l y have 0 < 4~¢(At)

< I hence the equation

= ~(At) + re 2 has two s o l u t i o n s ,

0 < to(At)

< it(At),

and limAt÷OC1(At)

is p o s i t i v e .

t i o n t h a t r ~ ( A t ) 2 - ~(At) + ~(At) ~ 0 we o b t a i n t h a t ~(At) ~ t 0 ( A t ) But ~(At) < ~1(At) f o r At > 0 s u f f i c i e n t l y tinuity

of 5. Now the a s s e r t i o n

+ @ satisfies

by V i e t a ' s

follows

small t h e r e f o r e

From the assump-

or ~(At) ~ ~ 1 ( A t ) .

~(At) ~ t 0 ( A t )

by the con-

observing t h a t the s m a l l e r root ~0 of r~ 2 -

root c r i t e r i o n

0 < ¢0 = ~ + r t ~ < ~ + F~0~ I = 2~.

With ( 4 . 1 . 5 )

the e r r o r equation is now w r i t t e n

P(T)en - At~(T)(Anen)

= At~(T)(fn(Yn)

as

- fn(Vn ) - Anen) + d ( A t ' Y ) n

where again e n = Yn - Vn and d ( A t , y ) n is the d i s c r e t i z a t i o n

~ dn' n = 0,1 . . . . .

e r r o r estimated

(1.1.8).

By means of Lemma ( 4 . 1 . 6 )

(4.1.8)

l ~ ( T ) ( f n ( y n) - fn(Vn) - Anenl ~ < < f ( ~ ) ( ~ k l E n + k 12 + IEn+k_112)

and i f

the i n i t i a l

(4.1.9) satisfies

0 < t < T, } ( 0 )

the assumption of Theorem ( 4 . 1 . 2 )

w i t h respect to the d i s c r e t i z a t i o n

= Y0'

then a s l i g h t

modification

w i t h Dv = (0 . . . . , 0 , ( ~ k l ~ ~[IEk_l[

of t h i s Theorem

error yields

IEnl ~ < e K * e n A t [ I E k - I I + ~n Lv=k e - < * e v A t I D ' v I ]'

or

,

value problem

~' = ( B f / @ y ) ( t , y ( t ) ) ~ ,

e-K*enAtlEnl

in Lemma

we o b t a i n

- BkAtAn)-1~_k)T. + ~=ke-<*0vAt(

n = k,k+l . . . . .

Therefore we o b t a i n At~f(~)(IEvl

2 + IEv_112) + d ( A t , y ) v _ k ) ]

91

e-K*enAt[En I S @(at) + 2at<~f(R)Z~:k e~*ev~t(e-~*evAtlEvl) 2, n : k,k+l . . . . . where @(At) ~ <[IEk_11 + ~f(~)IEk_ I j2 + at p iAte-K*O~ly(p+1) (T)IdT]. This i n e q u a l i t y

remains true i f on the r i g h t side the step number, n, is kept fixed

and on the l e f t

side n is replaced by ~, k c u c n. Then we obtain

max k~u~n ~e ~ -<*euAtl

(4.1.10)

,El}

s ~(&t) + r(maxk~sn{e -<*o~At lEvi )2

where r

~
<*OT K*G

- I >= At<
e

~*evAt •

Here, K, <*, and o are the constants of Theorem (4.1.2) with respect to the problem (4.1.9).

Now we assume that IEk_11 = C(At), At ÷ 0 then ~(at) and ~(At) = maxks~n{e-<*euAtlE

I}

f u l f i l the assumption of Lemma (4.1.7) Consequently, (4.1.10) yields IE J ~

for a certain At 0 > 0 defined by 4F@(At 0) = I.

2~(At)eK*0u~t,

= k,...,n.

The r e s u l t may be summarized in the following theorem. (4.1.11) Theorem. (Hackmack [81].)

( i ) Let the solution y o f the problem (4.1.3)

exist

uniquely and be (p+l)-times continuously differentiable; let f be two-times continuously differentiable with respect to y in a sufficiently large domain [0, T] × ~. (ii) Let the initial value problem (4.1.9) and the method (4.1.4) satisfy the assumption of Theorem (4.1.2) , and let ~, ~*, and g be the constants of Theorem (4.1.2) with respect to (4.1.9). (iii) Let the initial values Vo,... , Vk_1, satisfy IYk_ I - Vk_11 = ~(&t), At + O. Then there exists a At 0 such that for 0 < At ~ Ato, n : k,k+1,..., nat ~ T, : [Yn _ Vn[ < ~eK*enAt[IYk-1

Vk-11

+


,^ _ Vk-1 12 + AtPlnat b e -K U~ly(p+1)(T)Id~ ]

For a f u r t h e r e r r o r estimation of nonlinear problems and l i n e a r multistep methods we r e f e r to LeRoux [80].

92 4.2. The Modified Midpoint Rule

In chapter I I I and the previous section i t was shown t h a t f o r stable problems and s u i t a b l e l i n e a r m u l t i s t e p methods an i l l

c o n d i t i o n of the leading m a t r i x A does not

a f f e c t the e r r o r propagation s e r i o u s l y even i f t h i s m a t r i x varies with time. But the bounds are here no longer of t h a t uniform character as i t was obtained in the f i r s t chapters f o r systems with constant m a t r i x because the exponential m u l t i p l i c a t i o n depends now on A ' ( t )

two

factor

in some way. C e r t a i n l y , t h i s p e r t u r b a t i o n e f f e c t is p a r t l y due to

the f a c t t h a t a f i r s t

or second order d i f f e r e n t i a l

equation is approximated by a higher

order d i f f e r e n c e scheme whose c h a r a c t e r i s t i c polynomial has some 'spurious'

roots be-

sides the p r i n c i p a l ones. However, f o r some s i n g l e step m u l t i d e r i v a t i v e methods very s i m i l a r r e s u l t s were d e r i v e d , too, by Nassif and Descloux [77 ]. A l l these e r r o r bounds seem to be optimum i f no f u r t h e r c o n d i t i o n s are imposed on the a n a l y t i c problem. Theref o r e , i f one seeks f o r numerical schemes with e r r o r bounds being independent of the data of the d i f f e r e n t i a l

equation one would consider f i r s t

of a l l s i n g l e step s i n g l e

d e r i v a t i v e and s i n g l e step s i n g l e stage methods. The general c o n s i s t e n t s i n g l e step s i n g l e d e r i v a t i v e method has f o r (4.1.3) the form

(4.2.1)

Vn+I = v n + at[mfn+1(Vn+ I) + (I - m)fn(Vn)],

n = 0,1 . . . . .

and the general c o n s i s t e n t singe step s i n g l e stage method f o r (4.1.3) is Vn, I = v n + Atmf((n + m)At,Vn,1) Vn+ I = v n + A t f ( ( n + ~)At,Vn,1) ,

n = 0,I,...,

Vn+ I = v n + A t f ( ( n + ~)~t,mVn+ I + (I - m)Vn),

n = 0,1 . . . . .

or (4.2.2)

cf. also the f o l l o w i n g chapter. For ~ = 0 and m = I we obtain in both cases the explicit

and the i m p l i c i t

Euler method whereas f o r m = I / 2 (4.2.1) y i e l d s the t r a p e z o i d a l

r u l e and (4.2.2) the midpoint r u l e . The d i s c r e t i z a t i o n e r r o r of the method (4.2.1) f o l l o w s immediately from Section 1.1 or by d i r e c t v e r i f i c a t i o n .

Both methods, (4.2.1)

and ( 4 . 2 . 2 ) , a r e of order one and f o r ~ = I / 2 of order two and they are both A-stable f o r I / 2 ~ ~ s I . The t r a p e z o i d a l r u l e is the unique A(O)-stable l i n e a r k-step method o f order k+1 (Widlund [67 ]) and among a l l A-stable l i n e a r m u l t i s t e p methods of (maximum reachable) order two i t has the smallest e r r o r constant (Dahlquist [63 ] ) . In the remaining part of t h i s chapter the i n i t i a l is supposed to s a t i s f y the r a t h e r general s t a b i l i t y

value problem (4.1.3) with T = =

condition

93

(4.2.3)

(v - w ) T ( f ( t , v )

As two a r b i t r a r y dly(t)

- f(t,w))

s o l u t i o n s y and y * o f y ' = f ( t , y )

- ~*(t) i 2 = 2(y(t) dt

satisfy

- y*(t))T(f(t,y(t))

t h i s m o n o t o n i c i t y assumption i m p l i e s f o r m > 0 t h a t tially

V v,w E IRm, m > O.

s - ~Iv - wl 2

- f(t,y*(t))),

ly(t)

- y * ( t ) i decreases exponen-

for increasing t. If

(4.2.4)

( 4 . 2 . 3 ) holds then the e r r o r o f the i m p l i c i t

Euler method s a t i s f i e s

lyn+ I - Vn+ll ~ (I + ~ A t ) - 1 ( l y n - Vnl + I d ( A t , Y ) n l ) ,

n = 0,1 . . . . .

which i s optimum because in t h i s first order method the p r o p a g a t i o n f a c t o r is a first

order a p p r o x i m a t i o n o f the propagation f a c t o r o f the a n a l y t i c problem, e -sAt = (I + mat) - I + ~ ( A t 2 ) ,

At + O.

Let us now t u r n to the case of consistence o r d e r two. The c h a r a c t e r i s t i c

poly-

nomial o f the method ( 4 . 2 . 1 ) ,

x ( ~ , n ) = ~ - I - n(m~ + (I - m ) ) , has f o r m = I / 2 and

n = ~ the r o o t ~ = - I and i t

is well-known t h a t f o r A - s t a b l e methods the p r o p e r t y

E @S leads to undue o s c i l l a t i o n s

o f the numerical a p p r o x i m a t i o n s i f

n = AtX is very

l a r g e . N a t u r a l l y , also the m i d p o i n t r u l e s u f f e r s from t h i s drawback because both methods, t r a p e z o i d a l r u l e and m i d p o i n t r u l e , c o i n c i d e f o r the t e s t equation y ' = xy. I f m is chosen g r e a t e r than I / 2 then these o s c i l l a t i o n s

disappear but also the optimum

o r d e r two is l o s t since d(At,y)(t) i s the d i s c r e t i z a t i o n

: (~ - m ) A t 2 y ' ' ( t )

+ ~(At3),

e r r o r o f the method ( 4 . 2 . 1 ) .

At + O, So i t

suggests i t s e l f

to choose

slightly g r e a t e r than I / 2 and Kreth {81 ] has proposed to choose m = I / 2 + C(At) > I / 2 in o r d e r to preserve the optimum o r d e r . I t seems also in o t h e r methods very promising t o choose some parameters in a s u i t a b l e dependence o f the s t e p l e n g t h At. For the midp o i n t r u l e m o d i f i e d by t h i s way we then o b t a i n an e r r o r p r o p a g a t i o n which resembles t h a t of the i m p l i c i t

Euler method ( 4 . 2 . 4 ) but now holds f o r a method of o r d e r two:

( 4 . 2 . 5 ) Theorem. (Kreth [81 ] . ) I f the problem ( 4 . 1 . 3 ) satisfies ( 4 . 2 . 3 ) and m ~ { - I + Atm + (I + A t 2 m 2 ) I / 2 ] / 2 a t ~

then the error of the modified midpoint rule ( 4 . 2 . 2 ) satisfies

94

lyn+ 1 _ Vn+l I =< ' 1 -1 ~1 mat& - ~)At~

Proof. By an a p p l i c a t i o n writing

Yn - Vn [+~ld(at,Y)nl

of the monotonicity

c o n d i ti o n

' n = 0 ' 1 "'"

(4.2.3)

we obtain from (4.2.2)

e n = Yn - Vn and d n = d ( A t , y ) n (en+ 1 - en,men+ 1 + (1-m)e n) + At~lmen+ 1 + (1-m)en 12 s (dn,aen+ 1 + (1-m)e n)

which y i e l d s

after

some simple transformations l-2m(1-(1-m)At~) 2m(1+mAtm)

len+1 +

[(1-~)(1-(1-~)At~) ~(l+~At~)

enl

+ (

2

1-2m(1-(1-~)At~),2., ,2 2~(l+~Ata) ) Jlen[

(dn,men+ 1 + (1-m)e n) ~(1+~At~)

Now we observe t h at (1-~11-(1-~)At~) 1+~At~)

+ (

1-2~(I-(I-~)At~))2 2~(1+~Ata)

:

I 4 2(i+~At~)2

hence 1-2m(1-(1-m)At~) Zm(l+mAtm)

len+l +

en[

2

d < 1 12 + ( , n = 4~Z(1+~At~) 21en 1+~At~

1-2m(1-(1-m)At~)+1 en+ 1 +

2m(1+mAt~)

%)

or

1-2~(1-(1-m)At~) 2m(1+mAtm)

len+1 +

I en - 2(1+mAta) dnl

(4.2.6) I

I

12m(l+mAtm) en + 2(1+mAtmJ

dni2.

If (4.2.7)

I - 2~(1-(1-m)At=)

s 0

or

~ [- I + At~ + (I + A t 2 ~2 ) 1 / 2 ] / 2 A t ~ then the l a s t i n e q u a l i t y ]en+ll

~ ['

yields

2m(1-(1-m)At~)-1 2m(l+mAt~J

which is the a s s e r t i o n ,

1

+ 2m{l+~At~)] lenl

+ -f.~ldnl,

n = 0,1 . . . . .

95 The choice of (4.2.8)

~ = [ - I + At~ + (I + A t 2 ~ 2 ) I / 2 ] / 2 A t ~

yields

(I

I

~)nt~

=

I + mAta

I = I - Ata + ( A ~ + (I + (Ata)2) I / 2 + At~

We thus can s t a t e t h a t the s e c o n d o r d e r optimum with respect to the s t a b i l i t y

~((Ata)3).

method (4.2.2) with the parameter (4.2.8) is

because the propagation f a c t o r is a s e c o n d o r d e r

approximation of t h a t o f the a n a l y t i c problem, e -mAt . On the o t h e r s i d e , i f we only r e q u i r e t h a t the method is A - s t a b l e , i . e .

I / 2 ~ ~ ~ I , and not (4.2.7) then (4.2.6)

yields

(4.2.9)

len+ll

n = 0,1 , . . .

s I - mle n I + Idnl s

For m = I / 2 the damping disappears here completely in agreement with the above remark on the case ~ E aS. In the proof of Theorem (4.2.5) the i n e q u a l i t y (4.2.7) was obtained in a purely a l g e b r a i c way. We conclude t h i s section with a more h e u r i s t i c foundation of t h i s cond i t i o n and consider the i n i t i a l (4.2.10)

value problem

y' = Ay, t > 0, y(0) = Y0'

with diagonable matrix A, A = XAX- I . eigenvalues of A such t h a t

Let A = (X I , . . . , X m) be the diagonal m a t r i x of the

xm < . . . . <. x. 2 < x I = -a< 0, and l e t the eigenvectors of A,

i . e . the columns of X, be denoted by x 1 , . . . , x m. Then the exact s o l u t i o n of (4.2.10) is

y(t)

xt m = ~u=IX m e u x , YO = ~=lXuXu"

Here, a l l components xux u of y(0) = Y0 are damped at l e a s t as strong as the f i r s t

com-

ponent × i x i . I f we r e q u i r e the same property also f o r the numerical approximation (4.2.2) being here i d e n t i c a l with (4.2.1) then we obtain the c o n d i t i o n sup-~n~-Atm

11+-(Imn- m)n

=< I -i(I+ matsm)Ata. ,

This demand is very s i m i l a r to the concept of strong s t a b i l i t y [72 , p. 98] and a short c a l c u l a t i o n shows t h a t i t I - ~

I -

(I

- ~)At~

I + ~At~

which y i e l d s the i n e q u a l i t y ( 4 . 2 . 7 ) .

0 :< m ~< I . introduced by G r i g o r i e f f

is e q u i v a l e n t to

96 4.3. G - S t a b i l i t y and A - S t a b i l i t y

The midpoint r u l e is not r e a l l y a method of the class i n d i c a t e d in the t i t l e

of

t h i s chapter but a s i n g l e step s i n g l e stage method. Nevertheless we have studied i t

in

the previous section because of i t s i n t e r e s t i n g p r o p e r t i e s and i t s strong r e l a t i o n s h i p to the t r a p e z o i d a l r u l e . With respect to the s t a b i l i t y ,

Theorem (4.2.5) may serve as

a model f o r the e r r o r propagation in other methods but the d i s c r e t i z a t i o n e r r o r was not estimated at a l l

in t h i s r e s u l t as the reader has c e r t a i n l y remarked. Here we are

faced with an e n t i r e l y d i f f e r e n t s i t u a t i o n in genuine s i n g l e stage and in m u l t i s t a g e methods t h e r e f o r e we postpone a study to the f o l l o w i n g chapter on Runge-Kutta methods. In order to obtain e r r o r bounds f o r l i n e a r m u l t i s t e p methods comparable to t h a t of Theorem (4.2.5) Dahlquist [75 ] has introduced the concept of G - s t a b i l i t y . has considered only ' o n e - l e g ' methods, i . e . , (4.3.1)

But he

methods of the form

p(T)v n - Atf(o(T)tn,q(T)V n) : 0, t n = nat, o ( I ) = I ,

n : 0,1 . . . . .

of which the midpoint rule is an example. These methods are genuine multistep single stage methods because they can be w r i t t e n in the form

Vn, I = (s(T) - p(T))V n + A t f ( s ( T ) t n , V n , I) p(T)v n : A t f ( o ( T ) t n , V n , 1 ) , So they s u f f e r from the same d i f f i c u l t i e s

n : 0,1 . . . . . concerning a d a t a - f r e e e s t i m a t i o n of the d i s -

c r e t i z a t i o n e r r o r as Runge-Kutta methods. However, a f t e r some e s s e n t i a l pre-work of Dahlquist [78b], Nevanlinna and 0deh [81 ] have r e c e n t l y shown how to apply the concept of G - s t a b i l i t y

in m u l t i s t e p s i n g l e d e r i v a t i v e

methods.

We reconsider l i n e a r k-step methods (4.1.4) f o r the i n i t i a l under the general m o n o t o n i c i t y c o n d i t i o n ( 4 . 2 . 3 ) .

value problem (4.1.3)

R e c a l l i n g t h a t p and ~ are real p o l y -

nomials of degree not greater than k the f o l l o w i n g d e f i n i t i o n

is the above mentioned

G - s t a b i l i t y due to Dahlquist [75 ]. (4.3.2) D e f i n i t i o n .

The method

(4.1.4) is G-stable if there exists a real symmetric

and positive definite matrix G = [ g i j ] i Tk J = I such that for all w i E ¢, wi+ I = Twi i = 0, . . . . k - l , and Wn = (Wn_k+I . . . . ,wn) ,

W~GWk

-

W~_IGWk_I ~

I f wi E {m then l e t

2Re(~(T)w0P(T)w0).

97

W~GWk

~k Hw = ~i,j=igijwi j'

With this notation the following r e s u l t shows that a G-stable method s a t i s f i e s W~GWk - W~_IGWk_I s 2 Re((o(T)wo)Hp(T)w o) for arbitrary

vectors

wi E cm.

(4.3.3) Lemma. The real syn~netric matrix Q = [ q i j ]k+l i,j=1 is positive semidefinite iff Z~+! . q .zHz ~ 0 l,j=l ij 1 j

v z i E Cm.

Proof. The condition is obviously s u f f i c i e n t f o r the matrix Q to be p o s i t i v e semid e f i n i t e . On the other side, we have Q = X~XT, xTx = I, by assumption where ~ is the diagonal matrix of the eigenvalues of Q. Hence, i f Q is p o s i t i v e semidefinite then Lj=IXijZj zi , j = l qki j zHzi +j = L~k+l li = 1 ~ i JT~k+l

Recall now that a method (4.1.4) (4.3.4)

I

2

~ 0.

is A-stable i f f

{n E ~, Re(r) ~ 0} C S and {~} E S.

This is a slight modification of the frequently used original version of Dahlquist [63 ] who defined a method to be A-stable i f f (4.3.5)

{n E {. Re(q) < 0} C S.

Obviously, (4.3.4) implies (4.3.5) but the converse is also true since by remark ( i ) after (A.I.25) no point of S \ S can l i e on the straight line iIR u {~}.

(4.3.6) Theorem. (Dahlquist [75 ] . ) G-stability implies A-stability. Proof. With respect to the test equation y' = ~y, G - s t a b i l i t y

y i e l d s f o r the method

(4,1.4) V~GVn - V~_igVn_I ~ 2Re((o(T)Vn_k)H~tx~(T)Vn_k ) = 2AtRe(x)[~(T)Vn_k12, Hence the sequence {Vn}~=0 defined by p(T)v n - &tXo(T)vn = 0 is bounded for Re(h) ~ 0 and arbitrary start values. This implies A - s t a b i l i t y .

98

I t is r a t h e r astonishing t h a t the conversion of Theorem (4.3.6) is also t r u e . We s t a t e t h i s r e s u l t in a somewhat more general context: (4.3.7) D e f i n i t i o n .

(Dahlquist [ 7 8 b ] . ) A r a t i o n a l f u n c t i o n ~ i s an A - f u n c t i o n i f

I~I > I i m p l i e s Re(~(~)) > O.

This d e f i n i t i o n

is e q u i v a l e n t to the statement t h a t larg¢(~)l

< 3/2 f o r I~I > I . There-

f o r e an A-function is a n a l y t i c and d i f f e r e n t from zero f o r I~I > I and f o r ~ = =, because the v a r i a t i o n of arg¢(~) on a small c i r c l e around a zero or a pole (a large c i r c l e in the case ~ = ~) is a m u l t i p l e of 23. There may however be poles (and zeros) on the boundary of the u n i t c i r c l e .

D e f i n i t i o n (4.3.7) is thus also e q u i v a l e n t to the

statement t h a t l~I ~ I implies Re(¢(~)) ~ 0 or ~(~) = =. For instance, the f u n c t i o n ¢ = p/s associated w i t h the t r a p e z o i d a l r u l e has the values ¢(~) = 2(~ - I ) / ( ~ + I ) . Every pole ~0 with I~01 = I must be simple and ~(~) must behave l i k e y ( ~ + ~ O ) / ( ~ - ~ O ) with y > 0 in a neighborhood of ~0; cf. Dahlquist [78b].

(4.3.8) Lemma. A linear multistep method (4.1.4) is A-stable iff ~ : p/s is an A-function.

Proof. Observe t h a t n = ~(~) = p ( ~ ) / s ( ~ ) f o l l o w s from 3(~,n) = p(~) - n~(~) = O. I f the method is A-stable then Re(@(~)) ~ 0 implies I~I ~ I by (4.3.4) hence I~I > I imp l i e s Re(¢(~)) > O. On the other s i d e , i f

I~I > I implies Re(¢(~)) > 0 then Re(@(~))

0 implies I~I ~ I which is ( 4 . 3 . 4 ) . Now the f o l l o w i n g r e s u l t shows t o g e t h e r with Theorem (4.3.6) t h a t A - s t a b i l i t y stability

and G-

are e q u i v a l e n t f o r l i n e a r m u l t i s t e p methods.

(4.3.9) Theorem. (Dahlquist [78b].) Let p and s be two polynomials of degree k such that p/s is an A-function. Then there exists a hermitean positive definite (k,k)-matrix G such that for arbitrary Wi E ~, i : O , . . . , k ~ and Wn = (Wn_k+1,...JWn)T

W~GWk - W~_IGWk_I ~ 2Re(~(T)WoO(T)Wo). If w i C R, i = O,...,k,

or if p and ~ are real then G can be replaced by the real

symmetric matrix Re(G) : (G + GH)/2 which is also positive definite.

The proof of t h i s theorem y i e l d s at the same time a general device f o r the computation of the m a t r i x G. Because of i t s length we however must r e f e r to Dahlquist [78b] f o r the details. The next r e s u l t is w r i t t e n as a lemma only f o r convenience; c f . also the e r r o r equation ( 1 . 2 . 1 3 ) .

99 (4.3.10)

Lemma. Let - I < £0 ~ £ and let IIEnll 2 < (1 + Y)llEn_lll 2 + (llEnll + IIEn_lll)llDnll,

n = k,k+1,...,

n llEnll ~ (I + ~)n-k+111Ek_111 + K~v=k(1 + ~)n-v llDv II ,

n = k,k+1,...,

then

where ~ = £ if £ ~ 0 and

~ = r/2 if

£ < O,

and K ~ 2(1 + rO ) - 1 / 2 .

Proof. By induction we obtain f o r u = k - l , k , . . . , (4.3.11)

lIEla II2 < (1 + r)~-k+lllEk 11I2

+

~v=k(1 + I')~-u(IIE II + IIEv_III)IID II.

Hence, i f £ ~ 0 then (maxk_l~u~nllEult)2 s (1 + F)n-k+llfEk_lll 2 + ~=k(1 (maxk_lsu~nllE II)((1 + r)n-k+lllEk_ll[

+ 2~=k(1

+ r)n-V(llEvll + lIE _lll)llDvll + r)n-VllD II), n = k,k+l . . . . .

which proves the a s s e r t i o n in t h i s case. I f r < 0 then we observe t h a t by (4.3.11) (1 + r)-ullE

II 2

< (1 + r)-(k-1)llEk_1112 + ~vU:k((1 + r)-v/211Evll + (1 + r)-(u-1)/211Ev_IlI)E(1

+ I')-v/211Dvll.

Hence we obtain f o r ~ : k , k + 1 , . . . , IIE~II2 ~ I I1kE. writing

I12 + ~ : k ( l l E

= (I + r)-U/2E

and

II ÷ IIE~_II1)IIDvll = ~(I ÷ F)-V/2D . Now the f i r s t

part of the proof

yields IiEnll ~ (I + r)(n-k+1)/211Ek_111 + Z~:k(1 + F)(n-v)/2KllDvll which proves the a s s e r t i o n

in the second case because (I + r) I / 2 < (I + ?/2) i f r > - I .

With these aids we now are able to d e r i v e e r r o r bounds f o r A - s t a b l e m u l t i step methods and s t a b l e n o n l i n e a r d i f f e r e n t i a l equations. The f i r s t one concerns oneleg methods and the second one o r d i n a r y l i n e a r m u l t i s t e p methods which a l l o w an e s t i mation of the d i s c r e t i z a t i o n

e r r o r by means of Lemma ( I . 1 . 8 ) .

However, both types co-

i n c i d e f o r the t e s t equation y' = ~y and a repeatedly mentioned r e s u l t of Dahlquist [63]

1OO

says that the order of an A-stable linear multistep method cannot exceed two. Therefore both subsequent theorems concern only methods up to order two. (4.3.11) Theorem. (Dahlquist [75 , 78b].) Let the initial value problem (4.1.3) satisfy (4.2.3) and let the one-leg method (4.3.1) be A-stable. Then for n = k,k+Is...,

ly n - Vnl s <[IYk_ I - Vk_ll + Z~:kld(At,y)v_k I] where d ( ~ t , y ) ( t )

denotes the disoretization error.

Proof. The error en = Yn - Vn satisfies for n = 0 , I , . . . , p(T)e n - At[f(~(T)tn,~(T)Yn) - f(~(T)tn,~(T)Vn)] = d(at,y) n. We multiply from l e f t by o(T)e n and obtain using the monotonicity condition (4.2.3) (4.3.12)

(~(T)en)Tp(T)en ~ (o(T)en)Td(At,Y)n .

Now, by Theorem (4.3.9) there exists a fixed real symmetric and positive d e f i n i t e matrix G such that E~GEn - E~_IGEn_I ~ 2(~(T)en_k)Td(At,Y)n_k ~ ~(IEnl + LEn_11)Id(At,Y)n_k[ where ~ = 2(Z~:oIBiI2) I/2 and En = (en_k+ I . . . . . en )T. But IEn[ ~ IG-I/2[[GI/2En I ~ IG-I/2111EnIIG accordingly ]IEnll ~ s liEn_t11~ + (IIEnll G + IIEn_11TG)~IG-I/211d(At,Y)n_kl,

n = k,k+1,...,

and an application of Lemma (4.3.10) proves the result. This result is only stated to show the simple way of error bounding in A-stable linear one-leg methods. Naturally, i t cannot bear a comparison with the special result for the modified midpoint rule of Theorem (4.2.5) but i t can be improved substantially for two-step methods using the version of Lemma (4.3.10) for r < O. In the error e s t i mation of linear multistep methods additional d i f f i c u l t i e s arise in spite of the equivalence theorem (4.3.9). So, for instance, we have to introduce here the condition that E S which excludes the trapezoidal rule. (4.3.13) Theorem. (Nevanlinna and Odeh [81 ] . )

( i ) Let the i n i t i a l

value problem (4.1.3)

satisfy (4.2.3) and let the solution y be (p+l)-times continuously differentiable.

101 ( i i ) Let the l i n e a r ~ l t i s t e p

method ( 4 . 1 , 4 ) be consistent o f order p and A-stable.

o

(iii) Let ~ £ S, i.e., let o(~) have only roots of modulus less than one. Then for n = k,k+1 ,...,

ly n - Vnl < K[

max {ly i - v i l + A t l f i ( y i ) - f i ( v i ) I } 0<~:
Proof. The e r r o r e n = Yn - Vn of the method ( 4 . 1 . 4 ) p(T)e n - A t ~ ( T ) ( f n ( y n) - f n ( V n ) )

nat + ~tPf ]y(p+I)(T)]dT]. 0

satisfies

= d(At,Y)n,

n = 0,1 . . . .

Recall t h a t k i ~k S i p(~) = ~i=0~i ~ , ~(~) = Li=0 i t , and write

~(At,y),

: Z~:o~iei

- At~:0~i[fi(y

i)

fi(vi)],

, : 0 .....

k l,

(4.3.14) d(z~t,y) u = d ( A t , y ) , ,

~ = k,k+1 . . . .

Then the sequence {en}n=_. s a t i s f i e s (4.3.15) p(T)en - z~to(T)(fn(Yn) - fn(Vn)) = d(z~t,Y)n,

n = 0,1 . . . .

with en = d(At,y) n = 0 for - n £1N. We now consider (4.3.15). As o(~) has only roots of modulus less than one we can write

p(~)/o(~)

= Zj=0×j

-j

,

I~I > I ,

and then o b t a i n from ( 4 . 3 . 1 5 ) T - I p(T)e n - AteT(fn(Yn ) - fn(Vn)) = eno(T) T -I d(At,y) n, eno(T)

n = 0,1 . . . .

We apply the monotonicity condition (4.2.3) and write ~(m)-len = en' By this way we obtain (~(T)~n)Tp(T)~ n < (~(T)en)To(T)-Id'(z~t,Y)n ' which is now the same s i t u a t i o n (4.3.10) yield

as in ( 4 . 3 . 1 2 ) .

n = 0,1 . . . . .

T h e r e f o r e Theorem ( 4 . 3 . 9 )

in the same way as in Theorem ( 4 . 3 . 1 1 )

and Lemma

102 n = kik+l,...

I~n+kl~ < ~ [ ~ ( T ) - l d ( A t , y ) v _ k l , But (4,3.16)

lenl

= i~(m)~(m)-lenl

= io(m)~nl =< ( ~ = 0 1 6 i l 2 ) l / 2 ( ~ i =Io i ek n + i -

2)1/2

~ n+k -1"d(At,y)v_kl. =< ~( I ~ n+k I + I~n+k_ 1 ]) ~ 2<<~v=klO(¢) Now observe that k

~(~)-I : 6kI~(~_ i=I

~i)-I : 6kIT~Tc-1(li=1 - ( c i / c ) ) - 1 = ~j=k~j~ c-J

where Z~=kIOjl < ~ because ~(~)-I is analytic in the e x t e r i o r of a ball {n E 6, I~i =< r} with r < I. Therefore we obtain

lo(m)-Id(At,Y)vl

: IX~:kojd(At,y)v_jl

: l~-ko~d(At,y)v_.l,o_ ~ ~

v = k,k+1 .....

and accordingly zn+kv=2kiO(T)-Id(At'y)v-kl ~n

=< Z~=kZj=ki°jn ~ li~(At,y)v_jl

(~n-~

: ~u=k ~p=kl°~f)td(At'Y)v-k i <= ~=k I d ( A t ' Y ) v - k ] ' A similar estimation yields by (4,3,14) ~2k-I -I ~ v=k i~(¢) d(At,y)v_kl A substitution assertion,

~ Kmax0~i~k_1{leil

of these bounds into (4.3.16),

finally,

As an example we consider the modified trapezoidal (4.3,17)

+ A t l f i ( y i) - f i ( v i ) i } . and Lemma (I,1,8)

prove the

rule (4.2.1),

Vn+ I = v n + At[ufn+1(Vn+ I) + (I - m)fn(Vn)],

n = 0,1 . . . . .

with the parameter proposed by Kreth [81 ], (4.3.18) m = [- I + Atm + (I + At2m2)I/2]/2Atm = I/2 + ~(At) ~ I/2, Multiplying the modified error equation (4.3.15) from l e f t by a(m) -I and applying the monotonicity condition (4,2,3) we obtain at f i r s t for en = ~(m)-len (o(?)en)Tp(?)en + At~(~(?)en)mp(T)en ~ e ~ ( T ) - 1~d(At,y) n.

103 By Dahlquist [75 ] the only possible ' m a t r i x '

G is here G = I and u n f o r t u n a t e l y there

e x i s t s no p o s i t i v e constant a such t h a t

(~(T)en)T~(T)en ~ ~(l~n+112 + I~nl 2) therefore we have only Ien+ll2 T ~ _ lenl2 ~ ~ 2eno(T)-Id(At,Y)n ,

n = 0,1 ,...

Moreover, we deduce e a s i l y t h a t in the present case ° ( C ) - I = ~j=1~j ~

T

'

Zj=II~j I

= (2w - I) -I ,

and, for,~ = 1 , 2 , . . . , Id(At,y)(t)l Therefore, i f ~ : ~ +

= Id(at,y)(t)l

< 1½-~Iat21y"(t)

2 t+&t (3) + £at ~ lY (~)I dT, t

G(At) ->_½ then f o r v : 1,2 . . . . .

I o ( T ) - I d ( A t , Y ) ~ I = I~ ~' 1 o ' d ( A t , y ) v j l J= J

(v+1)At =< K(At21y ' [ + A t f vat

ly(3)(T)ld=),

and we obtain the f o l l o w i n g e r r o r bound f o r the method (4.3.17) w i t h the parameter (4.3.18)

in the case of stable n o n l i n e a r problems:

lYn - Vn[ s

K[lY0 v01 +

For the c l a s s i c a l

nat

Atlf0(Y0) - f0(v0)l

+ n~t2 max 0~n-1

ly#'l + At I

trapezoidal r u l e and n o n l i n e a r problems, i . e ,

= 1/2, no e r r o r bounds are known to the author besides the c l a s s i c a l

(~ lY'~'(T)Id~] •

(4.3,17) w i t h one ( c f , Henrici

{62 , Theorem 5,10]) even of t h i s form where the order of convergence is smaller than the order of consistence. But, in concluding t h i s section we quote a r e s u l t concerning the l i n e a r case. Recall t h a t A m 0 means f o r a real (m,m)-matrix A t h a t xTAx ~ 0 v x E IRm, i . e , ,

t h a t Re(A) = (A + AT)/2 is p o s i t i v e s e m i d e f i n i t e ,

(4.3.19) Theorem.

(i) Let the real (m,m)-matrix A(t) in (4.1.1) be symmetric, negative

semidefinite, and two times continuously differentiable; let the solution y be three times continuously differentiable. (ii) Let A"(t)

be positive semidefinite and At2111A'IIln/4 < 6 < I or let (At2111A'nln

+ nAt3111A"Illn)/4 < ~ < I . Then the method (4.3.17) with ~ = I/2 satisfies for n = 1,2 .... ,

ly n - Vnl ~ (1 -

a)-lK[ly0 - v0r

+ At[A0(Y 0 - v0) I + At 2 ~ A t l y ( 3 ) ( ~ ) I d ~ ] . 0

104

If A ' ( t )

is negative semidefinite and A ' ' ( t )

is positive semidefinite then this error

bound holds with ~ = 0 and assumption (ii) cancelled.

Proof. We change temporarily the notations and w r i t e E = (el . . . . . en )T' en = Yn - Vn' n and Dn = (e 0 + (At/2)Aoe 0 + d ( a t , y ) O, d(At,Y) I . . . . .

d(~t,Y)n_1) T.

A denotes the block diagonal matrix with the diagonal block elements A I the ' t r a n s l a t i o n ' m a t r i x , T = [tij]~,j=1,

ti+1, i = I, tij

,An, T is

= 0 else,

and I is the i d e n t i t y matrix of dimension both m and m-n. Then, w r i t i n g

4.3.17) f o r

= 1 , . o . , n as a large system we obtain f o r the e r r o r En the f o l l o w i n g equation,

(I - T)En - a t ~ ( l + T)A_~En = Dn,

n = 1,2 . . . . .

and a m u l t i p l i c a t i o n from l e f t by (I + T)E n y i e l d s

(4.3.20)

ET(I * T)T(I - T)En - At ½ E~(I + T)T(I + T)A_~En : E~(I + T)TDn.

Writing b r i e f l y By = Av + Av+1 we obtain f o r the real part of the matrix G = (I + T)T(I + T ) A the f o l l o w i n g representation because of the symmetry of A ( t ) , BI

B ~

I__I__I z

__I. . . . . .

!Bo

2Re(G) = G + GT = H +

B2',

B31

r. . . . . . . . . . . . . i • .

i .

0

.

•

.

i. n

.

.

.

.

•

i J

iBn-2 Bn-2 [+-!B-/-; [Bn- I

. . . . . . . .

Bn-I Bn- I

where H is a block diagonal matrix of the form (4.3.21)

H = ((2A I - A O-A2) + (A I+AO),2A 2 - A I - A 3 , . . . , 2 A n _ l - A n _ 2 - A n , A n - A n _ l ) .

But the matrix 2Re(G) - H is negative semidefinite because the marked submatrices have t h i s property and E~(I + T)T(I - T)E n : fen 12 therefore we have by (4.3.20)

1o5

(4.3.22)

fen 12 - - ~ ETnHEn < ETn(I + T)Dn =< 2(max1
Now observe t h a t the c l a s s i c a l Mean Value Theorem holds f o r the real scalar f u n c t i o n t ~ x T A ( t ) x where x E IRm is f i x e d . Therefore we have

e~(An - An_1)en : Ate~A'(Cn)en and

0<5

eT(Av+1 - 2A + Av_1)ev = At~veTA"(~ )ev, Accordingly, as -

eT(Ao + A1)eI

At ETHE

-'-4"

Lv=l

- ~-~[nAtllIA"llln

(4323)

-

+ IliA' IIIn]maxl__<~
~ O,

- - -At ~ E~HEn ~ - ~ or, i f A " ( t )

>__0, (4.3.21) yields

At2[~n-la eTA,,(C )e _ eTnA,(Cn)en]

n ->- T :>

or, if A"(t)


~ 0 and A ' ( t )

E HE n

llTA, ll[nmaX1~v~n[%12 ~ O,

0

By these three d i f f e r e n t

estimations,

(I - a ) m a x 1 ~ n l e l

(4.3.22) y i e l d s under assumption ( i i )

n-11d(At,y)vl] ~ 2UDnllI ~ 2 [ l e 0 + (At/2)Aoeol + ~ L~=O

where a = 0 in the t h i r d case, ( 4 . 3 . 2 3 ) . This proves the assertion a f t e r an a p p l i c a t i o n of Lemma ( 1 . 1 . 8 ) . This e r r o r bound contains as a p e c u l i a r i t y of the m a t r i x A ( t ) , A " ( t ) A"(t)

~ 0 and A ' ( t )

an assumption on the second d e r i v a t i v e

~ O, and c a r r i e s no r e s t r i c t i o n

of the time step At only i f

~ O. This is a somewhat s u r p r i s i n g r e s u l t because in the next

section e r r o r bounds of the above form with 6 = 0 are derived f o r s t r o n g l y A ( ~ ) - s t a b l e methods under the assumption t h a t A ' ( t )

~ 0 and A ( t ) ~ O.

The symmetry of A(t) seems to be an u n s u i t a b l e assumption in the trapezoidal but the author was unable to cancel i t .

rule

106

4.4.

Uniform Stability

under S t r o n g e r Assumptions on t h e D i f f e r e n t i a l

System

Thanks to Dahlquist's Theorem (4.3.9) we were able to establish a satisfactory s t a b i l i t y analysis for A-stable linear multistep methods and stable nonlinear problems without any data-dependent exponential growing factor. But these results concern only methods up to order two because methods of higher consistence order have no longer this property. This leads immediately to the question whether the error bounds of LeRoux, cf. Theorem (4.1.2), can be improved under stronger assumptions on the d i f f e r e n t i a l system. In the search for a generalization of the theory developed in the preceding section to the class of A(~)-stable methods, a very promising tool is the m u l t i p l i e r technique proposed by Nevanlinna and Odeh [81 ]:

(4.4.1) Definition. A rational function ~ = v/× is a multiplier for the linear multistep method with the characteristic polynomial ~(~jn) : p(~) - ns(~) if

-j

(i) u(~) = ~j=OUj~ (ii)

=

~

, I~I > I ,

u j E IR, j = 0,1 . . . . .

< ~,

(iii) (XP)/(v~) is an A-function (cf. Definition

(4.3.7)).

A multiplier is finitely supported if there exists a N E IN such that vj = 0 for j > N. Recall

that

an A - f u n c t i o n

is analytic

and nonzero i n ~ = - t h e r e f o r e

×p and vs must

have t h e same d e g r e e .

(4.4.2)

Definition.

fying (4.4.1)(i).

( N e v a n l i n n a and Odeh [81 ] . )

Let ~ be a rational function satis-

Then the differential system y' = f(tDy) satisfies the angle-bounded

monotonicity condition with respect to ~ if

k ~ : O [ ~ ( ? ) ( v n - W n ) ] T [ f n ( V n ) - fn(Wn)]

where Tv n = Vn+ I , n = 0,1 , . . . For i n s t a n c e ,

an A - s t a b l e

B e f o r e we d i s c u s s ral

~ 0 v Vn,W n E IRm,

N = 0,1 . . . . .

, and vn = wn = 0 for - n E IN.

linear

multistep

these notions

method has the m u l t i p l i e r

i n some d e t a i l

we f i r s t

~(~) ~ I .

p r o v e the f o l l o w i n g

gene-

result:

(4.4.3)

Theorem. ( N e v a n l i n n a and Odeh [81 ] . )

value problem ( 4 . 1 . 3 )

(i)

L e t the s o l u t i o n

y o f the i n i t i a l

be (p+2)-times continuously differentiable.

(ii) Let the linear multistep method (4.1.4) be implicit and consistent of order p, let ~ : ~/x be a multiplier for this method, and let v(~)s(~) have only roots of modulus less than one. (iii) Let the differential system in (4.1.3) satisfy the angle-bounded monotonicity condition with respect to ~.

107 Then the assertion of Theorem (4.3.13) holds for n = k , k + 1 , . . . :

]Yn - Vnl ~ ~[

nst + ~t p i I Y ( P + I ) ( ~ ) ] d ~ ] '

max { l y i - v i l + a t l f i ( y i ) - f i ( v i ) l } 0~isk-1

Proof. We consider again the modified error equation (4.3.15) with en = d(At,y) n = 0 f o r - n EIN, As ¢ ÷ 0(¢) -I and ¢ ÷ ×(¢)-I are a n a l y t i c in the e x t e r i o r of a ball {¢ ¢ ~, I¢I ~ r} with r < I we are allowed to m u l t i p l y (4.3.15) by o(T) -I and, a f t e r t h i s operation, s c a l a r l y by u(T)e n. The r e s u l t is (x(T)-Iv(T)en)Ta(T)-Ip(T)en - At(~(T)en)T(fn(Yn ) - fn(Vn)) = (u(T)en)T~(T)-Id(At,Y)n ,

n = 0,I . . . .

Writing en = x(T)-1°(T)-len we obtain by assumption ( i i i ) (v(T)a(T)en)Tx(T)p(T)en ~ (v(T)~(T)en)Tq(T)-1~(At,Y)n , Now ¢ = ( × p ) / ( ~ ) is an A-function by assumption ( i i ) .

n = 0,1 . . . .

Let xP and vo have the degree

r ¢ k, then there exists by Theorem (4.3.9) a real symmetric and positive d e f i n i t e ( r , r , ) - m a t r i x G such that for n = r,r+1 . . . . .

E~GEn - E~_IGEn_ I - At(u(T)en)T(fn(Yn ) - fn(Vn)) ~ (~(T)~(T)en_r)T~(T)-Id(At,Y)n_r , where En : (en-r+1 . . . . . en )T and En = 0 f o r n < r, But Iv(T)o(Tl~n_rl with

~ ~(Z~:01~n+r_il2) I/2 s <(llEn_111G

~nlIG = [G-I/2Enl hence assumption ( i i i )

+

ll~nll G)

y i e l d s a f t e r a recursive computation

ll~n+rll~ ~ ~(maXr~v~n+rll~'IG)~lo(T)-1~(At,Y)v_rl" This ~nequality remains true i f on the l e f t side n is replaced by ~, r ~ v ~ n+r, and we obtain a f t e r s i m p l i f y i n g IIEn+rllG

~ ~n+r I -I ~ maXr~v~n+r"EvllG :< <~v=r o(T) d ( A t , y ) v _ r l

The remaining part of the proof is the same as in Theorem (4.3,13). Let us f i r s t

turn to the monotonicity condition ( 4 . 4 . 2 ) . Nevanlinna and 0deh v e r i f y

assumption ( 4 . 4 . 3 ) ( i i i )

under several d i f f e r e n t assumptions on the d i f f e r e n t i a l

and derive also conditions f o r the m u l t i p l i e r

system

in the nonlinear case ( 4 , 1 . 3 ) . However,

108 we shall r e s t r i c t ourselves here to one of the most i n t e r e s t i n g results f o r l i n e a r i n i t i a l value problems (4.1.1). The following a u x i l i a r y r e s u l t is e.g. found in Ortega and Rheinboldt [70 , Lemma 3.4.4].

(4.4.4)

Lemma. Let 4: D C ~m ÷ IR be continuously differentiable on a convex subset DO

C D. Then @ is convex on DO, i.e., ~(~v

+ (I

- ~)w)

~ ~4(v)

+ (I

- ~)~(w)

V v,w E DO, 0 < m < I ,

iff (v - w)Tgrad(Q(v)) ~ 4(v) - @(w).

Notice that this lemma yields vTgrad(4(v)) (4.4.5) Lemma. (Nevanlinna and Odeh [81 ] . ) nonnegative with ~(t,O)

= 0 and let 4 ( t , . )

~ 4(v) i f moreover ¢(0) = O. (i)

Let

4: IRm+1 ) ( t , v ) ~ 4 ( t , v ) E IR be

be convex and continuously differentiable.

(ii) Let u(~) :~ j=O~j~ J be a multiplier with uj :< 0 and

:O~j ~ 0 for j = 1,2 . . . . .

Then

(~(T)vn)Tgradv(~n(Vn)) for all {Vn }~

n=-~ with

v

n

n 4 n(Vn-j ) - 4n_j(Vn_j)) >= u(T)~n(Vn) + ~j=lUj(

for - n E IN.

= 0

Proof. We have (u(T)vn)Tgradv(4n(Vn)) = ~=1(-~j)(Vn

n T = ~j=o~jVn_jgradv(Qn(Vn))

- Vn j)Tgradv(~n(Vn )) + (~0 + ~j : ~ lu~)v~grad,(4~(v~))

~:1(-uj)(¢n(Vn)

-

- 4n(Vn-j))

j

H

v

~

H

+ (UO + ~o:1~ )¢n(Vn)j

: ~(T)@n(V n) + Z~:lUj(@n(Vn_ j ) - ¢n_j(Vn_j)).

Now we reconsider the l i n e a r i n i t i a l value problem (4.1.1) and prove the f o l l o w ing modification of Theorem (4.4.3); cf. Nevanlinna and Odeh [81 , Example 3.16]. In order to avoid too many back-references all assumptions are once more collected:

(4.4.6) Theorem. (i) Let the (m,m)-matrix A(t) in (4.1.1) be real sym~netric and continuously differentiable, let A(t) ~ 0 and A'(t) ~ O; and let the solution y be (p+1)times continuously differentiable.

109 (ii) Let the linear multistep method (4.1.4) be implicit, consistent of order p, and let o(~) have only roots of modulus less than one.

-j

(iii) Let ~ = v/× : Zj=OVj~

be a multiplier for the method (4.1.4) with ~j <= 0 and J u >= Zv=O v O, j = 1 , 2 , . . . , and let v have only roots of modulus less than one. Then the assertion of Theorem (4.4.3)

holds.

Proof. We have only to verify assumption ( i i i ) of Theorem (4.4.3), i . e . , the anglebounded monotonicity condition (4.4.2) with respect to f(t,v) = A(t)v. Writing @(t,v) vTA(t)v/2 we obtain by Lemma (4.4.5)

=

-

- kNn:o(~(T)en)TAnen

=

kNn=O(U(T)en)Tgradv(¢n(en ))

(4.4.7)

>=

Z~=OU(T)@n(en)

N n + Zn=oZj=luj(@n(Vn_j) - Cn_j(Vn_j)).

But @n(en_j) - @n_j(en_j) where ~ depends on j ,

:

n, and en_ j ,

e~_jA'(~)en_ j ~ 0 and

2 ~ OU(T)¢n(en) vN vn eT A e = _ ~N (~N u )e~ A- e. > O. : = - Ln=OLj=OUj n-j n - j n-j ~n=O t j : O j m-n m-n m-n : Hence both terms on the r i g h t side of the i n e q u a l i t y in (4.4.7) are nonnegative which proves the r e s u l t . Assumption ( 4 . 4 . 6 ) ( i i i ) s t r u c t i o n of m u l t i p l i e r s

is r a t h e r r e s t r i c t i v e

but allows nevertheless the con-

in many important cases as shall now be e x p l a i n e d . Recall t h a t

a l i n e a r m u l t i s t e p method (4.1.4) is A ( ~ ) - s t a b l e f o r 0 ~ ~ ~ 7/2 i f f A(~) ~ {n E C, largn - ~I ~ a} u {~} C S. o

For 0 < ~ ~ 7/2 t h i s d e f i n i t i o n

o

is e q u i v a l e n t to the c l a s s i c a l one, A(a) c S, i n t r o -

duced by Widlund {67 ] because f o r ~ > 0 no points of S \ S

can l i e on the s t r a i g h t

l i n e s {n c C, ±argn : x - ~} or in n = ~ by Appendix A . I . The f i r s t

p a r t of the f o l l o w -

ing lemma includes f o r a = 7/2 Lemma (4.3.8) where we have proved t h a t a method (4.1.4) is A-stable i f f

¢ = p/s is an A - f u n c t i o n o r , in o t h e r words, i f f

I~[ ~ I implies

Iarg¢(~) 1 ~ 7/2 or ~(~) = ~. (4.4.8) Lemma. A l i n e a r m u l t i s t e p method (4.1.4) i s A ( m ) - s t a b l e , (i) iff ¢ = p/o satisfies larg¢(~)l (ii) iff A(~)~

Proof. The f i r s t

0 < ~ ~ 7/2,

~ ~ - ~ or ¢(~) = ~ for I~l ~ I;

~ m 0 and It[ = I implies larg¢(~)l

~ ~ - m or ¢(~) = ~.

a s s e r t i o n f o l l o w s in the same way as Lemma ( 4 . 3 . 8 ) . For the proof of

110 the second assertion i t suffices to show the A ( ~ ) - s t a b i l i t y . Recall that the root locus curve n = p ( e i ° ) / s ( e i e ) , 0 ~ e ~ 2~, divides the complex n-plane into several open and connected components ~ , ~ = I . . . . , r , and ~u c iff ~n S ~ 0. Therefore we obtain ° A(~) C ~ C ~ for some v if large( e ie )I ~ ~ - ~ or ¢(C) = ~ for 0 ~ e s 2~, and if o~

A(~) n S ~ 0.

Especially we obtain from ( 4 . 4 . 8 ) ( i i )

that ¢ = p/~ is an A-function - and the method

(4.1.4) is A-stable - i f f (4.4.9)

{n E { , Re(n) < O} ~ S ~ 0

and (4.4.10) Re(¢(eie)) ~ 0 or ¢(e i ° ) = - ,

v e EIR.

I f the method (4.1.4) is convergent and strongly D-stable in n = 0, i . e . , i f p'(1) = ~(I) = 0 and p(~)/(C - I) has only roots of modulus less than one then condition (4.4.9) ° i.e., iff is f u l f i l l e d , and condition (4.4.10) is f u l f i l l e d with ¢(eie) ~ ~ i f f ~ E S, ~(C) has only roots of modulus less than one. Hence, with respect to the function (xp)/(u~) we obtain the following c o r o l l a r y to Lemma (4.4.8). (4.4.11)

(i) Let the method (4.1.4)

Corollary.

be convergent, strongly D-stable in

o

n = O, and let ~ £ S. (ii) Let the real polynomials v and X have only roots of modulus less than one. Then ~ = ~/× is a multiplier for the method (4.1.4) iff (4.4.12)

Re((×p)/(uo)(eie)) z 0

v e E IR.

Notice that ¢ = (xp)/(~o) has real c o e f f i c i e n t s hence Re(¢(eie)) _> 0 V e EIR i f this is true f o r 0 < e < x. Moreover, as Re((p/~)(eie)) = lu(ei°)l-2Re(~(e-ie)(p/~)(ei0))

and ~(~)=Zj=0~j~

-j

, condition (4.4.12) is equivalent to

(4.4.13) Re[ ( Z j OUj eiJe)~( eie

>=0

O<e<~.

(4.4.14) Lemma. Let the method (4.1.4) and the multiplier ~ satisfy the assumption of Corollary (4.4.11) then the method is A(a)-stable if

(4.4.15)

largu(e-le) I s ~ - m

v e EIR.

111 Proof. Observe t h a t (4.4.12)

is e q u i v a l e n t to

l a r g ( ( p / s ~ ) ( e l e ) ) I ~ 7/2 because a r g ( ( p / o v ) ( e i e ) )

v e EIR.

= arg(u(e-ie)(p/o)(eie))

- ~ - larg(u(e-ie)) I $arg(~(eie))

we obtain

~ ~ + larg(~(e-ie)) 1

or ~ + ~ :< a r g ( ~ ( e l e ) )

-

< : ~ -

veEIR

and the assertion f o l l o w s by Lemma ( 4 . 4 . 8 ) ( i i ) . This r e s u l t shows t h a t a method (4.1.4) p l i e r u w i t h small a r g ( ~ ( e - i e ) ) .

is A ( ~ ) - s t a b l e w i t h large m i f i t has a m u l t i -

A converse of t h i s f a c t is stated in the f o l l o w i n g

lemma w i t h o u t proof. (4.4.16) Lemma. (Nevanlinna and Odeh [81 ] . ) Let the method (4.1.4) be A ( m r ) - s t a b l e and let it satisfy assumption

(4.4.11)(i).

Then there is a finitely supported multi-

plier ~ satisfying assumption

(4.4.11)(ii)

and (4.4.15)

for a 0 < ~ < ~'.

The next r e s u l t gives some more i n s i g h t in the form of m u l t i p l i e r s

and is also stated

w i t h o u t proof. (4.4.17) Lemma. (Nevanlinna and Odeh [81 ] . )

( i ) There e x i s t s f o r every k a k - s t e p

method of order p = k with the multiplier ~(~) = I - n~

-I

, 0 < n < I.

(ii) If v(~) = (~ - ~)/(~ - ~) is a multiplier for a method (4.1.4)

of order p ~ 2

then ~ ~ ~. (iii) If the method (4.1.4)

satisfies assumption

(4.4.11)(i)

Im(~(eie)) > 0 then there exists a multiplier v(~)

and

0 < e < : (~ - m)/(~ - X) with -I < ~ ~ ~ < I.

N a t u r a l l y , the e r r o r constant of the method can be large in ( i ) and the assumption in ( i i i )

cannot replace assumption ( i i i )

here t h a t ~ is nonnegative. By (4.4.13) a method (4.1.4) (4.4.18) if

~(~) = I - m - l ,

of Theorem ( 4 . 4 . 6 ) because i t is not claimed

has the m u l t i p l i e r 0 < m < I,

112 (4.4.19)

Re((1 - m e l e ) ~ ( e l e ) ) ~ O,

O<e<

n.

I f we i n t r o d u c e the modified root locus curve ( 4 . 4 . 2 0 ) n(e) = R e ( ~ ( e i e ) ) + i [ s i n ( e ) I m ( ~ ( e i e ) ) - c o s ( e ) R e ( ~ ( e i e ) ) ] ,

then t h i s c o n d i t i o n i s f u l f i l l e d

O
if

O~<e~<~,

Re(q(e)) + mlm(n(e)) ~ O,

o r , in o t h e r words, i f the l i n e Ren = - mlmn stays below the curve ( 4 . 4 . 2 0 ) . way i t can be e a s i l y checked whether a method ( 4 , 1 . 4 )

has a m u l t i p l i e r

By t h i s

(4.4.18).

Be-

s i d e s , as larg(l

- meie) 1 ~ arctan(m/(1 - 2 ) I / 2 )

= ~_

arccosm,

Lemma ( 4 . 4 . 1 4 ) says t h a t a method ( 4 . 1 . 4 ) w i t h the m u l t i p l i e r

(4.4.18) is at least

A ( B ) - s t a b l e w i t h 6 = arccosm. For the backward d i f f e r e n t i a t i o n

methods up to o r d e r

6 ( c f . Appendix A.4) the f o l l o w i n g t a b l e i s found in Nevanlinna and Odeh [81 3. Table I : Backward D i f f e r e n t i a t i o n k

m

Methods

arccos~

2

0

~/2

~/2

3

0.0836

85013 '

88o02 '

4

0.2878

73°16 '

73o21 '

5

0,8160

35o19 '

51050 '

6

5

17°50 '

The methods o f Cryer [73 ] ( c f . Appendix A.4) are k - s t e p methods o f o r d e r k defined uniquely by ~(~) = (~ + d) k. For d : - I + 2 / ( I Table I I :

+ 2 k+1) we o b t a i n the f o l l o w i n g t a b l e .

C r y e r ' s Methods k

arccosm

2

0

3

0.3000

72o33 '

~/2

88.8 °

7/2

4

0.6046

52o48 .

86.3 °

5

0.7952

37o20 '

83.6 °

6

0.8979

26007 '

81.0 °

7

0.9498

18°14 '

78,5 °

113

The m-values in both tables are optimum and were computed by Norsett [69 ] and Jeltsch [76 ] respectively, The modified root locus curves are plotted in Appendix A.4. F i n a l l y , by means of these data we can state the following consequence of Theorems (4.4.3) and (4.4.6): ,!4r4.21) C o r o l l a r y . Let the (m,m)-matrix A(t) in (4.1,1) be real syn~netric and continuously differentiable, let A(t) ~ 0 and A'(t) ~ O; and let the solution y be (k+l)times continuously differentiable. Then the error bound of Theorem (4.4.3) holds (i) for the k-step backward differentiation methods of order k up to order k = 5, (ii) for the k-step methods of order k defined by ~(~) = (~ + B) k and B = -I +

2/(I + 2 k+1 ) (Cryer's methods) up to order k = 7. For the backward d i f f e r e n t i a t i o n

methods up to order k = 6 a very s i m i l a r r e s u l t was

proved by Gekeler [82a] in a d i f f e r e n t way.

V. Run~e-Kutta Methods f o r D i f f e r e n t i a l

Systems of ~ ! r s t Order

5.1. General M u l t i s t a g e Methods and Run~e-Kutta Methods

Up to now we have considered l i n e a r m u l t i s t e p methods and m u l t i s t e p methods w i t h higher d e r i v a t i v e s in t h i s volume. These methods s u f f e r from two drawbacks i f the step number k is greater than one: They need a special procedure f o r the computation of the initial

values v 1 , ° . . , V k _ I and a change of the step length during the c a l c u l a t i o n is

complicated and d e t e r i o r a t e s the otherwise favorable r e l a t i o n between exactness and computational e f f o r t be d i f f i c u l t

per time step. Moreover, the computation of d e r i v a t i v e s of f can

f o r instance i f f is given in t a b u l a r form. These disadvantages are avoided

to some extent by the i n t r o d u c t i o n of intermediate time steps in which the numerical approximation is then to be computed by a d d i t i o n a l

recurrence equations and which on

the other side also augments the computational amount of work. Let us b r i e f l y

r e c a l l t h a t a m u l t i s t e p m u l t i d e r i v a t i v e method f o r the i n i t i a l

value problem (5.1.1)

y' = f ( t , y ) ,

t > 0, y(0) = Y0'

is a device in which one unknown, the approximation Vn+ k of y ( ( n + k ) A t ) ,

is computed in

each time step by one recurrence equation, (5.1.2)

_uZ? ^a ( A t e ) ¢ i f (n- 1 ) ( V n ) I=U 1

: 0, e : ~l~t

n = 0,1 . . . .

I f in t h i s equation some of the t r a n s l a t i o n operators T i : v n ÷ Vn+i = v ( ( n + i ) A t ) , say f o r i = k - r + 1 , . . . , k ,

i = 1,...,k,

I ~ r < k, are f o r m a l l y replaced by a r b i t r a r y

translation

operators, Cj: Vn ÷ Vn+T. = v ( ( n + ~ j ) A t ) , J

Tj ¢IN,

j : I ..... r,

then we obtain a ( k - r ) - s t e p device of the form (5.1.3)

Z~-[o ( a t e ) T i f ( - 1 ) ( V n ) I=U

1

n

+ Z~ 1o~(Ate)¢jf~-1)(Vn) J:

j

= 0

'

n = 0,1

'*''

( N a t u r a l l y , t h i s is an e n t i r e new formula and the polynomials ~ i ( q ) do no longer agree with those of ( 5 . 1 . 2 ) . )

Here we have to compute the unknown vectors

115 (5.1.4)

Vn+i, i = 1 . . . . . k - r , Tjv n = Vn+Tj, j = 1 . . . . . r ,

in each time step and we must therefore add r f u r t h e r equations of the same type as (5.1.3) to the recurrence equation ( 5 . 1 . 3 ) .

In t h i s c o n t e x t , r is c a l l e d the stage

number,

and the complete k-step ~ - d e r i v a t i v e r-stage method can be w r i t t e n as

(5.1.5)

k ~i=0~iv ( ~ t e ) ? i f (n- 1 ) ( V n ) + ~r3= 0 ~3v ( A t O ) T j f ~ - l ) (Vn) = 0,

v = 0 . . . . . r;

see e.g. Lambert [73 , Chap. 5] and S t e t t e r [73 , Section 5 . 3 ] . The data of t h i s method, i.e.,

the o f f - s t e p points ~j and the c o e f f i c i e n t s of the polynomials ~ i v ( n ) and ~

(n)

are to be chosen in such a way t h a t the method has the desired order of consistence and s t a b i l i t y

region S and, above a l l ,

t h a t the vectors (5.1.4) are determined in a

unique way. Let now t n , i = (n + T i ) A t , 0 ~ T i ~ I , Vn, i = V ( t n , i ) ,

fn,i(v)

= f(tn,i,v),

i=I ..... r,

then Runge-Kutta methods are s i n g l e step s i n g l e d e r i v a t i v e multistage methods of the form (5.1.6)

Vn,i = Vn + ~ t ~ = 1 ~ i j f n , j ( V n , j ) ,

(5.1.7)

Vn+ I = v n + ~ t ~ = 1 ~ j f n , j ( V n , j )

i = I ..... r, ,

n = 0,1

w i t h real c o e f f i c i e n t s ~ i j and Bj. Obviously, in each time step the unknown vectors kn,j = f n , j ( V n , j ) ,

j = I ..... r,

are to be computed hence the scheme (5.1.6) and (5.1.7) is frequently written as kn, i = f ( t n , i , vn + AtZ[j=1~i knj , j ) ,

i = I . . . .. r ,

Yn+1 = Yn + ~ t ~ = 1 ~ j k n , j '

n = 0,1

In this chapter we consider mainly the linear i n i t i a l value problem (5.1.8)

y' = A(t)y + c ( t ) , t >'0, y(O) = YO'

with a (m,m)-matrix A(t) and introduce the following notations:

~:

JAn,l,...

An, r ] block diagonal matrix of block dimension r,

116

~n = (Cn,l . . . . .

C n , r )T' -~n = (Vn,1 . . . . . T

~-n = (Vn . . . . .

Vn)

V n , r )T e t c , ,

and

block vectors of block dimension r ,

P : [ ~ i j ] ir, j = 1 ( r , r ) - m a t r i x , T = [~I ,. "" , Tr ] (r,r,)-diagonal matrix q = (81 , . . . . Br )T, z = ( I , . . . ,

I) T r-vectors,

r vn,j , qTp-z= Zri , j = 1 8 i ~ i j l z j (m,m)-matrix. = [ ~ i j l ] i r, j = I , qT.n = Zj=16j A Runge-Kutta method is called

semi-i~lioit or explioit i f - possibly a f t e r a suitable

permutation of rows and corresponding columns - the matrix P is lower t r i a n g u l a r or s t r i c t l y lower triangular the l a t t e r meaning that ~ i j = 0 for i ~ j . the method is called

In the other cases

implicit.

For the problem (5.1.8) the computational device (5.1.6), (5.1.7), i . e . , Vn, i = v n + A t ~ [O=l. m 1j . . ( A n , j .v n,o. + c n , j ) '

i = I, "" ,r,

Vn+I = vn + AtZ~=16j(An,jVn, j + Cn,j),

n : 0,1 . . . . .

can now be w r i t t e n

(_I -

in the f o l l o w i n g

AtP_&A)% : ~ + AtP_~c,

Vn+ 1 : v n + A t q T ( A ~ or,

if

(5.1.9)

(I_-

form,

+c),

n = 0,1 . . . . .

AtP.~A ) is i n v e r t i b l e ,

Vn+ I = G(AtA)nV n + r n,

n = 0,I,...,

w i t h the n o t a t i o n s (5.1.10) G(AtA)n : I +

~tqTA_n(i - ~tP_~nA)-Iz

and (5.1.11)

r n : A t q T ( z + AtA (Z - atP.~A )-1~)~.n = a t q T ( z - AtP_~A)-1.~n .

In p a r t i c u l a r , (5.1.12)

we o b t a i n

f o r the t e s t

e q u a t i o n y'

n = 0,I,...,

Vn+ I = G(At~)Vn,

where (5.1.13)

G(q) = I + n q T ( l - n p ) - I z

= xy

= - ~0(n)/~1(n).

117 The real polynomials ~O(n) and q1(n) = d e t ( l - riP) have degree not greater than r and s t ( n ) = I i f the method is e x p l i c i t . The e r r o r estimation of Runge-Kutta methods d i f f e r s

e n t i r e l y from t h a t of m u l t i -

step methods, Whereas in l i n e a r problems (5.1.8) w i t h constant matrix A the Uniform Boundedness Theorem does not come to a p p l i c a t i o n here, s u i t a b l e bounds f o r the d i s c r e tization

e r r o r are r a t h e r cumbersome to derive in the general case. Over a long period

the d i s c r e t i z a t i o n

e r r o r was only estimated using the Landau symbolic and neglecting

the i n f l u e n c e of an i l l - c o n d i t i o n e d

leading matrix A. However, due to Crouzeix [75 ]

who closed t h i s gap in the theory of numerical methods by his doctoral t h e s i s we have today e r r o r bounds f o r Runge-Kutta methods at our disposal which are of the same e f f i ciency as those f o r m u l t i s t e p methods at least as concerns l i n e a r i n i t i a l

value problems.

5.2. Consistence

Let y be again the s o l u t i o n of the i n i t i a l i = I,..., (5.2.1)

value problem (5.1.1) and l e t Wn, i ,

r , be the s o l u t i o n of the system Wn,i = Yn +

A

r

t~j=1~ijfn,j(Wn,j)'

i = I,

then the discretization error of the method ( 5 . 1 . 6 ) ,

(5.1.7)

d ( A t ' Y ) n = Yn+1 - Yn - A t ~ = I B j f ( t n ,J.,W n , j .), Because of the strong n o n l i n e a r i t y of the scheme ( 5 . 1 . 6 ) ,

...

,r,

is n = 0,I , . . .

( 5 . 1 . 7 ) the d i s c r e t i z a t i o n

e r r o r is defined here only f o r the exact s o l u t i o n y in opposition to m u l t i s t e p methods. (5.2.2) D e f i n i t i o n . A Runge-Kutta method ( 5 . 1 . 6 ) , rential

system y' = f ( t , y ) ,

(5.1.7) i s c o n s i s t e n t w i t h a d i f f e -

t > O, if there exist a positive integer p and a F > 0

not depending on At such that

l l d ( A t , y ) ( t ) l l S r a t p+I for every solution y C cP+I(IR+;IR m) of y' = f ( t , y ) .

The maximum p is the order of the

method.

Every r-stage method ( 5 . 1 . 6 ) ,

(5.1.7) can be associated w i t h r+1 numerical i n t e -

gration formulae, namely (5.2.3)

~i

i f(t)dt

r

~ ~j=laijf(Tj),

i

=

i,..,,r

i

118 I

(5.2.4)

i f ( t ) d t ~ Z;=IBjf(Tj) ,

and i t is convenient to introduce the following notations of which the f i r s t

one is

well-known and the second plays a p a r t i c u l a r role in the subsequent e r r o r estimation of i l l - c o n d i t i o n e d d i f f e r e n t i a l

systems.

(5.2.5) Definition. (i) A numerical integration formula (5.2.4) has order ~ if it is exact for all polynomials of degree less than or equal 4. (ii) For i = I .... Dr let ~i be the maximum order of

(5,2.3) then

= min1~isr{Li} is the degree of the Runge-Kutta method ( 5 . 1 . 6 ) ,

(5.1.7).

The f o l l o w i n g lemma shows that consistence order p implies order p-1 of the formula (5.2.4) in the case of the t r i v i a l why in ( 5 . 2 . 5 ) ( i i )

differential

equation y' = c ( t ) and so explains

the formula (5.2.4) is not included.

(5.2.6) Lemma. If a method (5.1.6), (5.1.7) has order y' = c ( t ) , (5.2.7)

p for every differential equation

c £ cP(IR+;IRm), then (5.2.4) has order p - l , qTTkz

=

~

I

,

k : 0 . . . . . p-1.

Proof. (Cf. Crouzeix and Raviart [80 ] . ) By assumption we have (5.2.8)

d(At,y) n : Yn+1 - Yn - AtZ;:16jC(Tj)

: C(AtP+I)

c E cP(IR+;IRm).

On the other side, a Taylor expansion of y' = c ( t ) provides (n+1)At S c ( t ) = Yn+1 - Yn = Z ~ nat

~

Atk+1 c(k) + ~(AtP+1) n

and

c (k) r = ~p-1~r ~ n ( t _ nAt)k ~(Atp) Vp-1(~r k,At k (k) + d(AtP). Zj=ISjCn,j ~k=0~j=1~j-~!" n,j + = ~k=0 ~j=ISjTj)-l~TC. n

A s u b s t i t u t i o n of these representations into (5.2.8) y i e l d s d(At,Y)n = Lk=0~T~-~ ~P-II I - ~rj = I B j kT jAt ) ~k+1 + ~(Atp+I) = ~(Atp+I) which proves the assertion.

119 Because of t h i s r e s u l t we henceforth assume that every Runge-Kutta method of order p

satisfies ( 5 . 2 . 7 ) . In p a r t i c u l a r we obtain f o r k = 0 ~;=IBj = 1 which is the well-known necessary and s u f f i c i e n t f(t.y)

~ 0 in ( 5 . 1 . 1 ) .

c o n d i t i o n f o r the consistence i f

Recall t h a t consistence implies convergence here in the c l a s s i -

cal meaning without f u r t h e r c o n d i t i o n s i f the basic problem (5.1.1) is w e l l - c o n d i t i o n e d ; see e.g. Henrici [62 , Theorem 2.1] and G r i g o r i e f f

[72 ] .

The next r e s u l t provides sufficient conditions f o r order p with respect to the general problem ( 5 . 1 . 1 ) .

I t has been proved by Butcher [64 ] in an a l g e b r a i c way and

by Crouzeix [75 . Theorem 1.2] in a more a n a l y t i c way. (5.2.9) Theorem. If the following three conditions are fulfilled: (i) The integration formula (5.2.4) has order p-l, (ii) the integration formulae (5.2.3) have order k-l, ~r

T.K+I 1

K

•

~..~.

r•

~..S.~.

0=I IJ J

=

~

<+I

,

K

=

O,...,k-1,

i

=

1,...,r,

(iii) ~

=

then the method ( 5 . 1 . 6 ) ,

1 8

I

(5.1.7)

~+I ),

= 0,...,~-I, i = 1,...,r,

has order

p = min{k + L + I , 2k + 2}. for the differential system (5.1.1) with y c cP+I(IR+;IRm). In p a r t i c u l a r , we obtain f o r ~ = 0 and k = p - l : (5.2.10) C o r o l l a r y . If the integration formula (5.2.4) has order p-1 and the formulae

(5.2.3)

have order p-2 then the method (5.1.6), (5.1.7) has order p for the general

problem ( 5 . 1 . 1 ) . A f u r t h e r consequence of Theorem (5.2.9) is due to Crouzeix [75 , C o r o l l a r y 1.2]: (5.2.1.1.) C o r o l l a r y . Let r * be the number of the different intermediate time steps ~i' i = I .... ,r, let the formulae (5.2.3) be of order r*-1, and let the formula (5.2.4) be of order p-l. Then the method (5.1.6), (5.1.7)

has order p for the general pro-

blem (5.1.1). With respect to the general linear problem (5.1.8) Crouzeix [75 , Theorem 1.3]

120

has deduced neoessary and s u f f i c i e n t conditions for consistence order p; cf. also Crouzeix and Raviart [80 , Theorem 3.1]. In order to present this r e s u l t we introduce some f u r t h e r notations. Let (5.2.12) d0(At,y) n : Yn+1 - Yn - AtZ[J= iB.y~ J ,j' (5.2.13) di(At,y) n = Yn,i - Yn - AtZ~=1~ijYn,j'

i :

l,...,r,

and

~(At,y) n = (d1(At,y) n . . . . .

dr(At,Y)n )T,

For the linear problem (5.1.8) we can write Yn,i = Yn + AtZ~=Imij(An,jYn,j

r A + Cn,j) + Cn,j) + Yn,i - Yn - AtZj=1~ij( n , j Y n , j

then, by (5.2.1), Yn,i - Wn,i = AtZ[J=l.m..A Ij n , j ( Y n , j and accordingly, Zn - ~

if ~-

AtP.~nA is regular,

: (l -

AtP_A~A )-I~.

For the d i s c r e t i z a t i o n

- wn ,j) + d i ( A t , y ) n

error d(At,y) n we thus obtain

d(At'Y)n = Yn+1 - Yn - AtZ[j=IB:(A n J ,jWn, j + Cn, j) (5.2.14)

= d0(At,y) n + A t ~j=l .B.A j niJ"(Yn ,j - Wn,j ) = d0(At,y) n + AtqTA~(Z- AtP.~nA)-1~(At,Y)n.

A s u b s t i t u t i o n of the Taylor expansion y'(t)

= ZP-~ ( t -

nat) k (k+1) + } ( ~ - nat) p-I y(p+1)(z)d z k! Yn nat (p-l)!

into (5.2.12) yields d0(At,Y)n

+

or

TP-IF(n+~)At(T - nat) k r ( t n , j - nAt)k = ~k:0L n i t k! d T - At~j=IB j k! ]y~k+1)

(n+1)At t ~t(T - nat) p-I .~r ^ ~'J(~ - nat) p-I ) nAtf n (p-l}! Y(P+I)(T)dTd° - a~Lj=1~Jn~t ( p - l ) ! y(p+l ( z ) d T

121 tp-1 r 1

~r

- k+l , (k+l) k/At

(5.2.15) d0(At,y) n = Lk=0Ll~T- Zj=16jjj .-ET--~n

+ 60,n

where

T.

I ~rTp_I (p+1 ~

(1 + ~ ; = l l e j l )

r

fJTp-ly(p+1)(nAt+T)dT

ly(p+l)(nat+T)[d~ s rat p nat

)(~)ld~.

The s t i p u l a t e d c o n d i t i o n ( 5 . 2 . 7 ) i m p l i e s t h a t the bracketed terms in the sum of ( 5 , 2 , 1 5 ) are zero hence

(5.2.16) Id0(At,Y)nl

~

l+0,nl.

In the same way as (5.2.15) we find k+1 ~p-IITi ~r kIAt k+1, (k+1) = (5.2.17) d i ( A t , y ) n A k = 0 L ~ - j=1 ~ . .ITj- j ] - ' k~ Y n

+ aijn D

i = 1,...,r,

where the 16i,n[ have the same bound as 160,nl, i . e . , (n+})At (5.2.18) max0
~ rat p

nAt

ly(p+1)(~)Id~.

(5.2.17) into (5.2.14) and obtain

d(At,y) n = d0(At,y) n + ~=IqTA ( Z -

AtP.~A)-1(~_ k - pTk-I)z~FTTy~k)A ~k+1

(5.2.19)

+ AtnT~(L- AtP~A)-1& But

~(L-

AtP_~A) -I : (L- At~P)-I~

and i f the method has degree p * - 2 then the numerical integration exact f o r the f u n c t i o n s x * x k, k = 0 , . . . , p * -

~T k - PTk-1 = 0, Therefore a substitution Id(At,Y)nl

formulas (5.2.3) are

2 which y i e l d s

k = I,...,p*-I. of the bounds (5.2.16) and (5.2.18) into (5.2.19) leads to

C r(1 + i(l_.- AtA P)-IAtA i)At p

(n+})Atly(p+1)(T)IdT nAt

(5.2.20) •

k+l

with the abbreviating notation

122

qT( L - ~tA_nL)-IA(~Z k

(5.2.21)

Uk, n z

- p.Tk-1)zy~k),

(5.2.22)

Theorem. (Crouzeix [75 ] . ) I n ( 5 . 1 . 8 )

k = p*,...,p.

let A E cP(IR+;IRm'm), c E cP(IR+;IR m) and

let _I - AtPA__n be regular for t > 0 and 0 < At =< At 0. Then the method (5.1,6) , (5,1,7) has order p for the linear problem (5.1.8) and 0 < At ~ At 0 iff

v c £1N v ~j E IN ~ { 0 }

=ixj

= p - ~

=>

(5.2.23) ~(x;~) where @(),;I)

=

-

qTTXIpTX2p

x c ~ -I PT ~z = T--T[(;~ - i + I) + Z j = i ~ j ] i=I

.-.

qTTXlz and ~(X;2) = qTT~IpTX2z.

For m > 2p this condition is also he-

cessary.

By (5.2.20) and (5.2.21) a Runge-Kutta method has order p i f f (5.2.7) holds and in (5.2.21) (5.2.24)

IqT(z - AtA~)-lAn(~k

- pTk-1)zl

s FAt p-k

where £ does not depend on At. (5.2.7) is (5.2.23) for ~ = I. For the proof of the equivalence of (5.2.23) for ~ > I and (5.2.24) we refer the reader to Crouzeix and Raviart [80 , Theorem 3.1]. Instead we prove below Theorem (5.2.22) for linear problems with constant matrix A (Corollary (5.2.27)). But l e t us quote before a further result of Crouzeix [75 ] without proof concerning the general nonlinear problem (5,1.1): (5.2.25) Theorem. (Crouzeix [75 ] . ) Let the solution y of the problem (5.1.1) be (p+l)-times continuously differentiable then the method ( 5 . 1 . 6 ) ,

(5.1.7)

has order p

for this problem and sufficiently small At if (5.2.23) is fulfilled and

(5.2.26)

1 TL+I z, PT~z = ~--~T

The conditions ( 5 . 2 . 2 3 )

= 0 . . . . . [~-~].

are necessary. If 6j > O, j = 1 , . . . , r ,

in ( 5 . 1 . 7 )

then the

conditions (5.2.26) are necessary, too.

(5.2.27) Corollary. Let the assumptions of Theorem (5.2.22) be fulfilled but let the matrix A be constant. Then the method (5.1.6),

(5.1.7) has order p for the linear pro-

blem (5.1.8) and 0 < At < At 0 iff

(5.2.28)

v k,~ EIN ,, {0}

k ~ < k < p-1 -,> qTpk-~T~z = T - ~ [ i i=~

+ I] -I .

123 Proof. A f t e r the above p r e l i m i n a r i e s we have only to v e r i f y the equivalence of (5.2.7) and (5.2.24) with (5.2.28) f o r A = A~. A s u b s t i t u t i o n of (Z-

AtAL ) - I = Z~=0(AtA--P)k + ~(At~+1)

into (5.2.24) y i e l d s the condition }]~-~-IAt~qTA(AP)~(~Tk - PT.Tk-1)z: Cf(z~tP-k),

k = 1.... ,p-l,

or

(5.2.29)

T~k,l~

q ~" t - ~

k EIN u{O}, ~ EIN, k+~ < p-l,

- P T ~ - I ) z = O,

or

l . ~ - pT~-1 )z = 0, (5.2.30) qT v. k - ~ ,t~L (5.2.28)

I < ~ < k < p-1.

implies (5.2.30) and, f o r k = 4, ( 5 . 2 . 7 ) . On the other side, (5.2.7) and

(5.2.29) define r e c u r s i v e l y qTpk-~T~z f o r k,z EIN u { 0 } , 0 ~ ~ S k s p - l , in a unique way. This proves the assertion. In order to derive bounds f o r the vectors Uk, n defined by (5.2.21) we s u b s t i t u t e

(I - AtA~P)-I : ~P-k-I(AtA_~P)J -

~j=0

+

(I - AtA.~p)-I(AtA~P) p-k -

and obtain

Uk,n : zP-~-IAtJqT(An~P)JAn(~ k - HI--k-l")Zyn(k) (5.2.31)

+ AtP-kqT(I _ AtA~p)-I(An~P)P-kA(~Tk - pTk-1)zy(k) --

- -

n

"

Recall that_~A is a block diagonal matrix with the diagonal elements An, j = A((n+Tj)At) and that in t h i s context ~ is a block matrix with the elements a i j l diagonal matrix with the diagonal elements T i l .

and Z is a block

We consider two cases:

( i ) I f A(t) = A is a constant matrix then A = AI commutes with P and T and we obtain ,(k) pTk-1)z Aj+1 Yn Uk,n = ~P-k-IAtJqTpj ~j=0 - (iT~ - k - -+ AtP-kqT(! _ A t A P ) - l p p - k ( ~ k - p~Tk-1)zAP-k+ly~ k) As the method is supposed to have order p the sum on the r i g h t side disappears by (5.2.26) hence we have in t h i s case (5.2.32)

lUk,nl ~ I(~ - AtA_p)-IIAtP-klAP+I-ky~k)I.

124 (ii) If A(t) varies with time then we can write (A P)JA as a function of the step --n-length At,

(A_~P)JA_n: [(An~P)JA_~](At). (5.2.33) Corollary.

If

the method

(5.1.6), (5.1.7)

has order p then

qT(A_~P) JA_n(~Tk- pTk-1)zy~k) = qT ~t(At-0 ''(p-k-j-1)1~)P-k-j-1[ (A~p)jA_a](p_k_j)(~)d~(~._T k _ pTk_1)zy~k),

j+k < p-l.

Proof. We have (X2) ...PTXJ+IA(Xj+1_~q) XI i! , ~ XI A (X I) PTx 2A_~ ~l+...+xj+1=i !'"~j+l" X ~N u {0}

[(A_~P)JA~j(i)(At) and

[(a_~PlJa_n](i)(0)

A( x I ) ... A(Xj+1 )TXl pTx2p-.. pTxj +I

: XI+...+Xj+I=i ~ I ! ' " X j + I X ~N u{0}

! --n

--n

As the method is supposed to be of order p, (5.2.23) yields for i + k ~ p - j -

I

qTTXIpTX2..-PTXJ+I({Tk - pTk-I)z = 0 and thus we have qT[(A_~P)JA~](i)(0)(~k - pTk-I)z : 0,

i+k s p - j - l ,

hence a Taylor expansion, (A-~P)JA-~(At) : ~i=0?P-k-j-1~] t A( ~ t A_P i ~)J A_ (i)(0) + ~t(At-T)P-k-j-I[(A.~P)JA~](P-k-J)(T)d~,(p_k_j_Ij, 0 proves the assertion. Now, (5.2.31) yields by Corollary (5.2.33) lUk,n i < F[AtP-k~P-~-Imax0<~
atA_n~P)-11I(A_n~P)P-kA(~Tk - pTk-I)zw~k)I]

125 or

lUk,nl

< rmax{1, i(l_ - AtA_~_P)-I l}At p-k

(5.2.34) × ~P.-~max0< < J=

i[(A P)JA ](P-k-J)(t)(~=Tk

:T=A'C

--

- pTk-1)zy~k) I

--

~

"

If A(t) = a(t)A in (5.1.8) with a scalar-valued function a and a~ = (an,1 I, . . . . an,r I) denotes the diagonal matrix with the elements a n , j l = a ( ( n + T j ) A t ) l then [(A-~P)JA~](P-k-J)(T)({ T - k -

- --pTk-1)zy(k)n : [(aP_.)J~n](P-k-J)(T)(~T_k

- __PTk-1''j+1)za yn(k)

and thus we obtain in this case

lUk,nE s £max{1,

I(L - AtAL)-ll }Atp-k

(5.2.35) X

•

p-k . j + 1 . (k)

maxosi~p-k+Imaxo~jsp-klli(al)(J)llln+IZj=O a

Yn I.

We substitute the bounds (5.2.34), (5.2.35), and (5.2.32) successively into (5.2.20) and assemble the result in the following lemma. (5.2.36) Lemma. (Crouzeix [75 ].) ( i ) Let the (m,m)-matrix A(t) in (5.1.8) and the solution y be respectively p-times and (p+l)-times continuously differentiable. (ii) Let the method (5.1.6), (5.1.7) be of order p for this problem and of degree p*- 2. (iii) Let _I - AtA..v_ P be regular for 0 < At ~ At 0 and w = 0,...,nl and let

(5.2.37) ?AtA,n ~ £max{1 + i ( Z -

AtA-p)-IAtA-I,.~..~,

(k - AtA-p)-II }

where F is a generic positive constant depending only on the data of the method. Then, for 0 < At ~ At 0 and n = 0,Ii... i

Id(At,Y)nl S £AtA,nAtP[(n+})Atly(p+1)(T)IdT + AtZ~=p.Wk(At,y)n] nat where

(5.2.38) Wk(At,y) n : Z~-~max~< <.~I[(A=~P)JA_~](P-k-J)(T)(~zk Wk(At,Y)n = if

A(t) = a(t)A,

max max Ni(ai)(J)i[In+]Z~IAJ+ly~k)i 0sisp-k+1 0~j~p-k

and

Wk(At,y)n : IAP+1-ky~k)i if

A(t) = A is

- HI--k-1")Zyn(k) I

a constant matrix.

126 5.3. Error Bounds f o r Stable Linear D.iffe.rential._Systems

A general Runge-Kutta method of stage r and a general s i n g l e step r - d e r i v a t i v e method do not d i f f e r

from each other f o r the homogeneous d i f f e r e n t i a l

system y' = Ay

w i t h constant matrix A, they both have the form ~1(AtA)vn+ I + ~0(AtA)vn = 0,

n = 0.I . . . . .

~O(q) and ~1(n) being polynomials of degree not greater than r ; c f . ( 5 . 2 . 1 3 ) . The region of absolute s t a b i l i t y

(5.2.12) and

S is therefore defined f o r Runge-Kutta

methods in the same way as in Chapter I , D e f i n i t i o n

(1.2.7).

However, the c h a r a c t e r i s t i c

polynomial is now always l i n e a r in ~,

~(;,n) : ~I(~)~ hence the s t a b i l i t y

+

o0(n),

region S is always closed in C.

In t h i s section we consider the l i n e a r i n i t i a l (5.3.1)

y' = A ( t ) y + c ( t ) + h i t ) ,

value problem

t > 0, y(0) = Y0'

w i t h the p e r t u r b a t i o n h t) which is again omitted in the numerical approximation, (5.1.6),

( 5 . 1 . 7 ) , as in the former chapters. W r i t i n g the Runge-Kutta method in the

form (5.1.9) we then obtain the f o l l o w i n g equation f o r the e r r o r e n = Yn - Vn by definition

of the d i s c r e t i z a t i o n

(5.3.2)

e n : G(AtA)n_len_ I + d(At,Y)n_ I + AtqT(~ - AtP_.AnA) - l h _ i ,

I f the 'constant' (5.3.3)

e r r o r d ( A t , y ) n and by ( 5 , 1 . 1 1 ) ,

rAtA, n defined by (5.2.37)

r *AtA,n : max{rAtA,n'

n : 1,2 . . . .

is modified s l i g h t l y ,

I ( -I - AtPA ) - I I }

then Lemma (5.2.36) y i e l d s immediately

lenl ~ (5.3.4)

IG(AtA)n_iIien_ll nat

+ YAtA,n-I* ( A t P ( n _ I ) A t i Y ( P + I ) ( T ) I d T + A t p + I ~ : p * w k ( A t ' y ) n - I

+ Atlilhllln)"

For a constant and diagonable matrix A ( t ) = A = XAX-I we can w r i t e instead of (5.3.2) (5.3.5)

x - l e n = G(AtA)X-Ien_I + X - I d ( A t ' Y ) n _ I + AtqT(l-- - AtPA)-Ix-Ih-~n-I -, n= 1,2,. .. ,

127 and a slight modification of Lemma (5.2.36) yields IX-Id(At,Y)n + AtqT(z - At_PA)-Ix-Ih_~I

r~tAEIX-lI

(Atp(n+})Atly(p+1 nat

)(~)ld~ + Atillhll[n) + ~tP+IZ~:p, lX

-I p+1-k (k) A Yn

I].

Recall that G(n) = - oo(n)/~l(n) and that A = [ X l , . , , , xm] is the diagonal matrix of the eigenvalues of A. Therefore we have f o r Sp(AtA) c S

IG(~tA)I = maxlsusmlG(atx)l

s SUPne slG(n)I

~ 1,

and (5.3.5) yields nAt IX-lenl ~ [ x - l e o I + r ~ t A [ I X - I I ( A t P f lY(P+I)(T)Ed~ + nAtlllhllln) (5.3.6)

+ nAtP+l~=p *lllx-lAp+l-kyn(k)lll] So we are faced with two problems here: an estimation of rAtA, n*

' and, in the general

case ( 5 . 3 . 2 ) , with an estimation of IG(AtA)ni. We begin with a c o l l e c t i o n of estimations of F* under d i f f e r e n t assumptions. AtA,n one makes no assumptions on the matrix P.

The f i r s t

(5.3.7) Lemma. Let P* : [ l a i j l ] ri , j : 1 and let IA(t) I ~ ? for £spr(P*) < I

•

Then

r* A,n

t

> 0

such that

< :
Proof. Cf. also Crouzeix [75 , Proposition 3.3]. As IA(t)i is bounded i t suffices to consider (I - A(t)P) - I . Let U = (u I . . . . , Ur)T, uj E {m, and l e t W : A(t)PU. r At , j ~ i j u j Then we have wi = Zj=I (5.3.8)

, i = I , . . ,. r ,

and hence

lwil ~ Z ; = 1 1 A t , j i i ~ i j l l u j l s r Z ~ = l i = i j I I u j i .

I f IUI~ = maxlsi~rlUil and I~I~ is the associated matrix norm then we obtain by this way recursively

I(~(t)L)nl= ~ rnII(p*)nII. By e.g. Stoer and Bulirsch [80 , Theorem 6.9.2] there exists for P* and c > 0 a norm II.II, such that liP*,, s spr(P*) + ~ and ,QII ~
128

1(~(t)~)nl~

s £nll(p.)nll ~ ~ ~£nll(p.)nll * ~
Choosing ~ s u f f i c i e n t l y I(l - a(t)p)-II~'

small such t h a t × = F(spr(P*)

~ limn_~Z~=Ol(~(t)~)nl~

•

n

+ s)n.

+ £) < I we thus f i n d

~ mllmn+~Zv=O×

~ m(1 - r ( s p r ( P * )

~))-I.

+

Let now I~11 be the m a t r i x norm associated w i t h the v e c t o r norm IUII = -.~=11uil then we d e r i v e in the same way a bound f o r

I(l

- A(t)p)-III

depending only on P*, c, and £.

With these two bounds f i n a l l y I(I - a(t)p)-112

= spr((l

I(I - A(t)p)-H(l proves the a s s e r t i o n

for

- A(t)p)-H(I

- a(t)p)-11~

I(I - A(t)p)-II.

- a(t)P) -I)

~ I(I - a(t)p-1111(l For I ( l

- PA(t))-II

- a(t)p)-11~

~ mr

the proof f o l l o w s

in an

analogeous way. Let now P = WJW-I be the Jordan canonical diagonal of J, i . e . , diag(J).

decomposition of P, l e t d i a g ( J )

be the

the diagonal m a t r i x of the eigenvalues of P, and l e t Q = J -

Then Q is a n i l p o t e n t

m a t r i x w i t h Qr = O. The next two r e s u l t s

case of a c o n s t a n t diagonable m a t r i x A ( t )

concern the

= A. Here we have AP = PA and hence in

(5.3.6) F~t A = r a t A ~ £(I + m a x = o , 1 1 ( ~ - A t A ~ ) - I ( A t A ) ~ I ) ~ F(1 + sup n E Sp(~tA) I ( I - n P ) - I n ~ l ) "

(5.3.9) Lemma.

(i) Let Re n ~ 0 and let all nonzero eigenvalues of P have positive real

part or let Re n < 0 and let all eigenvalues of P have nonnegative real part. (ii) Let P be regular or let the dimension of the kernel of P be equal to the multiplicity of the eigenvalue 0 of P. Then I - n P

is regular and

maxv=o,11n~(I

-

nP)-ll

s m

where m depends onZy on P.

Proof.

If

P is

r e g u l a r then

In~(l - np)-II where J is r e g u l a r .

= In~W(l - nd)-1)W-11

I f P is s i n g u l a r

~ mlnV(l - nd)-1[

then J can be chosen of the form

129

where ~ is r e g u l a r , and we have

nV(I _ nj)_ I W-If s <(I + InU(l - n 5 ) - I I ) hence i t s u f f i c e s to consider the case where P is r e g u l a r . Then, as O is r e g u l a r , l(l

-

nO)-11 =

l[(I -

nO) -I

l]O-11

~ <(I

and i t s u f f i c e s to f i n d a bound of I ( l - nJ) - I ] Re(n - I )

+ l(I

- nJ)-11)

f o r n m 0, But i f Ren~ 0 (< 0) then

~ 0 (< 0) and t h e r e f o r e I - nJ = n(n-11 - J)

is r e g u l a r because Re(n-11 - d i a g ( J ) ) < 0 by assumption. Moreover. [ ( I - ndiag(J))-11 is bounded in a neighborhood of ~ = 0 hence bounded and (5.3.10)

I(I - nJ)-II

~ I(I - ndiag(J))-IIl(l

- (I - ndiag(J))-IQ)-11

~ <

proves the a s s e r t i o n .

(5.3.11) Lemma.

(i) Let Re n ~ 0 and let all nonzero eigenvalues of P have positive

real part or let Re n < 0 and let all eigenvalues of P have nonnegative real part. (ii) Let n C R where R is bounded in ~. Then I - n P

is regular and

max =0,11nU(l - np)-11 ~ KR.

Proof. Obviously i t s u f f i c e s to consider (I - n P ) -I and the r e g u l a r i t y of t h i s m a t r i x f o l l o w s in the same way as above. Then

I - ndiag(J) is r e g u l a r and the a s s e r t i o n f o l -

lows from ( 5 . 3 . 1 0 ) . The r e s u l t s received h i t h e r t o f o r l i n e a r problems with constant diagonable m a t r i x can be assembled in the f o l l o w i n g theorem: (5.3.12) Theorem. (Crouzeix [715 ] . )

( i ) L e t the ( m , m ) - m a t r i x A i n (5.3.1) be c o n s t a n t

and diagonable, A = XAX-I , and let the solution y of (5.3.1) be (p+l)-times continu-

130 ously different,able. (ii) Let the method (5.1.6),

(5.1.7) be consistent of order p and of degree p* - 2

with the stability region S. (iii) Let Sp(AtA) C S ~ {n E ~D Re n < 0} and let all eigenvalues of P have nonnegative part or let Sp(AtA) C S n {q E C, Re n ~ 0} and let all nonzero eigenvalues of P have positive real part. (iv) Let P be regular or let the dimension of the kernel of P be equal to the multiplicity of the eigenvalue 0 of P. Then for n = 1,2, .... nat

,

•

{X-1(yn - Vn) I ~ K IX-11[ly0 - v01 + At p ~ l y t P + i ) ( T ) I d T + nAt[ilhliln 0

+ ~tPnAtZ~:p.lllx-lAP+l-ky(k) llln] . If assumption (iv) is not fulfilled then K depends on AtlA ]. Without any assumption on P the estimation holds for AtlA I < spr(P*) -~

with P* = [l~ijl] riij:1 and K depend-

ing on AtlAI. In the sequel, A > 0 means again t h a t the hermitean (m,m)-matrix A is p o s i t i v e d e f i n i t e and Re(A) = (A + AH)/2 denotes the hermitean part of A. I f the constant matrix A in (5.3.1)

is not diagonable then we have to modify s l i g h t l y

to obtain uniform bounds. (In Lemma ( 5 . 3 . 1 1 ) ,

Lemma (5.3.9)

in order

IAI ~ K f o l l o w s u n i f o r m l y from spr(A) ~

only i f A is normal hence diagonable.) (5.3.13)

Lemma.

(i) Let A be regular with Re(A) ~ 0 and let all nonzero eigenvalues

of P be positive or let Re(A) < 0 and let all eigenvalues of P be (real and) nonnegative. (ii) Let P be regular or let the dimension of the kernel of P be equal to the dimension of the eigenvalue 0 of P. Then ~ - A~ is regular and

max=0,11( Z

- A~)-IAV

i ~ <..

Proof. As A commutes w i t h P we can f o l l o w Lemma (5.3.9) and have to f i n d a bound f o r i(l

- Aj)-II

only. I f A is r e g u l a r w i t h Re(A) ~ 0 or i f Re(A) < 0 then Re(A - I ) ~0 or

ReTA-I)-< 0 hence I - AJ = A ( A - I I - J) is r e g u l a r because Re(A-II - d i a g ( J ) ) < 0 by assumption. Moreover we have IA-TI ~ - I i f Re(A) ~ ~I > 0 and Re( Z - Adiag(~)) ~ I m

because diag(J) is real and nonnegative therefore the assertion f o l l o w s from (5.3.10) w i t h n replaced by A. I f A(t) varies with time then A ( t ) does no longer commute w i t h P and we have to - -

impose s t i l l

more conditions on P:

m

131 (5.3.14) Lemma. Let Re(A(t)) s 0 for t > 0 and let there exist a regular diagonal matrix Re(DPD- I ) > O. Then r *~tA,n = < ~ where ~ depends only on P (and D).

D such that

Proof. As ~ ( t ) and ~ are block diagonal matrices they commute with each other and we have I - A(t)P = D-I(I - A(t)DPD-I)D hence i t s u f f i c e s to prove the a s s e r t i o n f o r Re(P) > O. Then Re(P- I )

> 0 and, as P is

regular, I - A(t)P : (~-I _ ~ ( t ) ) ~ with Re(~-I - ~ ( t ) )

~ Re(P-t ) z ~I > 0

whence

l(I

-

A(t)p)-11 s Ip-111(~-~

~(t))~l ~ IP-~l~-~ ~ ~.

Now the assertion follows from (I - A(t)P)-IA(t) = [ ( I - A(t)P) -I - I]P -I and I - A(t)P = P-I(I - PA(t))~.

I f P is a lower triangular matrix then, choosing D = [E, 2 , . . . ,

on] with s u f f i -

c i e n t l y small ~ > O, the assumption on P in this lemma is revealed to be equivalent to the condition diag(P) > O. Now we turn to the estimation of the i t e r a t i o n operator G(AtA)n under the assumption that the matrix A is not necessarily diagonable. The main tool is here the f o l lowing result due to J. von Neumann:

(5.3.15) Theorem. Let @ be regular in a neighborhood of the unit disk {n E {, lql ~ I} and let B be a (m,m)-matrix with [BI ~ 4. Then

I~(B)I ~ su%c{,[~l

~II~(~)I"

Proof. See e.g. Riesz and Nagy [52].

132

(5.3.16) Lemma. (Crouzeix [75 ] . ) Let Re(A) z al, ~ E IR, and let G be a rational function which is bounded in the half-plane {q E 6, Re n ~ a}. Then

IG(A) I ~ suPnc $,Re n B~IG(n)t.

P r o o f . The m a t r i x

B = (I - (A - ~ I ) ) ( l + (A - ~I) -I satisfies I B I s I by Lemma (1.5.5). Writing I n) ¢(q) = G(m +T-~'~'n -

we have

~))

- (n G(q) = ~( I + (n - a) and n ÷ ~ + (I - n ) / ( 1

+ n) is a b i j e c t i v e

{n E 6, R e n ~ ~} ~ { - } .

suPlnI ~ 11m(n)l

mapping o f the u n i t d i s k onto the h a l f - p l a n e

Thus we have

: sUPReq ~ ~IG(n)l

and the rational function@ has no poles in the unit disk. ¢ is therefore regular in a neighborhood of the unit disk and because G(A) = ¢(B) Theorem (5.3.10) proves the assertion. Before we prove the next result l e t us note once more that Runge-Kutta methods and single step multiderivative methods coincide for the test equation y' = xy. In particular, a Runge-Kutta method which is consistent with y' = ~y must also be consitent in the sense of Definition (1.1.7) (and Lemma (1.1.12)). Accordingly, the principal root of a consistent method (being here the only root at a l l ) has the same form as in (1.3.6), ~(~) : I + ~ + C ( 2 )

q~Oi

and Lemma (A.I.41) yields that every consistent Runge-Kutta method has a s t a b i l i t y region S containing a disk {~ E $, In + Pl ~ P}, P > O. (5.3.17) Lemma. (Crouzeix [75 ] . ) Let Re(A) > O, let G be a rational function which is bounded in the disk D = {n E {, In - Pl ~ P}, P > O, and let

(5.3.18) 0 s At spr[(A--~--~)-IAHA] ~ 2p.

133 Then

IG(AtA) I S SUPnEDIG(n)l.

Proof. By straightforward computation we v e r i f y that IAtA - Pl ~p is equivalent to the condition (Aw)HAw SUPw~ O ~

~ 2p,

and this condition is equivalent to (5.3.18) because ,,A + AH,-I.H . . . . A + AH,-I/2.H_,A + AH,-I/2~ A(A + AH)-I/2 sprt£---,2---) a Aj : sprLk---,2----) A J = I I. Writing now B = (AtA - p l ) / p , ¢(n) = G(o(q + I ) ) , and G(n = ¢((n - p)/p),

we have ¢(B) = G(AtA), ¢ is regular Theorem (5.3.10) proves the r e s u l t .

n a neighborhood of the u n i t disk, and hence

With respect to the linear problem (5.3.1) with constant but not necessarily diagonable matrix A we can now estimate ?&tA* in (5.3.4) by Lemma (5.3.7) or Lemma (5.3.13) and find that IG(AtA) I S I under the assumptions of Lemma (5.3.16) or Lemma (5.3.17). The result can be assembled in the following theorem: (5.3.19) Theorem. (i) Let the (m,m)-matrix A in (5.3.1) be regular and constant, and let the solution y be (p+l)-times continuously differentiable. (ii) Let the Runge-Kutta method be consistent of order p with the problem (5.3.1), of degree p* - 2, and A-stable. (iii) Let Re(A) < 0 and let all eigenvalues of P be real and nonnegative or let Re(A) ~ 0 and let all nonzero eigenvalues of P be positive. (iv) Let P be regular or let the dimension of the kernel of P be equal to the multiplicity of the eigenvalue 0 of P. Then for n = 1 , 2 , . . . ,

ly n - Vnl S K [ ly 0 - v0' + at pnat I IY(P+I)(T)Idt + nAtI"h]"n + nAtP+I~=P*IIIAP+I-ky(k)IIIn ] There exists a p, 0 < p < s p r ( P * ) - I / 2 ,

P* = [ I ~ i j l ] ri , j = l '

I

such that A-stability, assumption (iii) and (iv) can be replaced by Re(A) < 0 and IAtA + pl I ~ p but then K depends Qn

p.

134 With respect to the general l i n e a r problem (5.3 I) we can estimate r * by • AtA,n Lemma (5.3.7) or Lemma ( 5 . 3 . 1 4 ) . In order to f i n d a bound f o r IG(AtA)n{ we suppose t h a t P is r e g u l a r and obtain G(AtA)n z I + AtqTA_n(l- AtP~nA)-Iz = I + qT[-1[( Z-

AtP_~nA) - I - Z]z = I - q T[ -I z + q T p - I ( I

- AtPA ) - I z

(5.3.20) = G(AtAn ) + q T [ - 1 [ ( ~ _

AtPA ) - I _ ( Z -

= G(AtAn) + AtqTp-1(l

AtP--An)-1]z

_ AtP_~A)-I[[(A.A n - An~.l)Anl]An( Z - AtP_An)-Iz.

Here, G(AtA n) = - o1(AtAn)-la0(AtAn ) is a r a t i o n a l function with the argument AtA n. Hence Lemma (5.3.14) and Lemma (5.3.16) y i e l d f o r an A-stable method and Re(A(t)) ~ 0 IG(AtA)nl ~ I + <*en+iAt with the notation (5.3.21)

en :

max {IA(t)-I(A(T) 0 ~ t ~ ( n - 1 ) A t , t~Tst+At

- A ( t ) ) I, I(A(~) - A ( t ) ) A ( t ) - 1 1 } A t

-I

and

lqllAt~( Z-~tP_A~A)III(I

~tP_anllltz I ~<*.

The complete result reads as follows: (5.3.22) Theorem. (Crouzeix [75 ] . ) ( i ) Let the (m,m)-matrix A(t) in (5.3.1) and the solution y be respectively p-times and (p+l)-times continuously differentiable; let A(t) be regular and ReIA(t)) < 0 for t > O.

(ii) Let the Runge-Kutta method be of order p for the problem (5.3.1), of degree p * - 2 , andA-stable,

S = {n E {, Ren < O} v {~}.

(iii) Let there exist a regular diagonal matrix D such that Re(DPD -I) > O. Then for n = 1,2,..., <*OnnAt [

ly n - Vnl < e

[[Y0 - v0[

nat

where IIWk(At,y)IIn = maxn_ _w.(z~t,y) and w.(z~t,y) u~v
KI

v

K

n

is defined in (5.2.38). For regular

P exist~ a p, 0 < p < spr(P*)- /2, such that A-stability and ass. (iii) can be replaced

by Re(A(t)) < 0 and IAtA(t) + pl I < p for t > 0 but then < depends on p.

135 The assumption of regular P is somewhat unsatisfactory here because not only exp l i c i t methods but also some i m p l i c i t methods have a singular matrix P. However, i f there exists a vector g such that q = Pg then we can write instead of (5.3.20) (5.3.23) G(Ata) n = G(AtAn) + g T ( z - AtP~A )-Ip[(A

- an~I)anl]AtAn(Z-

At~An)-Iz.

Here, IG(AtAn) I ~I follows as above, IAtAn( ~ - AtLAn)-11 ~ < follows from Lemma (5.3. 13) but for r*AtA,n (cf. (5.2.37) and (5.3.3)) we have to find a new bound under the assumption of Lemma (5.3.13): (5.3.24) Lemma. Let the assumption of Lemma (5.3.13) be fulfilled, let <, be the constant of Len~na (5.3.13), and let

(5.3.25) <,IPIOnAt 0 ~ r 0 < I,

n = 0,1 . . . .

Then for 0 < at ~ ~t 0

m a x { l ( Z - AtPA_nA)-II, [ ( L I(Z-

AtA~p)-IAtA_ml

stA-~P)-11 } s <(I - r0 ) - I ,

~ <(I - r0 )-In~t~(n+1)Atlmax anIA(~)l.

Proof. We write (I_ - AtPa._.n) = (I_ - At~An)(l_ - Ba) where B

= At( Z - AtP_An)-IP(A - An~I) = ( Z -

~t-PAn)-IAtAnP[An1(A-~ - An~I)]

hence by Lemma (5.3.13), as An~I commutes with ~, IBm1 ~ <,IPIenAt 0 ~ r 0 < I. Accordingly we obtain again by Lemma (5.3.13) I(Z - AtP_~nA)-II

~ I(Z - AtP..An)-111(Z- ~n)-11 ~ <,(I - r0)-1.

This bound holds also for I ( ~ -

AtA - I I ,P) ~

furthermore

136

(Z - AtA~R)-IAtA_~ = (Z - B - ~ ) - I [ ( z - AtPAn)-IAtAn](AnIA_~) which proves the second r e s u l t a f t e r a f u r t h e r application of Lemma (5.3.13). By means of t h i s lemma, Theorem (5.3.22) can be modified as f o l l o w s , see also Crouzeix [75 ] . (5.3.26) Theorem. Let ass. (i) and (ii) of Theorem (5.3.22) be fulfilled, and let the assumption of Lemma (5.3.13) be fulfilled for A(t) and t > O, Let At 0 > 0 be defined by (5.3.25) and let there exist a g such that q = Pg, Then the error bound of Theorem

(5.3.22) holds for 0 < At < At 0 with K and K* multiplied by (I - F0 )-I and K depending

on m a x 0 s t ~ ( n _ 1 ) A t , t ~ t + A t l A ( t ) - I A ( T ) l . There exists a p > 0 such that A-stability and the assumption of Lemma (5.3.13) can be replaced by Re(A(t))

< 0 and IAtA(t) + pl i ~ p for t > 0 but then ~ depends on p.

In concluding t h i s section we note that instead of (5.3.20) we also can w r i t e without any assumption on P G(AtA) n = G(AtAn) + AtqT[An - A I ] ( Z - AtP_~nA) - I z + At2qTan(Z - At_PAn)-I~[A - an~l]( Z - AtP_~nA) - I z . This leads to f u r t h e r e r r o r bounds in the case where IAtAn( Z -

AtP..An)-I [

and

i[A=n - An~l]( Z -

AtP~nA) - I i

is bounded, e . g . , i f A(t) = a(t)A with a scalar function a.

5.4. Examples and Remarks

The c h a r a c t e r i s t i c polynomial ~(~,q) = ~1(n)~ + ~0(n) of a single step multistage or m u l t i d e r i v a t i v e method has only one root ~(q) which is the a m p l i f i c a t i o n f a c t o r G(n) with respect to the t e s t equation y' = xy, Vn+ I = G(n)Vn, c f . (5.1.12). Hence the s t a b i l i t y

n = 0,I,...,

n = AtX,

region S is closed in ~ as we have already mentioned

137 above. In p a r t i c u l a r , we have 0 E S i f the method is c o n s i s t e n t and (5.4.1)

c(n) : G(n) : I + n + O(n 2) : en + ~(n2),

Therefore, an e x p l i c i t meaning, i . e . , and f i n i t e

~÷0.

c o n s i s t e n t Runge-Kutta method is convergent in the c l a s s i c a l

f o r w e l l - c o n d i t i o n e d problems with L i p s c h i t z - c o n t i n u o u s f in (5.1.1)

time i n t e r v a l s . Because of ( 5 . 4 . 1 ) , Lemma ( A . I . 4 1 ) or a simple d i r e c t v e r i -

f i c a t i o n shows: (5.4.2) C o r o l l a r y .

I f the general Runge-Kutta method ( 5 . 1 . 6 ) , (5.1.7) is consistent

then there exists a p > 0 such that

p

={nee,

In+~I-
Obviously we have r j G(n) = I + n + ~ ~j=2~jn in e x p l i c i t

consistent methods of stage r. Jeltsch and Nevanlinnna {78 ] have proved

the f o l l o w i n g r e s u l t : (5.4.3) Lemma. The stability region

S of an explicit r-stage method satisfies "~'r C_ S

iff G(n) = (I + ( n / r ) ) r.

A g e n e r a l i z a t i o n o f t h i s r e s u l t to n o n l i n e a r m u l t i s t e p methods is found in Jeltsch and Nevanlinna [81 , Theorem 3 . 1 ] . For e x p l i c i t

r-stage Runge-Kutta methods the maximum a t t a i n a b l e order pf has been

derived by Butcher [65 et a l . ]

in a tedious work. The r e s u l t can be represented in the

f o l l o w i n g t a b l e ( c f . also Lambert [73 , p. 122]): (5.4.4) Table. r pf

1

2

3

4

5

6

7

8

9

r ~ I0

I

2

3

4

4

5

6

6

7

p* s r - 2

I t can be shown t h a t there e x i s t e x p l i c i t

Runge-Kutta methods of a r b i t r a r y order. For

a given stage r = I . . . . ,4 the a m p l i f i c a t i o n f a c t o r G(n) of an e x p l i c i t of (maximum) order p = r has the form

r-stage method

188

G(n) = Zj=0 r ~ qJ

p = r = I,...,4.

A c c o r d i n g l y , a l l these methods have the same s t a b i l i t y are given e.g. in G r i g o r i e f f

region f o r f i x e d p = r. Plots

[72 , p. 109], Lambert [73 , p. 227], and S t e t t e r [73 .

p. 176], and we r e f e r also to these books f o r f u r t h e r d e t a i l s and special examples. A f t e r these b r i e f remarks on e x p l i c i t

methods l e t us now turn to i m p l i c i t

which deserve more i n t e r e s t in the s o l u t i o n of s t i f f

methods

problems.

The general c o n s i s t e n t single-stage method is the method (4.2.2) with m E IR. We consider the l i n e a r problem ( 5 . 3 . 1 ) , (5.4.5)

y' = A ( t ) y + c ( t ) + h ( t ) ,

t > 0, y(0) = Y0'

and obtain the computational device (5.4.6)

Vn+ I = (I - mAtAn+w)-1[(l + ( 1 - w ) A t A n + ) v n + AtCn+ ] ,

n = 0,1 . . . . .

where Cn+m = c ( ( n + ~ ) A t ) . Let 0 < m < I and l e t A(t) be r e g u l a r w i t h Re(A(t)) ~ 0. Then Lemma (5.3.13) y i e l d s i(l

- mAtan+ ) - I I ~ ~,

I ( l - ~AtAn+ ) - I ( I

+ ( 1 - ~ ) A t A n + ) i ~ K.

The degree of the method is p* - 2 = 0 f o r a l l ~ EIR hence (5.2.19) and Lemma (5.3.14) y i e l d f o r the d i s c r e t i z a t i o n

error ot+At

Id(At,y)(t)l ~ 1½ - ~IAt21y(2)(t)I + r[ At~ f t

Io~

lYt°J(T)I dT

+ At31(l - m A t A ( t + ~ A t ) ) - I A ( t + ~ A t ) y ( 2 ) ( t ) i ] where the t h i r d term on the r i g h t side cannot be cancelled. We state the r e s u l t as follows:

(5.4.7) Corollar£. Let the real (m,m)-matrix A(t) in (5.4.5) be regular with Re(A(t)) 0 for t > 0 and let the solution y be three times continuously differentiable. Then

the error of the method

(5.4.6)

satisfies for 0 < ~ < I and nat

ly n - Vnl < = Iy 0 - v0I + £[At 2 + n([~-

ml

+

lil(l

/3)

i Iy~

n = I12 .... i

(T)id~ + nAt[llhilln

- mAtA('))-IAtA(')li[n)At21[lY(2){iinl.

J

With exception of the second row t h i s e r r o r bound is r a t h e r simple in comparison w i t h

139 the corresponding r e s u l t f o r the trapezoidal r u l e , Theorem (4.3.19). For m = I/2 the method has convergence order two f o r well-conditioned problems. However, i f A(t) is i l l - c o n d i t i o n e d then

l(I - mAtA(t))-IAtA(t)l

~

by Lemma (5.3.14) and the uniform convergence order with respect to (5.4.5) is only one.

A s i m i l a r remark holds obviously i f m is n~odified s l i g h t l y f o l l o w i n g the proposi-

tion of Kreth, (4.2.8). In Section 1.5 the s t a b i l i t y of diagonal Pad~ approximants has been proved in a d i r e c t way which however seems d i f f i c u l t

to apply to other methods. But Lemma (5.3.16)

generalizes the second part of Lemma (1.5.5) to every A-stable single step method and, by Corollary ( 5 . 4 . 2 ) , Lemma (5.3.17) applies to every consistent single step method. Recall now that o0(n) and o1(n) are always supposed to have no common f a c t o r and w r i t e G ,v(n)

= - ~0(n)/o1(n)

i f deg(~0(n)) ~ ~, deg(o1(n)) s v. Then we have by Lemma (1.3.5) G

(n) : en + ~(nP+1),

n ÷ 0, p ~ I ,

in the case of consistence,and p = u + v is the maximum a t t a i n a b l e order of the corresponding single step method f o r the test equation y' = xy.

(5.4.8) Definition. G

(n) is a (v,~)-Pad¢ approximant (of the exponential function

near~ : O) iff G(n)

= e n + ~(n ~+~+I ),

n +0.

A Pad~ approximant determines uniquely a single step m u l t i d e r i v a t i v e method but not a Runge-Kutta method f o r the general problem (5.1.1). Nevertheless we say b r i e f l y that a method is a Pad~ approximant i f the corresponding Gu,v(n) is a Pad~ approximant. By d e f i n i t i o n these methods have maximum order f o r the test equation which j u s t i f i e s their popularity. (u,v)-Pad~ approximants of an a r b i t r a r y function are determined uniquely i f they e x i s t and can be computed e x p l i c i t e l y , see Hummel and Seebeck [49 ]. For the exponent i a l function we obtain the f o l l o w i n g r e s u l t ; cf. also G r i g o r i e f f [72 ] . (5.4.9) Lemma. L e t ~! =

=

(~+~-j)[ nj Tr-Fc

T

7

140

and

G~, (~)=o ,~(n)l%, (-n) then

en : Gu,v(n) + (-1)VKn ~+v+1 + ~(nu+v+2),

where ~ > O, and

su,w(n), o(-n)

n + O,

have no common factor.

Crouzeix and R a v i a r t [80 , Theorem 2.4.3] have shown in a simple way t h a t every ( ~ , ~ ) Pad~ approximant is A(O)-stable f o r u ~ v. The f o l l o w i n g r e s u l t has been proved by Wanner, H a i r e r , and Norsett [78a] by a study of the order s t a r : (5.4.10) Lemma. (~,v)-Pad~ approximants are A-stable iff ~ ~ v ~ ~ + 2. For ~ = u, u + I t h i s r e s u l t is also found in G r i g o r i e f f

[72 ].

I f v = ~ + I or ~ =

+ 2 then we have = c S which is f a v o r a b l e f o r the e r r o r propagation in i l l - c o n d i tionend problems, cf. e.g. Section 4.2. Lemma (5.4.10) implies t h a t no poles of G in other words, t h a t a l l Therefore the f i r s t (5.4.11)

roots of the denominator of G ( n )

p a r t of Lemma ( I . 5 . 5 )

I~ , ( - A t A ) - I I

(q) l i e in the l e f t

half-plane or,

have p o s i t i v e real p a r t .

is v a l i d here, t o o ,

~ K,

Re(A) ~ O, ~ ~ ~ ~ ~ + 2.

Furthermore, Lemma (5.3.16) y i e l d s (5.4.12)

IG ,~(AtA) I ~ I ,

Re(A) ~ O, ~ ~ ~ ~ ~ + 2.

(5.4.11) and (5.4.12) t o g e t h e r provide the f o l l o w i n g r e s u l t : (5.4.13) C o r o l l a r y . Let the (m,m)-matrix A in (5.4.5) be constant, let Re(A) ~ O, and let the solution y be (~+~+l)-times continuously differentiable.

Then the error of

a (~,w)-Pad~ approximant satisfies for ~ ~ ~ ~ ~ + 2

ly n - VnI s Iy 0 - Vo{ + ~[At "+~

nit[y(U+v+1)

An a p p l i c a t i o n of t h i s r e s u l t to the f i r s t (5.4.14) z' = A*z + c * ( t )

+ h*(t),

of the second order problem

( z ) I d z + nAt maxo~i~max{~,v}_1111h(i)liln].

order t r a n s f o r m a t i o n ( I . 4 . 2 ) ,

t > O, z(O) = z O,

141

(5.4.15) y "

: Ay + By' + c ( t ) + h ( t ) , t > O, y(O) = YO' y'(O) = y~,

yields a f t e r the transformation (1.4.10) immediately the following generalization of Theorem (1.5.9): (5.4.16) Theorem. In the initial value problem (5.4.15) let A, B be real symmetric, A ~ - yI < O, B ~ O, and let the solution y be (~+~+2)-times continuously differentiable. Let V*n : (Vn' Wn )T~ n = 1,2,..., be obtained by a (v,~)-Pad¢ approximant with ~ ~ ~ ~ + 2 applied to the transformed problem (5.4.14). Then

lY n -

vnl

+ At~+~

+ ly~ - Wnf ~ K(1 + y - 1 ) 1 / 2 [ l ( - A ) l / 2 ( y 0 -

vo)l

nat I (I(-A)I/2y(U+v+I)(T)I + lY(~+~+2)(~)l)d~ + n~t

+ ly~ - Wol

max llIh(i)llln]. Osi~max{~,u}-1

Up today Runge-Kutta methods haven't lost anything from t h e i r a t t r a c t i o n for numer i c a l analysis and application. On the contrary, methods in which the matrix P has only one eigenvalue are an essential subject of current research. A thorough presentation of the results available here in the meanwhile would go f a r beyond the scope of this volume. For a concise treatment and some interesting existence and uniqueness statements we refer to the forthcoming book of Crouzeix and Raviart [80 ].

VI. Approximation of I n i t i a l

6.1. I n i t i a l

Boundary Value Problems

Boundary Value Problems and Galerkin Procedures

Unlike e l l i p t i c

and p a r a b o l i c problems there are in hyperbolic problems b a s i c a l l y

two d i f f e r e n t ways of numerical approximation in dependence of the underlying form of the d i f f e r e n t i a l

equation and the given i n i t i a l

and boundary c o n d i t i o n s : The method of

c h a r a c t e r i s t i c s and the method o f lines. In the former method the s o l u t i o n is computed along the c h a r a c t e r i s t i c curves which implies a strong connection between time and space d i s c r e t i z a t i o n whereas in the l a t t e r method time and space are d i s c r e t i z e d in a separated way. On each time l e v e l t = n&t an ' e l l i p t i c ' finite

problem is solved here by a

d i f f e r e n c e method or a Galerkin procedure. The connection between time and space

d i s c r e t i z a t i o n consists i f at a l l in a Courant-Friedrichs-Lewy c o n d i t i o n which guarantees t h a t the spectral radius of the i t e r a t i o n operator with respect to the time d i r e c t i o n is not greater than one. The method of l i n e s has the advantage t h a t numerical methods f o r e l l i p t i c differential

problems and methods f o r i n i t i a l

value problems with o r d i n a r y

equations can be applied in space and time d i r e c t i o n r e s p e c t i v e l y w i t h o u t

much p r e l i m i n a r y work. In t h i s chapter we study the numerical approximation of l i n e a r p a r a b o l i c problems and hyperbolic problems of second order by the method of l i n e s choosing Galerkin procedures f o r the d i s c r e t i z a t i o n in the space d i r e c t i o n . Some s p e c i f i c assumptions are then made f o r the e r r o r estimations which are f u l f i l l e d and f i n i t e

by a large class of problems

element methods.

The d e s c r i p t i o n of the a n a l y t i c problems to be considered needs some f u r t h e r notat i o n s which are l i s t e d up f o r shortness: c IRr bounded and open domain; (f,g)

= Sf(x)g(x)dx,

Ifl 2

= (f,f);

llfll s = (Zl~l~sIDSfI2) I / 2 , s £1N, Sobolev norm with the standard m u l t i - i n d e x nota~r t i o n , o = (o I . . . . . Or), oi ELN, D°f = B l ° I f / ~ x ~ 1 . . . B x r , Iol = o I + . . . + o r , t

111fllls,n

: m a x o s t ~ n A t l l f ( . , t ) 11s;

wS(~) = { f c L2(R), asf c L2(~), v o, Is[ ~ s} Sobolev space, W~(~) = { f c ws(R), f ( x ) = 0 v x E @R}, H C ws(~) H i l b e r t space with W~(~) C H; a: HxH ~ (u,v) ~ a(u,v) E ]R symmetric b i l i n e a r form such t h a t a ( v , v ) I / 2 defines

143 a norm which is e q u i v a l e n t to II,IIs over H, 0 < yIIvIIs _-
b: IR+xHxH ) (t,u,v) ~ b(t;u,v) E IR symmetric bilinear form in u and v such that 0 < b(t;v,v)

CAx,I' " " '

< BIIvII~ v 0 m v E H or b = 0.

~Ax,m(Ax) denote l i n e a r independent functions which span the Galerkin

subspace G~x c H and d e f i n e the Galerkin procedure. In f i n i t e notes the maximum diameter of a l l omit the index Ax i f

it

element a n a l y s i s ax

de-

'patches' by which the domain ~ is p a r t i t i o n e d . We

is not necessary and use the f o l l o w i n g f u r t h e r n o t a t i o n s :

= (~I . . . . . Cm)T, M = [(~u,tv)]m,v~ = I ' K = [ a ( t

~

~

-

E G R i t z p r o j e c t i o n of w E H defined by a(w - ~,v) = 0 v v E G; E G L 2- p r o j e c t i o n of w E L2(~) = W0 (Q) defined by (w - ^w,v) = 0 v v £ G. In order to make a d i f f e r e n c e between s c a l a r - and v e c t o r - v a l u e d functions we denote in the sequel vectors of dimension m by c a p i t a l s and w r i t e f o r instance w(x) = wT~(x) E k~:lWu¢ ( x ) , w(x) = wT~(x). A f t e r these p r e l i m i n a r i e s we can define an e l l i p t i c (6.1.1)

model problem by

a(w,v) = ( c , v ) v v E H,

c E L2(~).

The R i t z p r o j e c t i o n ~ £ G of the exact s o l u t i o n w E H is the f i n i t e

element a p p r o x i -

mation of w by the main theorem of the R a y l e i g h - R i t z - G a l e r k i n theory, cf. e.g. Strang and Fix [73 , Theorem 1.1]. only i f G is a f i n i t e

( N a t u r a l l y , we can speak of a f i n i t e

element approximation

element subspace and not a general Galerkin subspace.) A - p r i o r i

estimations of w - w take a large place in modern f i n i t e

element a n a l y s i s . In the pre-

sent work we however do not t r e a t e r r o r estimations f o r e l l i p t i c

problems but r e f e r the

reader e.g. to the book of C i a r l e t [79 ] f o r a d e t a i l l e d a n a l y s i s of f i n i t e methods. Instead we suppose here t h a t the e l l i p t i c procedure, i . e . ,

element

problem (6.1.1) and the Galerkin

the subspace G c H, have the f o l l o w i n g property in which Ax denotes

again the small parameter introduced in the d e f i n i t i o n of G. (6.1.2) Assumption. Let W E H be the solution of (6.1.1) and let w E G be the Ritz projection of w then there exist positive integers q and q. such that for all c£ L2(~) Iw - wl =<
144 The announced parabolic model problem now reads (6.1.3)

( u t ( . , t ) , v ) + a ( u ( . , t ) , v ) = ( c ( . , t ) , v ) v v E H, t > 0, u(-,0) = u0,

where ut = Bu/~t, u0 E H, and c ( . , t ) E L2(Q) for t > 0. We assume that a l l data are s u f f i c i e n t l y smooth such that Assumption (6.1.2) holds with respect to the underlying b i l i n e a r form a and that there exists a unique solution u with u ( . , t ) E H and u t ( . , t ) E L2(~) for t > 0. The Galerkin approximation UG(.,t) E G of u ( . , t ) is the solution of (6.1.3) for a l l v E G. We substitute

(6.1.4)

UG(X,t) = UG(t)T~(x) = c^ ( x , t )

Z~=IUG, ( t ) ~ ( x ) ,

: ^C(t) m@(x), ( c ( . , t )

- ~(.,t),v)

: 0 V v E G,

^ ^T u0(x) = U0~(x), (u 0 - u0,v) = 0 v v E G, into (6.1.3) and set v = Cu, u = I . . . . ,m, successively. Then we obtain an i n i t i a l value problem with an ordinary d i f f e r e n t i a l system for the unknown function UG: [0,~] +IRm: (6.1.5)

MU~ + KUG = M~(t), t > 0, UG(0) = ~0"

The hyperbolic model problem we consider reads (utt(.,t),v) + b(t;ut(.,t),v) + ×a(ut(.,t),v) + a(u(-,t),v) : (c(.,t),v)

(6.1.6) v v E H, t > 0, u(.,0) = u0, u t ( . , 0 ) = u~.

Here we assume t h a t x ~ 0 and t h a t in a d d i t i o n to the above assumptions u~ E H and the exact s o l u t i o n u s a t i s f i e s

utt(.,t)

£ L2(Q) and u t ( . , t )

E H i f x > 0. The special

form of the damping term is chosen here to enclose a l l cases which are considered below. Let

(6.1.7)

N(t) = [ b ( t ; ¢ ~ , ¢ v ) ] ~ , v : I

then the semi-discrete problem associated w i t h (6.1.6) reads a

(6.1.8)

A

MU~' + (N(t) + xK)U~ + KUG = MC(t), t > 0, UG(0) : U0, U~(0) : U~.

Note t h a t M, N, and K are real symmetric and p o s i t i v e d e f i n i t e matrices. In engineering mechanics t h i s system is c a l l e d the e q u i l i b r i u m equations of dynamic

145 f i n i t e element analysis and plays a fundamental r o l e . The basic p a r t i a l d i f f e r e n t i a l equation is however not available in matrix s t r u c t u r a l analysis. Instead the o r i g i n a l body is p a r t i t i o n e d into more or less small c e l l s of which the equations of motion can be approximated in a simpler way. These interdependent equations are then assembled to a large system which has the form (6.1.8). M, N + ×K, and K are then the mass, damping, and s t i f f n e s s matrix, and C(t) is the external load vector. See e.g. Bathe and Wilson {76 ] , Fried [79 ] , and Przemienicki [68 ]. In the meanwhile, these notations have also become customary in numerical analysis. I f damping is not disregarded then i t is frequently of the above form; cf. e.g. Przemienicki [68 , ch, 13], Clough [71 ] , and Cook [74 , p. 303]. A f t e r having d i s c r e t i z e d the problem in the space d i r e c t i o n i t remains to solve the semi-discrete problem (6.1.5) or (6.1.8) numerically. For t h i s we always w r i t e the d i f f e r e n t i a l system in e x p l i c i t form, e.g. instead of (6.1.5) (6.1.9)

U~ = - M-IKuG + ~ ( t ) ,

and then t r y to avoid the e x p l i c i t computation of M- I as a r u l e . For instance, the multistep m u l t i d e r i v a t i v e method (1.1.3) has for (6.1.9) the form (1.2.2) with A = M-IK and c = C, -

k k (6.1.10) ~i=0oi(-AtM-Im)TiVn = - ~i=0~j=1oij(-AtM - I ~

K)AtJTIC(J-l) ^ .

,

n = 0 m1,..e

Of course, t h i s scheme is m u l t i p l i e d by M again. As M~(t) = C * ( t ) , C*(t) = ( ( c ( . , t ) , ~ I) . . . . . ( c ( . , t ) , ~ m ) ) T, we get for l i n e a r multistep methods the computational device n = 0,I , . . .

(6.1.11) MP0(T)Vn - AtKPI(T)V n = AtPI(T)C~, In the general case (6.1.10) a l i n e a r system of the form (6.1.12) MOk(-AtM-IK)Vn+k = Rn

is to be solved in every time step. The computation of Rn requires f o r ~ > I some matrix-vector m u l t i p l i c a t i o n s and the s o l u t i o n of l i n e a r systems with the mass matrix M. The matrix on the l e f t side of (6.1.12) is regular i f Sp(-AtM-I/2KM-I/2) C S because M~k(-AtM-IK ) = MI/2~k(_AtM-I/2KM-I/2)MI/2 and the d e f i n i t i o n of the s t a b i l i t y

region S, (1.2.7).

I f the polynomial Ok(n) is non-

146 linear, h °k(n) = ~hk ] ~ ( n j=1

- n j ) , nj c ~, I < h _-< ~,

then (6.1.12) can be written as mhk(&tK + Mnl)M-I(AtK + Mn2)...M-I(&tK + Mnh)Vn+k = (-1)hRn , and the solution Vn+k is computed by solving successively the h linear systems ~hk(AtK + Mnl)Z I = (-1)hR n, (AtK + Mnj)Zj = MZj_I , j = 2 . . . . . h - l , (6.1.13) (&tK + Mnh)Vn+k = MZh_1. Here i t is advantageous for an application of the Cholesky decomposition to use methods in which the leading polynomial Ok(n) has only roots nj with positive real part. This requirement is e.g. f u l f i l l e d by the diagonal and subdiagonal Pad~ approximants presented in Section 1.5, and by the methods of Enright given in table (A.4.5). Obviously, i f the nonlinear Ok(n) has the form Ok(n) = ~hk(n - nl )h, I < h ~ ~, then (6.1.13) leads to the solution of h linear systems with the same matrix AtK + Mn I. Methods with this property (and Ren I > O) are e.g. Calahan's method, cf. A.4.(iib), Enright's methods II given in table (A.4.6), and the restricted Pad~ approximants presented e.g. in the forthcoming monograph of Crouzeix and Raviart [80]. Naturally, the same arguments concerning the computational amount of work hold also for the numerical solution of second order initial value problems (6.1.8).

6.2. Error Estimates for Galerkin-Multistep Procedures and Parabolic Problems

In this section we use Theorem (1.2.12) and (1.2.18) to derive a - p r i o r i error bounds for the parabolic model problem (6.1.3). Instead of (6.1.9) and the numerical approximation (6.1.10) we write (6.2.1)

M~/2U~ = AMI/2u G + M1/2~(t)

and k ~k ~ I/2^(j-I) (6.2.2) ~ ~i=O~i(AtA)TiM1/2Vn : - Zi=OZj=IOij(AtA)AtJTiM Cn ,

n = 0,1,

147 w i t h the leading matrix A = - M-1/2KM-1/2 being real symmetric and negative d e f i n i t e .

The f u l l - d i s c r e t e

approximation U A ( . , t ) E G of the exact s o l u t i o n u ( - , t )

scheme ( 6 . 2 . 2 ) y i e l d s an

of the form

UA(X,t) : V ( t ) T ~ ( x ) , t : nat,

n = k,k+1 . . . .

By the fundamental r e l a t i o n (6.2.3)

IM1/2Wl = IwT~(-)I ~ lwl v w = wT~(.) E G

we then obtain immediately an e r r o r bound f o r the Galerkin approximation uG defined by (6.1.4),

(6.1.5),

i.e.,

a bound of

IMI/2(UG,n - Vn) i = i(u G - u A ) ( - , n A t ) l , when an estimation of Section 1.2 is applied to the p a i r ( 6 . 2 . 1 ) ,

n = k,k+1 . . . . . (6.2.2).

However,

an e r r o r estimation via the decomposition (6.2.4)

u - uA : (U - u) + (~ - UA)

denoting the Ritz p r o j e c t i o n of u again d i s t i n g u i s h e s more e x a c t l y between space and time d i s c r e t i z a t i o n

and moreover an estimation of u - u G needs also the approximation

properties of ~; c f . e.g. Fairweather [78 ]. Therefore we use the decomposition (6.2.4) in t h i s chapter. Assumption (6.1.2) y i e l d s immediately (6.2.5)

i(u - U A ) ( . , t ) i ~ ~GAXqIlu(.,t)llq, + i(~ - U A ) ( . , t ) I

hence i t s u f f i c e s to deduce e r r o r bounds with respect to the Ritz p r o j e c t i o n u in the sequel. I f the data are s u f f i c i e n t l y

smooth then the parabolic problem (6.1.3) y i e l d s

a(u(~)(.,t),v) = (c(")(.,t) - u(~+l)(.,t),v) v v E H writing shortly u(u) = ~Uu/~tu and Assumption (6.1.2) yields again (6.2.6)

lu(~)(.,t)

- u(~)(.,t)i

~
AS U(~) = ~(u) in the present case of a time-independent b i l i n e a r form a, time d e r i v a t i v e s of ~ in the e r r o r bounds can be replaced by corresponding t i m e - d e r i v a t i v e s of

148

u using the triangle

inequality and (6.2.6) or, in a more direct way, by

lu(~)(.,t)] ~ (F/y)I/211u(~)(.,t)lls which results from [w I s Ilwlls and ¥11Wll~ ~ a(~,w) ~ a(w,w) ~ rllwll~ v w E H. Now, the Ritz projection u s a t i s f i e s (ut(.,t),v)

+ a(u(.,t),v)

= (c(-,t),v)

- ([ut - ut](.,t),v)

v v E G, t > O.

We substitute u ( x , t ) = U(t)T@(x) and obtain in the same way as above the following d i f f e r e n t i a l system for the unknown function ~: [0,~] ~IR m, (6.2.7)

MI/2u ' = AMI/2u + MI/2c(t) - MI/2H(t),

where h ( . , t ) denotes the L2-projection of h ( . , t ) = (u t - ~ t ) ( . , t ) and h ( x , t ) = ^ T H(t) @(x). For instance, Theorem (1.2.12) then yields immediately the following error bound:

IM1/2(~n - Vn)l

~ ~R [ Z~IM1/2(Ui~ -

vi)l

+ nAtmaxo~i~_ III[

÷ AtP n i t ,M1/2~(p+1 )(T)IdT

M1/2~(i)

llln].

But, by (6.2.3), IMI/2(Un~ _ Vn) I

=

l(u - u A ) ( . ) n [ ,

IMI/2u(P+I)(~)I

= I~(P+I)(.,~)I

and the Projection Theorem together with Assumption (6.1.2) yields IM1/2H(i)(t)l

= l~(i)(.,t)l

s l[u t - ~ t ] ( i ) ( . , t ) l

(6.2.8) = I[u ( i + I ) _ u ( i + 1 ) ~ ] ( . , t ) l

~
For an application of Theorem (1.2.18) and Corollary (1.3.16) we have to v e r i f y that the (m,m)-matrix A = - M-I/2KM-1/2 in (6.2.1) is negative d e f i n i t e . For this l e t again 0 ~ w E G, w(x) = wT@(x), then the e l l i p t i c i t y condition 0 < yllvll~ ~ a(v,v) v v E H implies 0 < yIMI/2wI 2 = y[w[ 2 ~ a(w,w) = wTKw.

149

Thus we have 0 < ywTMw s wTKw

v

0 ~ W E IRm

and a s u b s t i t u t i o n of W = M-I/2z proves the desired r e s u l t , A ~ - y l . We summarize the r e s u l t s of t h i s section in the f o l l o w i n g two theorems:

(6.2.9) Theorem. (i) Let the parabolic problem (6.1.3) and the Galerkin subspace G C H satisfy As~umption ( 6 . 1 . 2 ) ;

let the exact solution U satisfy U(i) ( . , ) t

E H, i = 0 , . . .

...,~, U(£+1)(-,t) E L2(£) for t > O, and let the Ritz projection u be (p+l)-times continuously differentiable with respect to t. (ii) Let the method (6.1.10) be consistent of order p ~ ~ with stability region S. (iii) Let Sp(-AtM-I/2KM-I/2) C R C S where R is closed in ~. Then f o r

n = k,k+1, . . . .

I(u - uA)(')nl --< ~¢xqlllulllq.,n [k-1 nat + KR ~i=01(~- UA)(')il + AtP I lu(P+1)("~)Idt

] + KGnAtAxqmax1
(6.2.10) Theorem. Let the assumptions of Theorem (6.2.9) be fulfilled and let the method (6.1.10) fulfil Assumption (ii) of Corollary (1.3.16). Then for n = k~k+1,...~

r :~¥(n-k)At

t(u - uA)(.)nl

~ ~GAXqtljulllq.,n

n A t - ~ y ( ( n - k ) A t - T ) ~(p+l)( + AtP i e ~ I

+

~sLe

k 1

~i~ol(~- %)(.)i I

.,T)Id~

n -<~y(n-~)At + KGAtZ~=ke max(v-k)Ats~svAt m a x 1 ~ i ~ l l u ( i ) ( ' , ~ ) U q . ] •

The strong assumptions on the smoothness of the data can be released s u b s t a n t i a l l y . They are only introduced here in t h i s form in order to make the statements not too complicated. With respect to the system (6.2.1) Runge-Kutta methods d i f f e r from single-step m u l t i d e r i v a t i v e methods only in the treatment of the time-dependent vector C ( t ) . However, by Theorem (5.3.12), the e r r o r bound of Theorem (6.2.9) contains here the addit i o n a l term ~p . . A.p+l-k.m1/2~(k)illn" At pnAt~k=p

150 6.3. Error Estimates f o r Galerkin-Multistep

The Ritz projection (6.1.6) s a t i s f i e s

Procedures and Hyperbolic Problems

u of the exact solution u of the hyperbolic model problem

(utt(.,t),v)

+ b(t;ut(.,t),v)

= (c(.,t),v)

- ([utt

+ xa(ut(.,t),v)

+ a(u(.,t),v)

(6.3.1) - utt](.,t),v)

- b(t;[u t - ut](.,t),v)

I f b ~ 0 then l e t w t ( . , t ) E G be the projection of w ( . , t ) p o s i t i v e d e f i n i t e and symmetric b i l i n e a r form b ( t ; - , . ) , b(t;w(.,t)

- wt(.,t),v)

v v E G, t > O.

E H with respect to the

= 0 v v E G,

and l e t

(6.3.2)

A2 : - M-I/2KM-1/2, B(t) = - M-I/2N(t)M-I/2.

Then the time-dependent part U of ~ = ~T@ s a t i s f i e s MI/2u ' ' = A2MI/2u + (B(t) + ×A2)MI/2u ' + M1/2~(t) (6.3.3) -

MI/2HI(t)

- M-I/2N(t)H~(t )

where the matrix N(t) is defined in (6.1.7) and h1(x,t)

= [utt - utt~]^(x,t)

h~(x,t)

: [ut - u~]~(x,t)

= H1(t)m~(x),

= H~(tlT~(x).

In the following lemma a l l outstanding bounds are assembled which are needed in this section. (6.3.4) Lemma. Let the solution

U

of (6.1.6)

be sufficiently smooth and let Assumption

(6.1.2) be fulfilled, Then IM1/2Hli)(t) I S mG&Xqllu(i+1)(.,t)llq,; iM-I/2N(t)H~(i)(t)l

~ B~G&Xqllu(i+1)(.,t)IIq,

[AMI/2u(t) I ~ rl/211u(.,t)IIs.

i f i = I or N(t) = N constant;

151 Proof. The f i r s t assertion follows from (6.2.8). to consider the case i = I. Because IM-I/2N(t)H~(t)I

For the second assertion i t suffices

~ IM-I/2m(t)I/211N(t)|/2m~(t)l

and !M-1/2N(L)1/21

= IN(t)1/2M-1/21

= maxlv]= I vTM-1/2N( t ) M - I / 2 v

BmaxlvI=IVTM-I/2MM-I/2v = this r e s u l t follows from

=< b C t ; h 2 ( " t ) ' h 2 ( " t ) )

i
=< ~lh2 ( ' ' t ) 1 2 0

utl t>

follows from = a(u(.,t),u(.,t))

~ a(u(.,t),u(.,t))

s rllu(.,t)ll~.

For the solution of the semi-discrete problem (6.1.8) by an i n d i r e c t multistep method, we have to w r i t e the d i f f e r e n t i a l system of second order as an e x p l i c i t d i f f e r e n t i a l system of f i r s t order, Z' = - M-IK*(t)Z + C ( t ) , where Z = (UG, U~) T, C(t) = (0, C(L)) T, and

K*(t) =

10 K

I

N(t) + ×K

cf. Section 1.4. I f we suppose here that the damping does not depend on time then N(t) = N, K*(t) = K*, and the computational device is again (6.1.10), (6.3.5)

Z~:0oi(_AtM-IK.)~iv~:_~ ~i:0oj:1oij(~tMk ~ -IK.)~tJTic~Jl),

n:01

. .

But now we have V~ = (VTn, WT ). and the functions u.a,1 and uA,2 with ua,1(x,nat) = VnT~(x), uA,2(x,t) W~¢(x) represent approximations of u(x,nAt) and u ' ( x , n A t ) respectively.

152 For the e r r o r estimation we consider the modified system (6.3.6)

Z*' = A'Z* + C*(t) - H*(t)

with the notations Z* = (MI/2u, MI/2u') T, C*(t) = (0, MI/2c(t)) T, H*(t) = (0, MI/2HI(L) + M-I/mNH~(t))T, and A* =

cf.

(6.3.2).

(6.3.7)

[0 A2

i ] B + ×A2 '

Then, i f (6.3.5)

~=0oi(Ata.)TiM1/2V.

is also w r i t t e n as = _ ~k g (AtA*)AtJTiMI/2c*(J-I) -i=0~j=1~ij n

n

'

n = 0,I,..

"'

the pair ( 6 . 3 . 6 ) , (6.3.7) of analytic equation and numerical method is the same as that considered in Section 1.4 and 1.5: I f the damping in the hyperbolic equation (6.1.6) is independent of a, i . e . , × = 0, and b is independent of space and time, b(t;u,v)

= 6-(u,v),

6 ~ 0,

then B(t) = B = - Bl commutes with A2. I f then moreover damping is less than c r i t i c a l damping, B s 2v~-y, then Theorem (1.4.7) yields immediately the following e r r o r bound: lMI/2(~ n - Vn) i +

KR[(¥+ I ) / ( 2 y -

IAI-IIMI/2(~ v~y6)] I

_ Wn) l

/ 2 [ k-1 MI/2 Zi=0([

(Ui - Vi)l

+ MI ~ I /2(u~ -

wi)l)

+ At p nit(IM1/2~(p+1 )(T)I + IMI/2u(P+m)(~)I)d~ + ~GnAt max0si~z_ I lilM1/2H * ( i )

llln].

I f x > 0 and b does not depend on time then Theorem (5.4.16) yields a s i m i l a r bound f o r diagonal and subdiagonal Pad~ approximants. Both e r r o r estimations can be w r i t t e n with the arguments u, UA,I, UA, 2, and u instead of MI/2u, MI/2v, MI/2w, and H* by the fundamental r e l a t i o n in the following theorems.

(6.2.3) and Lemma (6.3.4).

The results are summarized

153 (6.3.8) Theorem. (i) Let the hyperbolic problem (6.1.6) and the Galerkin subspaoe G C H satisfy Assumption (6.1.2); let the exact solution u satisfy u(i)(.,t) E Hp i = 0~... ...,~+I, u(~+2)(-.t) E L2(Q) for t > O, and let the Rit8 projection u be (p+2)-times continuously differentiable with respect to t. (ii) Let

X = 0

in (6.1.6), b(t;u,v) = 6.(u,v),

and

0 ~ 6 <

2v~y where

y

is the ellip-

ticity constant.

(6.3.5) be consistent of order p ~ ~ with stability region S. Sp(-AtM-I/2KM -I/2) C R C S where R is closed in ~.

(iii) Let the method (iv) Let

Then for n = k,k+1 . . . . .

[(~ - u&,1)(.)nl

+ IM-I/2KM-I/21-1/21(~t

+ At p nit(l~(P+1)(.,~)l

_ uA,2)(.)nl

+ l~(P+2)(.,~)I)dr

+ ~G(I + 6)natAxqmaxlsis~+iIIIu(i)llIq,,n].]

(6.3.9) Theorem. (i) Let assumption (i) of Theorem (6.3.8) be fulfilled. (ii) Let X ~ 0 in (6.1.6) and let the bilinear form b be independent of time, b(t;u,v) = b(u,v). (iii) Let the method

(6.3.5)

be a diagonal or subdiagonal Pad~ approximant of order p.

Then for n = 1 , 2 , . . . ,

I(u - uA,1)(')n[
-I/2[

+ ](Gt - ua,2)(')nl

~ rl/2u(~ _ uA,1)(.,0)ll s + l(u t - %,2)(-,0)I

nat + AtP I (rl/211u(P+1)("~)fls

+ I~(P+2)(''T)I)dT

+ ~G(I + s)nat~xqmaxlsi~+1[llu(i)l[lq.,n].

Now we turn to the d i r e c t methods studied in Chapter I I and consider at f i r s t hyperbolic problems (6.1.6) without damping, i . e . , b z 0 and × = O. Obviously, the multistep multiderivative method (2.1.33) and the Nystr~m type procedure (2.4.9), (2.4.11) y i e l d computational devices for the semi-discrete problem (6.1.8) i f we subs t i t u t e A2 M-IK and cn = Cn" Theorem (2.1.34) and (2.1.39) apply in the same way as above. The result is stated =

-

in the following theorem:

154 (6.3.10) Theorem. (i) Let the hyperbolic problem (6.1.6) and the Galerkin subspace G C H satisfy Assumption (6.1.2); let the exact solution u satisfy u(i)(-,t) E H~ i :

0~...,2~, u(2~+2)(.,t) E L2(~) for t > O, and let the Ritz projection u be (p+2)-times continuously differentiable with respect to t. (ii) Let × : 0 and b ~ O. (iii) Let the method (2.1.33) be consistent of order p ~ 2~-I with stability region S. (iv) Let Sp(At2A 2) C R C S where R is closed in ~ and A 2 : - M-I/2KM -I/2. Then for n : klk+1,...,

I(~ - UA)(-)nl ~ < R n A t ' A t - I z ~ I ( ~

- UA)(')il

+

where

:
+ ~GnAtAxqmax2sis2Lillu(i)lIIq,,n].

If the method is strongly D-stable in R = [-s,

i(u-UA)(')nl

0] (and
~
-

+

Theorem (2.4.14) and (2.4.17) lead to the following result where A2 and ~ have the same meaning as in Theorem (6.3.10). (6.3.11) Theorem. Let the assumption of Theorem (6.3.10) be fulfilled for the Nystr~m type procedure (2.4.9). Then for n = 1,2,..., max{i(~ - UA,1)(.)nl , Ati(u t - UA,2)(-)nl}
+ i(~ t - UA,2)(-)01 ] + ~.

If the method is strongly D-stable in R = I-s, 0] then

I(~ - uA,1)(')nl s ~R[(I + nAtlAl)(~ - UA,I) (.)01 + natl(~ t - UA,2)(-)01 ] + ~, Ati(~ t -

UA,2)(.)nl~ AtKR[nAtIA21[(u

- UA,I)( • )01 + ( 1 + n A t l A [ ) l ( ~ t - UA,2)(.)01] +~.

In Section 2.3 and 2.4 we have considered linear multistep methods and different i a l systems of second order with damping. The scheme (2.2.3) yields for the semi-discrete problem (6.1.8) the computational device

155

(6.3.12) MP0(T)Vn - At2KPI(T)Vn - AtZ~=0Bi(Nn+i + xK)zi(T)V n : - 6t2pI(T)C~,. n : 0,I ..... c f . the corresponding scheme (6.1.11) f o r the d i f f e r e n t i a l (6.1.9).

system of f i r s t

order

I f the b i l i n e a r form b does not depend on time then we can w r i t e k

~(~) : Zi:0Si~i(~) and obtain (6.3.13) MP0(T)V n - At2KP1(T)Vn - At(N + xK)~(T)V n = - At2p1(?)C~, cf.

n = 0,I . . . . ,

( 2 . 3 . 3 ) . For the e r r o r estimation again the p a i r of a n a l y t i c equation and nume-

r i c a l procedure, (6.1.8) and ( 6 . 3 . 1 2 ) , is s l i g h t l y

modified i n t o (6.3.3) and

P0(T)MI/2vn + at2A2p1(T)M1/2Vn + AtZ~=06i(Bn+i + ×A2)ti(T)MI/2v n = _ At2p1(T)M1/2~n, with the notations ( 6 . 3 . 2 ) .

n = 0,I,...,

For b ( t ; u , v )

= 6 . ( u , v ) we have the case of orthogonal

damping and Theorem (2.3.13) and (2.3.15) y i e l d (6.3.14)

Theorem.

(i) Let assumption (i) of Theorem (6.3.10) be fulfilled for ~ : I.

(ii) Let X ~ 0 and b ( t ; u , v )

= 6"(ujv).

(iii) Let the method (6.3.13) be consistent of order p with the two-dimensional stability region S 2. Let Sp(At2A2)xSp(&t(-SI+×A2)) .M-I/2.

C R C S z where R is closed in $~ and A 2 = - At2M-I/2K •

Then f o r n = k , k + 1 , . . . ,

r nat [(~ - ua)(')n[ ~ E + KRngt[AtP ~ (lu(P+2)(',T)I + ¢(T))dT + ~GnAtAxqll[uttlllq., n where

: KRnAt'At

-1 k-1

Zi:01(u- %)('1iI

and

¢(T) : (8 + ×IA21)I~(P+I)(.,T)I. If the method is strongly D-stable in R = [-s, O] x [-r, O] (and K R > I) then

: KR[(I + nat(6 +

×I/21AI))~-~[(~

- %)(')ii

+

156 + nAt'At-1Z~---111(~- U A ) ( ' ) i

- (~-

%)(.1i_II]

If T(~) has exact degree k and

v u E cP+2(IR;IRm)

[~(T)U(t) - AtP1(T)u'(t) I _-
~(T) =- 0.

I f the b i l i n e a r form b in the hyperbolic model problem (6.1.6) does not depend on time but depends on the space variable x E IRr such that the matrix N =

]m

[ b ( t ~ ' ¢ v ) v,v:1 does no longer commute with the s t i f f n e s s matrix K then Theorem (2.2.10) y i e l d s the following r e s u l t where suppose that × = 0 because K is an i l l - c o n ditioned matrix.

(6.3.15) Theorem. (i) Let assumption (i) of Theorem (6.3.10) be fulfilled for ~ = I. (ii) Let × = 0 and let b be independent of time, b(t;u,v) : b(u,v). (iii) Let the method (6.3.13) be consistent of order p with stability region S. (iv) Let Sp(At2A 2) C R C S where R is closed in ~ and A2 = - M-1/2KM-1/2. (V) Let ~(~) be a polynomial of degree not greater than

k-1 and let Po(O) + n2p1(O)

0 v n 2 E R.

Then for n = k , k + 1 , . . . ,

l(u-uA)(.)nl

< KRnz~texp{K~6nAt}[At-Iz~--~I(~-

+ At p nit(l~(P+2) (.,T) I + f31~(P+1) ( ' , ~ ) l ) d T

UA)(')i I

+
F i n a l l y , i f b has the general time-dependent form stipulated in Section 6.1 then Theorem (2.2.18) applies. However, we assume again that x = 0 because in the other case the exponential m u l t i p l i c a t i o n c r e t i z a t i o n becomes sm~ll.

f a c t o r grows up i f the parameter Ax of the space dis-

(6.3.16) Theorem. Let assumption (i), (iii), and (iv) of Theorem (6.3.15) be fulfilled, let X = 0 in (6.1.6), and let the method (6.3.12) fulfil assumption (ii) and (iv) of Theorem (2.2.18) f o r R = [ - s ,

l(u - UA)(')nl

0] and B(t) = - M - I / 2 N ( t ) M - I / 2 .

--< ~ - ~ I ( ~

Then f o r n = k , k + 1 , . . . ,

- UZ~)(')il

k-2 ~ + KRnAtexp{<~6nAt} [AIZi=01(U - Uz~)(')il

+

At-1~k-1 Z.i=11(~ - Uz¢(')i

" (~-

UA)

(')

+ Atp nit(,~( P+Z)(.,T)[+ 61~(P+I)(.,~)[)dT + mGnz~taxqHuttlllq.,n].

i-I

I

Appendix

A . I . A u x i l i a r y Results on Algebraic Functions

Let ~(~,n) be a polynomial in ~ and n with real or complex c o e f f i c i e n t s , k ~(~,n) : ~i=0oi(n)~ I = k~ 0Pj(C)n j , Ok(n) ~ 0.

(A.I.1)

Then the roots ~i: n ÷ ~ i ( n ) , i = I , . . . ,

k, are regular

functions in C with exception

of the c r i t i c a l points. These are the s i n g u l a r i t i e s nO where ~k(n0) = 0 and the branching points of the algebraic function ~ defined by (A.I.2)

x(~(n),n)

m 0,

i . e . , the points n I in which some roots ~i(n I) of x(~,n I) coincide. Obviously, there e x i s t at most ~ poles and i f ~(~,n) is i r r e d u c i b l e , i . e . , i f there e x i s t no polynomials ~1(~,n) and ~2(~,~) such that ~(~,n) = ~1(~,n)~2(~,n), then there is only a f i n i t e number of branching points. See e.g. Ahlfors [53 ] and Behnke and Sommer [65 ]. n0 £ is a removable s i n g u l a r i t y or a branching point i f f

a l l branches ~i of ~ remain bounded

in a neighborhood of nO. Accordingly, the behavior of ~ at the point n = ~ is ruled by the f o l l o w i n g simple r e s u l t .

(A.I.3)

Lemma. Let the coefficients o i of

Then all roots of

~(~,n)

(A.I.1)

be polynomials of degree

are bounded in a neighborhood of

n

= ~

deg(~i).

iff

deg(o k) ~ max0si~k_Ideg(oi ).

Proof. The r a t i o n a l functions o i ( n ) / o k ( n ) are the elementary symmetric functions of ~1(n), . . . .

~k(n) by Vieta's Root Criterium hence they are bounded near n = ~ i f ~1(n),

. . . . Ok(n) are bounded. But lim l ~ i ( n ) / o k ( n ) I < ~ holds only i f deg(~ i ) ~ deg(o k) therefore the condition is necessary. On the other side, i f the condition is f u l f i l l e d and i f ~ = deg(o k) then the polynomial n~(C,n - I ) in C has polynomial c o e f f i c i e n t s

in

n and the leading c o e f f i c i e n t n~ok(n - I ) is unequal zero f o r n = 0. Thus the roots Ci(n - I ) of x(C,n - I ) cannot possibly have a genuine s i n g u l a r i t y in n = 0. This proves the s u f f i c i e n c y . I f some roots ci(n) of ( A . I . 1 ) , say r, coalesce in the point n I which is not a pole then these branches of the algebraic function ~ are regular

in a neighborhood of n I

158

or they behave l o c a l l y l i k e (n - n l ) P / q where p,q E IN and I s q ~ r ~ k. This is a c l a s s i c a l r e s u l t of a n a l y t i c f u n c t i o n theory which reads more e x a c t l y as f o l l o w s ( c f . e.g. Ahlfors [53 ] , H i l l e [62 ] , and Behnke and Sommer [65 ] ) . ( A . I . 4 ) Theorem. If Sk(nl) m 0 and if r roots of ~(¢,n) coincide in n l , say ¢ i ( n i ) = ~2(n I) = ... = ~r(nl) , then there exists a n e i g h b o r h o o d ~ o f

zero such that - possibly

after some permutation -

(A.I.5)

~ (n) = ~ i ( n I ) + 7 ® ' [e2~i~/q( n - n l ) I / q ] ~, .u=p¢p

where p,q E IN have no common factor,

n - n I E jI~', u : 0 . . . . . q - l ,

I ~ q ~ r ~ k, and (n - nl )I/q is a fixed branch

of Cq - (n - n I) : O.

I f a simply connected d o m a i n ~ is given which contains no branching points or singularities

in i t s i n t e r i o r

then t h i s r e p r e s e n t a t i o n shows t h a t a l l roots of ~(¢,n) can

be numbered in such a way t h a t they are continuous i n 4 . Let

s* : {n E ~,

Ici(~) I ~ I,

be the general ' s t a b i l i t y

i

=

I .....

k}

region' o f ~ ( ¢ , n ) . We s h a l l show in t h i s section t h a t f o r

n I E aS* the values of p, q, r , and @p associated with unimodular roots in n I by ( A . I . 5 ) determine the shape of S* near n l . But at f i r s t

we derive some a u x i l i a r y r e s u l t s f o r

the f o l l o w i n g case: Let ~(~,q2) be the characteristic polynomial of a convergent m u l t i s t e p m u l t i d e r i v a t i v e method f o r d i f f e r e n t i a l

systems of second order then there e x i s t at l e a s t two

r o o t s , ¢I(n) and ¢2(n), which coalesce to a double roots of modulus one f o r n = 0. Without loss of g e n e r a l i t y l e t a l l pairs with t h i s property be c j ,

¢j+I' j = 1,3,...,

k . - 1 , k. < k. (As the method is supposed to be convergent ~(¢,0) = p0(¢) can have only simple or double roots of modulus one.) Then ( A . I . 5 ) y i e l d s (A.I.6)

cj,j+1(n)

= Cj(0) ± ~jn +

d(Inl2),

q ÷0,

where (A.I.7)

A2 Xj = - 2PI(¢j(O))/P~'(~j(O))

^2 and hence ×j m 0 i f the method is l i n e a r . Recall t h a t the s t a b i l i t y

region S is de-

f i n e d here with respect to n2 ( D e f i n i t i o n ( 2 . 1 . 1 4 ) ) . ^2 ( A . I . 8 ) Lemma. Let I - s , O] C S, 0 < s ~ ~, and let Xj ~ 0 for j = 1 , 3 , . . . , k , - I .

Then

159 -sin

max1
2

<0.

Proof. Because of ( A . I . 5 ) and ( A . I . 6 ) we can w r i t e (A.I.9)

Cj,j+1(n)

= ~j(0) ± ~jn(1 + ~j(±n))

where ~j(n) = ( c j ( n ) - C j ( 0 ) ) / ~ j n

- I is bounded in [ - s , 0] because I c j ( n ) l

~ I in t h i s

i n t e r v a l , and Sj(0) = 0. Accordingly, the assertion follows with

<s = maxj=1,3,... ,k,-1 sup_ssn2s0

I×~jI11

+ mj(±n)l < ~.

(A.I.10) Lemma. Let the assumption of Lemma ( A . I . 8 ) be fulfilled but let all roots <

~i{n) of ~(~,n 2) be simple for -s ~ n 2 : 0 with exception of n (i)

m a x l s i s k l ~ i ( n I) - ~i(n2)I ~ % I n I - n21,

(i<)

maxj=1, 3 . . . . . k,_1{ICj(n)

2

= 0. Then

-s ~ n2v :< 0, v = 1,2, s < ~.

- Cj+1(n)I - I } ~ KsmaX{1,1nl - I }, -s _-< n2 < 0, s < ~.

Proof. ( i ) I f ~i(0) is a simple root of ~(~,0) = p0(~) then I ~ ( n ) l finite

interval

is bounded in the

I - s , 0] and the assertion is an immediate consequence of the Mean Value

Theorem. I f ~j(0) is a double root of ~(~,0) then i t can be w r i t t e n in the form ( A . I . 9 ) where now the function @j is regular

on the e n t i r e i n t e r v a l I - s , 0], i . e . ,

domain containing I - s , 0], because ~j(n) is regular

in an open

in I - s , 0) by assumption. Hence

the assertion follows again by the Mean Value Theorem applied to ( A . I . 9 ) (ii)

By ( A . I . 6 ) there is a a > 0 and a ~I > 0 such that J~j(n) - ~j+1(n)J ~ mlJnJ,

s n2 ~ 0, j = 1 , 3 , . . . , k , - I ,

and in [ - s , -8] a l l roots are bounded away from each other by assumption hence <s = minj:1,3 . . . . . k,-Imin_s~n2~_aI~j(n) - ~j+1 (n)l > 0. Both bounds together y i e l d l¢j(n) - ¢j+1(n)l m min{Ks,<1}min{1,1nl},

- s ~ n2 ~ 0, j : 1,3 . . . . . k,-1.

Next, we have to prove a modification of Lemma ( A . I . 8 ) f o r l i n e a r multistep

160 methods and d i f f e r e n t i a l systems with orthogonal damping. In this case the character i s t i c polynomial has the form (A.1.11)

=(~,n2,~)

= pO(~) + q 2 p l ( ~ ) + ~ ( ~ ) .

The method defined by this polynomial is consistent i f f (A.I.12)

(2.3.8) is f u l f i l l e d , i . e . ,

I

tt

P0(1) = p~(1) = P0 (I) + 2Pi(I) = ~(I) = • (I) - PI(I) = 0,

and i f 0 E S for u = 0 then p~'(1) = - 2Pi(I) = 0. Recall that the s t a b i l i t y region S= is defined here with respect to (n2,~) (Definition (2.3.12)). By a s l i g h t modification of Theorem(A.1.4) the principal roots ~1(n,~) and ~2(n,~) are now regular functions of n and ~ in a neighborhood of (n,u) = (0,0) and we obtain instead of (A.I.9)

~1,2(n,u)

= I + (±~n + m~)(1 + e ( ± n , u ) )

^2 where x = - P 1 ( 1 ) / p ~ ' ( 1 )

as in ( A . I . 7 )

and m = I / T ' ( I ) .

Again ~ denotes a f u n c t i o n

which is r e g u l a r in a neighborhood o f (n,u) = (0,0) w i t h ~(0,0) bounded in the s t a b i l i t y

= 0 and which is

region S2. Hence we can s t a t e :

(A.1.13) Lemma. Zf (A.1.12) holds with P1(1) = 0 then the principal roots of a method defined by

(A.I.11) satisfy I1 - ~1.2(n.~)1

~ ~r.s(Inl

÷ I,I)

v (n2,u) E I-S, 0 ] x [ - r , 0] C S2.

After these estimations we now assume again that ~(~,n) is an irreducible but otherwise

arbitrary

polynomial (A.I.1) of degree k in ~ and degree ~ in n and consider

the behavior of a root ~i(n) in a neighborhood of a point q* E C where i~i(q*) [ = I . For s i m p l i c i t y we set n* = 0 and write ~(n) = ~ i ( n ) , ~(0) = ~*, i . e . l~*I = I . Then Theorem (A.I.4) yields in a neighborhood of n = 0 (A.I.14)

~(n) = ~*(I + ×n p/q + ~ ( n S ) ) ,

and a simple c a l c u l a t i o n (A.1.15)

I~(n)l

n + O, s > p / q ,

shows t h a t

= 1 + Re(×n p/q) + ~ ( [ n l m i n { 2 p / q ' s } ) ,

n ÷ O.

In t h i s equation we i n s e r t (A.I.16)

xn p/q = pe i e ,

0 =< e < 2~, p > O,

161

and obtain Ic(~)l

= I ÷ ocose

÷

Hence, i f ~ > 0 is s u f f i c i e n t l y Ic(~)l

>

l

if

Ioi

C(pmin{2,s*}),

n÷0,

s*>1.

small then there e x i s t s a pE > 0 such t h a t

~ ~ - ~,

Ic(n)i

< I if

Io - ~I ~ ~ - ~, o < p < p,

But from ( A . I . 1 6 ) we f i n d t h a t n : (pl×)qlPei°q/Pe 2~ijq/p,

j = 0,I,...,p-I,

and, a c c o r d i n g l y , argn : ~ ( - argx + O + 2 x j ) . We w r i t e c f o r ~q/pandassemble the r e s u l t in the f o l l o w i n g lemma which is stated at once f o r general n* E { ; c f . also Jeltsch [77 ] and Wanner, H a i r e r , and Norsett [78a]. ( A . I . 1 7 ) Lemma. Let (A.I.18)

¢(n) = ~*(I + x(n - n*) p/q + ~((n - n * ) S ) ) ,

n ÷ n*, s > p/q,

be a root of ~(~,n) with I~*l = I, x ~ O, and p,q E IN having no co,~non factor. Then there exists for each small ~ > 0 a PE > 0 and branches ~p(q), ~v(n) of (A.I.18) such that [~p(q)l > I for (A.I.19) and [%(~)I (A.I.20)

n : n* + pe i ° ,

0 < p < PE' Io - q(2j~p- ar~x) I : < ~ p -

~, j = 0,I . . . . . p - l ,

< I hr n : n* + oe i0 m 0 < p < P E '

le

_ ~rr<4~2j+ij~

P

-

ar~x)i I ~ .

_ ~,

The number q-1 is c a l l e d the ramification index of the roots ( A . I . 1 8 ) .

j

:

0,1, . . . .

p-1.

I f q > 2 then

the angular domains ( A . I . 1 9 ) overlap each o t h e r . For q = 2 only the h a l f - r a y s q = i3" + pe 1 8 '

@ =

( 4 j + I ) ~ - 2ar~× p

j = 0,1 . . . . . p - l ,

are not contained a s y m p t o t i c a l l y in the set defined by ( A . I . 1 9 ) , and f o r q = I the angular domains ( A . I . 1 9 ) and ( A . I . 2 0 ) a l t e r n a t e f o r increasing e. Thus we can s t a t e the f o l l o w i n g c o r o l l a r y where [x] denotes the l a r g e s t i n t e g e r not g r e a t e r than x.

162 (A.I.21) Corollary. Let q* E @S*, let ~ i ( n ) , i = 1 , . . . , k + , be the roots of ~(~,n) with I~i(n*) I = I, and let qi - I be the ramification index of ~i in q* and ×i the growth parameter defined by (A.I.18). (i) If q i > 2 for some i then there exists a d i s k ~ w i t h

~)\{n*}

center q* such that

c C \ S*.

(ii) If qi = 2 for some i then there exists no angular domain

(~,6,p) with a

and

> 0

= {n E ~, 0 < In

p > 0

(iii) For each ~,

<

p, I~

-

arg(n

-

there exists an angular domain

qi : I, I ~ Pi ~ min{~,[~/2a]},

(A.I.22)

n*l

n*)l

~ ~}

~.(~,6,p) C S*.

such that

0 < ~ ~ a,

-

.~I'(~ -

~,6,PE) C S*

iff

and

i = 1,...,k,.

In -pi 6- argxil s ~ - pi ~,

Proof. We have only to v e r i f y the t h i r d assertion. I s p s {~/2~] and (A.I.22) f o l l o w immediately from ( A . I . 2 0 ) . In order to show p ~ ~ we observe that C(q) = C*(1 + ×np + ~(nP+l)) holds i f f

n ~ O,

the algebraic function n(c) defined by ~(C,n(C)) = 0 has p branches ni(~)

of the form r~ -

~*~I/p

q i ( ~ ) = ,x--CTr-,

+

C((~

-

~.)~),

~

-

~.

+ 0, ~ > I / p ,

according to the chosen branch of n P - ((~ - C*)/×~*) = O. But n(~) cannot have more than ~ branches hence I ~ p ~ c. (A.1.21)(iii)

is i l l u s t r a t e d by Example ( A . 4 . 7 ) .

In the s i t u a t i o n of ( A . 1 . 2 1 ) ( i i )

always two segments of the boundary curve ~S* of S* emanating from n* are tangent to each other in n* (or coincide near n*). This is i l l u s t r a t e d in a p a r t i c u l a r way by the examples in Appendix A.6. The general r e l a t i o n between p, q, and the m u l t i p l i c i t y somewhat complicated because the i r r e d u c i b i l i t y irreducibility

r of ~* in (A.I.18) is

of ~(~,q) does not imply the paiz~aise

of the polynomials p j ( ~ ) , j = I , . . . , c ,

with exception of ~ = I , i . e . ,

of l i n e a r multistep methods. However, i f r = I , as always in single step methods, then obviously q = I . The f o l l o w i n g two cases concern single step methods and l i n e a r m u l t i step methods and do not involve the Puiseux diagram. Buts n a t u r a l l y , they are contained in the general r e s u l t , Lemma ( A . I . 4 0 ) .

163

(A.I.23)

~*

C o r o l l a r £ . Let

be a simple root of

~(5,n*).

Then, in ( A . I . 1 8 ) ,

q = I and

p : min{u E]N, ~nv~ ,n ) = 0}.

Proof. Here we o b t a i n by a T a y l o r expansion × = ~*-tc(P)(n*)/p!

and a d i f f e r e n t i a t i o n ~'(P)

~* = ~ ( ~ * ) ,

of ~(~(n),q)

w i t h respect to n y i e l d s o m i t t i n g the argument

+ ~P~ = 0

"~ if

,

an p

[BxV/@nV](~*,n *) = O, ~ = 0,1 . . . . . p-1.

(A.I.24)

Cor o l l a r £ .

~*

Let

be an r-fold root of

~(~,n*).

Then, in ( A . ~ . 1 8 ) ,

q = r and

p = I if

(A.I.25)

~-~n£~ ,n ) = O.

Proof. Let w i t h o u t loss of g e n e r a l i t y ~1(n*) = . . .

= ~r(n*)

= ~*

and l e t ~ be a f i x e d branch of ~q = n - q* in a neighborhood o f ~ = 0 w i t h e x c e p t i o n o f an a r b i t r a r y (A.I.4)

but f i x e d h a l f - l i n e

I s q s r branches ~ j ( n ) ,

w i t h endpoint in ~ = O. Then t h e r e are by Theorem j = I . . . . . q, which can be w r i t t e n

in t h i s s e t , ~ r ,

as r e g u l a r f u n c t i o n s ~j in ~, ~j(~)

= Z~=O@ ( e 2 x i j / q ~ ) u,

j = I . . . . . q,

where ¢0 = ~I (n*) = ~*" For the computation o f 41 = ~ i ( 0 ) we w r i t e ~(~,~)

=

~(~,~q

then the values ~ j ( 5 ) , an r - f o l d

+

q*)

~ ~q ~ _ ( )~qJ = Zj=OPj(~)( + n*) j = Zj=OPj

j = I . . . . . q, are roots of ~(~,~) f o r 5 E,~t'. Because ~1(n *) is

r o o t o f =(~,q*)

on the one side and because ~(~,~) is a f u n c t i o n o f ~q on

the o t h e r side we o b t a i n ~q (A

261

+

a

(0),01

:

@q . +

: 0

164

But

aq

aq

[~'~](~I(0)'0):

[ = O, I s q < r ,

[T~](~1(n*)'n*)

I m o, q = r

and aq ~ I Z * a [~-~-~](~i(0),0) = q]~1(~1(n*)) = q. Zj=oPj(¢1(n ) ) i n *j-1 = [-~n ] ( ~ 1 ( ~ * ) , q * ) q ! hence q < r leads to a contradiction by ( A . I . 2 6 ) . Therefore we have q = r and then (A.I.26) y i e l d s ~i(0) m 0 which implies p = I in the expansion ( A . I . 1 8 ) .

Now we observe that in l i n e a r multistep methods the i r r e d u c i b i l i t y

of ~(~,n) implies

[B~/an](~1(n*),n*) = p1(~1(n*)) m 0 and obviously the same conclusions can be drawn f o r c h a r a c t e r i s t i c polynomials x ( { , q 2 ) of l i n e a r multistep methods f o r d i f f e r e n t i a l systems of second order replacing (A.I.25) with ( A . I . 2 5 ) * ,-]T-~-C(~1(n*),n an .2) a(n )

= p (~

I

I

(~*))

= 0

because i t doesn't matter whether we w r i t e n2 instead of n. Thus a combination of Corollary

(A.I.21) and C o r o l l a r y (A.I.24) y i e l d s f o r Zimea~ m u l t i s t e p methods with i r r e d u -

c i b l e c h a r a c t e r i s t i c polynomial: ( i ) I f S is the s t a b i l i t y

region of a method f o r d i f f e r e n t i a l

systems of f i r s t

order

( c f . D e f i n i t i o n ( 1 . 2 . 7 ) ) then S \ S consists of the points n* g C where ~(~,n) has double unimodular roots. In a point n* E S \ S

both segments of @S emanating from n*

are tangent to each other by Corollary ( A . 1 . 2 1 ) ( i i ) . (ii)

I f S is the s t a b i l i t y

region of a method f o r d i f f e r e n t i a l

systems of second order

( c f . D e f i n i t i o n (2.1.14)) then S = S. Therefore, the constants mR in Theorem (2.1.34) and (2.2.10) and Ks in Theorem (2.1.39) depend only on the data of the method. I f [ - s , O] is a p e r i o d i c i t y i n t e r v a l then i t is a subset of aS and double unimodular roots can only l i e in the endpoints of the i n t e r v a l . Hence the Frobenius matrix of the method is diagonable in the i n t e r i o r of the p e r i o d i c i t y i n t e r v a l , c f . assumption ( i i i )

of

Theorem ( 3 . 3 . 3 ) . Following Hensel and Landsberg [02] or H i l l e [62] l e t now (A.I.27) oi(n) = ~;=niajin3 ' ~n i , i

~ O, ~Ok = 0 ( c f .

(1.1.5))

i = O, . . . . k,

and w r i t e instead of ( A . I . 1 8 ) near n = 0 E

(A.I.28) ~(n) = Z~=oX n u, x 0 = O,

~0 < ~I < . . . .

For the computation of ~0 we s u b s t i t u t e (A.I.27) and (A.I.28) i n t o ~(C(q),n) and as-

165

semble equal powers o f n then we o b t a i n

( A . I . 2 9 ) ~ ( C ( n ) , n ) = c0~ If

(A.I.28)

Y0

+ Cln

YI

+ ....

Y0 < YI < . . . .

represents some r o o t o f ~(~,n) near n = 0 then the c o e f f i c i e n t s

disappear i d e n t i c a l l y .

c v must

Assembling on the o t h e r side a l l components whose exponent con-

t a i n s E0 we f i n d

:(~(~),n) (A.I.30)

no

n1+~0

= ~n0,0n

+ ~n1'1×0n

2 n2+2~0 + ~n2'2×0n + ...

Y0 Con is obtained from t h i s equation by c o l l e c t i n g

all

k nk+kC0 + a n k ,kX0 n + ....

terms in the sum w i t h the same

minimal exponent, say n g + g~0'

n g + gc 0 = min0si~k{n i + i~0} ,

g EIN v { 0 } .

I f g is the only number w i t h t h i s p r o p e r t y then Y0 COn

q n_+gcn =

an

g

.x~n~u

y

u

and thus c O does not disappear hence c 0 must be chosen in a way t h a t a t l e a s t two numb e r s , say g and h, s a t i s f y

¥0 = ng + g~0 = nh + hE0 = min0~i~k{ni_ + i ~ 0 } '

g = h.

Then we have

~0 = ( n g and, by t h i s way,

- n h ) / ( h - g) a finite

geometrical version o f t h i s

number o f a d m i s s i b l e values f o r ~0 can be d e r i v e d . A more idea is the ~ i s e ~ r diagram:

Let in a ( x , y ) - p l a n e

zi = (i,n i)

z~1 = (0,n i +

t

i - t g ( x - ¢))

i

z~ is the i n t e r s e c t i o n o f the o r d i n a t e a x i s and a s t r a i g h t I

angle ~ with r e s p e c t to the p o s i t i v e real l i n e in ( A . I . 3 0 )

t h e r e f o r e we have to compute a l l

through a t l e a s t two p o i n t s z. such t h a t a l l 1

C ~v)

=

tg(~

-

~u), v

=

•

=

0j,,,jk,

l i n e through z i w i t h the

The o r d i n a t e s of z~ are the exponents 1

chords C w i t h angle ~ , v = 1 , . . . , s , o t h e r p o i n t s l i e on o r above C . Then

1 . . . . , s , are the a d m i s s i b l e exponents,

Let us now assume t h a t Cu is such a lower boundary chord of the p o i n t set {z i }

166 and that i 0 < i I < . . . < i V,

{Zio, zi1,...,z iV} C C , then ¥0

• = n~o

+

i OsO =

nl° I

+

ilc 0

=

...

=

n.~

+

i

s0

whereas a l l other points z i l i e above but not on C , Accordingly, the c o e f f i c i e n t c O belonging to ~0 in (A.I.29) is by (A.I.30)

(A.I.31)

i0 iI i Co = a n i o ' i o ×0 + ~ n l z , i i X o + " ' " + ~ni ' i ~ xO

=

~(x0),

i . e . , ×0 ~ 0 must be a root of the polynomial of degree iv - lo," il-i 0 (A.I.32)

P~(X) = an

io

,i

By t h i s way the i n i t i a l

0

+ an. , i l X

+ "'" + an

~

i

iu-i 0 ,i~ X '

terms in (A.I.28) of a l l roots of ~(~,n) can be found.

A f t e r these p r e l i m i n a r i e s we turn to our actual problem namely to compute p and ×p* in (A.I.33)

~(n) = g* + ~ :~=p x*n u ~ g* + ¢(n)

where x(~*,O) = 0 and xu* ~ O. For t h i s we have to modify s l i g h t l y

the above device, A

s u b s t i t u t i o n of (A.I.33) into x(~(n),n) y i e l d s (A.1,34)

k

"

"

'-'

"

x(~* + @(n),n) : Z i : 0 ~ i ( n ) Z ] : o ( ] ) ~ . I J@(n)J k k " k , = Zi=O[~m=i~m(n)(?)~*m](,(n)/~*) 1 ~ Zi=O~i(n)(~(n)/~*) i

where oT(n)

k ~ j (m) ~ k m ,m j = Zm=i(Zj=0ajm n ) ~,m = Zj=o[Zm=iajm(i)~ ]n .

But ~i ~k ( ~ ) i ' m-i ~-'-TPlj (~) : -m=i-1 "ajmC

hence (A.I.35)

~#(~)

:

~L t (i)(~.)~l~.')nJ &j=o~Pj

~

Now we observe that •

o

Pj(i) (C*) - ~T_L~._~(c.,0)/j !

~L a .., ~ J J=n i j1

L"

i = Oi...jk.

167

t h e r e f o r e we o b t a i n f o r the c r u c i a l

(A.I.36)

n i = m i n { j EIN,

With r e s p e c t t o ( A . I . 3 3 ) ,

i = 0,...,k.

[~1+Jxla~1BnJ]((*,0) ~ 0},

the Puiseux diagram suggests the f o l l o w i n g

(A.I.37)

i

(A.I.38)

Pv = - ( n i v - n i v _ 1 ) / ( i ~ - i v _ l ) ,

v

=

values n i

max{arg m i n { ( n i - n

lv_ I

)/(i

- i _i),

i > i _I }},

notations:

i 0 = 0,

and

(A.I.39)

I v : { j EIN, ( n j - n i

w-1

)/(j

say f o r v = I . . . . , v * . Then C o r o l l a r y (A.I.34)

and ( A . I . 3 5 )

yield

(A.I,40)

Lemma. For each ~,

{n E ~, 0 < Inl

(cf.

- iv_ I ) = - p } , (A.1.21)(iii)

also Jeltsch

[76a, 77 ] ) :

there exists

0 < ~ ~ mj

< P , i n - argnl

a

p

> 0

such that

< e - ~} C S*

C

i f f f o r each unimodular root ~* of ~ ( ~ , 0 )

and the Puiseux diagram a p p l i e d t o

= p0(~) the associated i ,

m > O,

Pv' and I v satisfy

for u = 1 , . . . , v * (i)

(ii)

I ~ p~ ~ m i n { ~ , [ ~ / 2 m ] } and pv EIN, I(I

- p )~ - arg×i

i+n.

ziiL Naturally,

S ~ - p ~ for all roots × ~ 0 of the polynomial

"

ni

n~]~ *i ×

these necessary and s u f f i c i e n t

edge o f angle 2m i n n = 0 can be c a r r i e d culty.

The n e x t r e s u l t

also Jeltsch (A.I.41)

i

algebraic c o n d i t i o n s

f o r S* having an

over t o general 0 ~ n* £ @S* w i t h o u t d i f f i -

concerns the case o f a d i s k i n s t e a d o f an a n g u l a r domain; c f .

[77 ] .

Lemma. There exists a p > 0 such that

= {hE

¢, IN+ pl

s p} c S*

iffevery root ~ j ( n ) of ~ ( ~ , n ) with I ~ j ( 0 ) I (A.I.42)

•

~j(n)

: ~*(I

+ Xn + ~ ( n S ) ) ,

I~*I

= 1 has near n = 0 the form : I,

x > 0, s ~ 2.

168 Proof.

If

yields

by ( A . I . 1 5 )

(A.I.42)

holds then a s u b s t i t u t i o n

[~j(n)I

o f n = p(e le

: 1 - ×p(1 - cose) + ~ ( [ p ( t

-

I),

p > 0, 0 ~ e < 2~,

- cose)l/2Is),

X > O,

because Inl 2 = 2p2(I - c o s e ) . This proves t h a t ciently

s m a l l . On the o t h e r s i d e ,

and, by C o r o l l a r y

(A.1.21)(iii),

let ~p ~j(n)

~ C S* f o r s ~ 2 i f p > 0 i s s u f f i p C S* then ~S* has in n = 0 an 'edge' o f angle

must be a branch o f ~(n) having near n = 0

the form ~(n)

More e x a c t l y ,

= ~*(1

let

+ xn + ~ ( n s ) ) ,

x>O,s>l.

s = p/q > I where p CIN and I < q E IN have no common f a c t o r ,

q - p l a n e be c u t along the p o s i t i v e

real axis,

and l e t

t h a t branch o f ~q - n = 0 w i t h ( - I ) I / q = e i ~ / q . ration

l e t the

n I / q be h e n c e f o r t h i n t h i s

Then we have a f t e r

a suitable

proof

renume-

by Theorem ( A . I . 4 ) ~j(n)

or, writing

~

: ( * ( 1 + xn + ~=q+1¢

= × e

ter-2~ij/qnl/q'uj ) ,

j = 1,...,q,

' ×u = I~pl ~ O, and

@(J,u,m) = ~u + (2~j + m ) u / q , ~j(n)

= ~*(I

+ ×n + ~ u = q + 1 × u l n [ U / q e i ~ ( J ' ~ ' a r g n ) ) ,

j = 1,...,q,

and hence

Icj(n)l 2 Into this

=

I

+ 2×Ren+

×21n]2+

e q u a t i o n we s u b s t i t u t e Ren : -

2Z~qq+iX~InlU/qcos(@(j,u,argn))

+ ~(InlS*),

s* > 2.

again n = p(e l e - I ) and observe t h a t f o r these c i r c l e s

In12/2p

then we o b t a i n

Icj(n)[ 2

~r2q-1 ~ i n l U / q c o s ( @ ( j , u , a r g n ) ) = I + LL~=q+I× + in12(×2 - ×p - I + 2×2qCOS(~2q + 2 a r g n ) ) + ~ ( I n l s* ) ,

S~

> 2,

Obviously,

f o r every f i x e d p > 0 the terms in the sum dominate here the o t h e r terms

in Inl f o r

o ÷ 0. Thus, i f

,~p C S* f o r some p > 0 then we have e i t h e r

xlj = 0, ~ =

169

q + 1 , . . . , 2 q - 1 , or i f ×~ (A.1.43)

O, ~

q + 1 , . . . , q + v - 1 , and ×q+~ > 0 then e + O, j = 1 , . . . , q .

cos(@(j,q+~,argn)) < O,

But argn ÷ ~/2 i f

e ÷ O+ and argn ÷ 3~/2 i f

0 ÷ 0

hence, w r i t i n g q+~

~1(¢,j,v)

= ¢ + 2~j~ + ~ . q

, ~2(~,j,v)

we obtain from ( A . I . 4 3 ) t h a t c o s ( ~ 1 ( ~ , j , ~ ) ) ~ O

= ¢ + 2xj~+ ~

and c o s ( ~ 2 ( ~ , j , v ) ) ~ O .

order to prove t h a t .~p c S* f o r some p > 0 implies ×~ to prove t h a t f o r a l l v E { I , . . . , q - I }

q+v q '

O, u

Accordingly, in

q + 1 , . . . , 2 q - 1 , we have

and a l l ~ £ [0, 2~) there e x i s t s a j E { 1 , . . . , q }

such t h a t (A.I.44) cos(~1(~,j,v))

> 0

or (A.I.45) cos(~2(¢,j,v))

> O.

For t h i s l e t u be given and l e t ~, ~ s a t i s f y ~/~ = v/q but having no common d i v i s o r . Then apparently I ~ ~ < q ~ q and I < ~. We consider two cases: (i)

I f ~ > 2 then ~ 1 ( t , j , v )

differ

modulo 2~ occur f o r j = 1 , . . . , q .

by m u l t i p l e s of 2~/q and a l l d i f f e r e n t m u l t i p l e s

Hence there is at l e a s t one j E { 1 , . . . , q }

such t h a t

( A . I . 4 4 ) is f u l f i l l e d . (ii)

I f ~ : 2 then % : I and (q+v)/q = I + (~/~) = 3/2. In t h i s case ~ 1 ( ~ , j , v ) and

~ 2 ( ~ , j , v ) have only d i f f e r e n t values f o r j = 1,2 and we obtain modulo 2~

Wl(~'l,v)

= e - {

~2(~,1,v) = ~ -

3~

+~r"

,

~1(~, 2,v) = ~

,

~2(~,,2,v) = ~ + ~ •

Hence there e x i s t s also here a j such t h a t ( A . I . 4 4 ) or ( A . I . 4 5 )

is f u l f i l l e d .

We conclude t h i s section with an algebraic c h a r a c t e r i z a t i o n of the ' d i s k s t a b i l i t y ' near ~ : 0 described in the l a s t lemma. For t h i s we need some f u r t h e r aids and define ~ ~

= {0 ~ z E C, l a r g z

- arg~ I < ~ / 2 } ,

denoting the closed h u l l of ~ .

170

(A.I.46)

Theorem, ( L u c a s . )

roots of

p'(z)

of

p(z)

n z ~ be a non-constant polynomial then all Let p ( z ) = ZV=oav

lie in the convex hull ~

are not collinear, no root of

multiple root of

p(z).

of the set of the roots of

pt(z)

lies on the boundary o f ~

If the roots

unless it is a

p(z).

Proof. See Marden [66 , p. 22]. (A.I.47) Lemma. (Jeltsch[77 ] . ) Let p(z) = ~v=0avz n v satisfy for 0 ~ k < m - I < n ak ~ 0, av = 0, v = k + 1,...,m - I , am = 0. Then there exists for every 0 = ~ E ~ a root Z* E ~

{0} of

p(z)

and Z* E ~

for

k<m-2. Proof. Assume that a l l roots of p(z) l i e in £ \ ( ~ . ~ \ { 0 } )

then by Theorem (A.I.46)

a l l roots of p(k)(z) l i e in this set. But then a l l roots of q(z) = zn-kp(I/z) = bn_kzn-k + b zn-m + + b0 n-m "'" l i e in { \ ( ~ I / X \ { 0 } ) obtain by assumption

and hence a l l roots of q(n-m)(z) l i e in this l a t t e r set. We

q(n-m)(z) = ci zm-k + c 0,

c o = 0, c I = 0, m - k ~ 2,

thus at least one root of q(n-m)(z) l i e s outside ~(~i/~\

{0)) = {z E C, largz - arg(I/X) I > ~/2} u {0}

which is a contradiction. In the same way i t is shown that a root of p(z) lies in ~.x f o r k < m - 2. Now we introduce the polynomials (A.I.48)

u (s) Qij(~,×) = Zs=tPi_s(~)~s(

)xS-J/s!,

t = max{i - ~, j } , u = min{k, i } ,

and deduce the algebraic version of the disk Lemma (A.I.41) in two steps: (A.I.49) Lemma. (Jeltsch [76a].) L e t ~* be a u n i m o d u l a r r o o t o f ~(~,0) = p0(~) w i t h multiplicity r . Then every root ~i(n) with ~i(0) = ~* has near n = 0 the form (A.I.50)

~(n) = ~ * ( I

+ xn + ~ ( n S ) ) ,

x > O, s > I ,

171

iff (i)

root of pj(~),

~* i s a ( r - j ) - f o l d

j = I . . . . . r , and p j ( ~ * )

m O, j : r + I . . . . .

£,

(ii) all roots of Qro(~*,×) are real and positive.

Proof. We have to reconsider the Puiseux diagram for the polynomial (A.I.34),

~(~(n),n) w i t h the r - f o l d

~k ~9~ p(i)(~.)~Ti nJ(¢(n)/~*) i

= ~(C* + ~ ( n ) , q )

= ~i=O~j=O j

r o o t ~(0) = 0 f o r n = O. I f

(i)

h o l d s then ( A . I . 3 6 )

yields

ni = r - i

and we o b t a i n

two l o w e r boundary chords i n t h e P u i s e u x d i a g r a m as t h e f o l l o w i n g

describes for

r = 4 and k = 7.

(A.I.51)

figure

Figure:

r x x x x I

x

x

r

x k

The chord with the ascent rate -p = -I belongs to the root @(0) = 0 and the chord with the ascend rate zero belongs to the k - r nonzero roots of ~(¢(n),n). In the f i r s t case the growth parameters x are the roots of the polynomial (A.I.32), •

.i

.i

which are by assumption ( i i ) real and positive. Therefore (A.I.50) holds. On the other side, i f (A.I.50) holds for a l l roots ~i(q) with ~i(O) = ~* then the Puiseux diagram must necessarily have exactly the two lower boundary chords described in Figure (A.I.51) and ni ~ r - i , i = I , . . . . r - I. But as a l l roots × of the polynomial (A.I.52) are real and positive by assumption, this polynomial cannot have a zero coefficent by Lemma (i) (A.I.47). This implies that Pr_i(~ *) ~ 0, i . e . , Pi( r - i ) ( ~ , )

~ 0 ' i = 0,1 " ' " , r .

Hence

( i ) and ( i i ) are necessary conditions for the expansion (A.I.50). (A.I.53) Lemma. (Jeltsch [79 ].) There exists a p = {neC,

p > 0

such that

In + p l - < _ p } c S *

iff the following three conditions are fulfilled for unimodular roots ~* of (i)

rj = r0 - j, j = O,...,ro,

pj(~)

and p j ( ~ * )

where

rj

= 0, j = r 0 + I . . . . ,~.

(ii) all roots of Q r o o ( ~ * , × ) are real and positive. (iii) If X* is a root of

Qroo(~*,x)

~(~,0):

denotes the multiplicity of ~* as a root of

of multiplicity K ~ 2 then

172

Qij(~*,×*) : 0 for all integers i , j

w i t h r 0 < i < - j + ~ + r0, j = 0 , . . . , K

Proof. N o t i c e t h a t by the l a s t r e s u l t ~(n) = ~*(I + x*q iff

the polynomial ( A . I . 5 2 )

disk lemma ( A . I . 4 1 )

- 2.

I s ~ ~ r 0 roots ~ i ( q ) have the form

+ ~(nl/~)), has a r o o t x* o f m u l t i p l i c i t y

~. Hence, because o f the

and the above lemma we have o n l y to show t h a t the t h i r d

is necessary and s u f f i c i e n t

f o r s ~ 2 in ( A . I . 5 0 ) .

condition

For t h i s we s u b s t i t u t e

~(n) = ~*(I + ~ [ x * + ~ ( ~ ) ] ) i n t o ~(¢,q) = 0 and o b t a i n

~(~(~) ,~) = Z~=0nJPj(C*(I~ + n [ x * + ~(n)])) (A.I.54) ~k ~L+k- . . .~ i = Lj=0Li=0~ij~C ,x )n ~(n) j As r j = r 0 - j we have f o r a l l Qij(~*,×*)

j

= 0,

i < r0,

and we f i n d e a s i l y t h a t f o r a l l

~-~x i j ( ~ , x )

i and j

= (j + 1 ) Q i , j + 1 ( ~ , x ) .

Thus, i f x* is a r o o t o f Qro0(~*,×) o f m u l t i p l i c i t y

Qr00(~*,x

*)

= 0,

j

= 0 .....

< ~ 2 then

~ - I , Q r o ~ ( ~ * , x * ) ~ 0,

ro and ( A . I . 5 4 )

yields after division

by q

Q r o K ( ~ * , x * ) ~ ( n ) ~ + Zj=K+IQr0j(~ * ,x * ) ~ ( n ) j ~

+ ~i=iZj=iQro+i

,j(~.,x.)~i~(n)J

= o.

Applying here once more the Puiseux diagram we f i n d t h a t c o n d i t i o n ( i i i ) and s u f f i c i e n t

f o r ~(n) having the form

is necessary

173

~(n) = mn + O(n

S~

),

s* > I ,

with some m E { being nonzero or not. A s u b s t i t u t i o n of t h i s r e s u l t i n t o ( A . I . 5 4 ) y i e l d s ~(n) = 5 " ( I + ×% + mn2 + ~ ( n s ) ) , which is the necessary and s u f f i c i e n t

s > 2,

c o n d i t i o n of the disk lemma ( A . I . 4 1 ) .

A l l c o n d i t i o n s of t h i s lemma are empty i f x(5,0) has no unimodular roots a t a l l . The t h i r d c o n d i t i o n is empty i f Qroo(~*,×) has only simple roots. As Qro(5*,x) is a l i n e a r polynomial f o r ~ = I or k = I , the t h i r d c o n d i t i o n can be omitted f o r l i n e a r m u l t i s t e p methods or s i n g l e step m u l t i d e r i v a t i v e methods. From the present s t a b i l i t y p o i n t of view the l a t t e r class encloses here also the Runge-Kutta methods.

A.2. A u x i l i a r y Results on Frobenius and Vandermonde Matrices

The Frobenius m a t r i x F (n) associated with the polynomial x(~,n) is defined in ( 1 . 2 . 5 ) . This matrix has the c h a r a c t e r i s t i c polynomial ~ ( ~ , n ) , ~k(n)det(~l - F (n)) : ~(~,n),

and so the roots o f ~(~,n) are the eigenvalues of F (n).

I f x(~,n) has k d i s t i n c t

roots

then F (n) is t h e r e f o r e diagonable but the converse is also t r u e ; see e.g. Stoer and B u l i r s c h [80 , Theorem ( 6 . 3 . 4 ) ] . F

Omitting the argument ~, a diagonable Frobenius m a t r i x

has the Jordan canonical decomposition

(A.2.1)

F ~T

= WZW- I

where Z = (~1

~k ) is the diagonal m a t r i x of the eigenvalues of F

Vandermonde matrix,

El . . . . . . . . . (A.2.2)

W=

Ck "

-1 . . . . . . . .

~-1

, det(W) = .~-~.(~i - ~ j ) " l>j

and W is a

174 Let Wji r e s u l t from W by cancelling the j - t h row and i - t h column then we obtain by Cramer's rule (A.2.3)

W-I = [ w T j ] ki , j = 1 '

w~. IO = (-1)i+Jdet(Wji)/det(W).

The elements of W and W- I are thus r a t i o n a l functions of ~ I " ' "

~k without s i n g u l a r i -

t i e s i f no roots ~i(n) coalesce in some point n of the considered domain. Observing that W(nl)-IW(n2 ) = I + (W(nl) -I - W(n2)-1)W(n2 ) we can state the f o l l o w i n g r e s u l t .

(A.2.4) Lemma. Let R C ¢ be a closed domain and let ~ I " ' " Ck be k distinct holomorphic functions in R with I~i(n) I ~ I. Then the associated Vandermonde matrix satisfies SUPnE RIW(n)l s <' suPnE RIw(n)-ll ~ ~R' IW(nl)-Iw(q2)I

Let now s

~ I +
v q1' q2 E R.

be the m-th elementary symmetric function in the k v a r i a b l e s ~ 1 , . . . , ~ k ,

s (C I . . . . ,C k) = Z1~v1
su~ = s~(~1 .... '~-I'~+I""'¢k

)'

s~v = sJ¢i,...,¢ _1,Cu+1,...,C~_1,Cu+1,...,Ck),

I ~ ~ < v ~ k,

be the elementary symmetric functions in the indicated k-1 and k-2 variables respectively.

We w r i t e henceforth b r i e f l y s (n) = s ( ¢ i ( n ) . . . . . ok(n)).

Then t e.g. by Muir [60 ] or Gautschi [62 ] , the elements w~. l J of the inverse of the Vandermonde matrix W can be w r i t t e n as (A.2.5)

wTj = ( - 1 ) J - l s ~ ~ / l ~ T ( ~ v - ~i ). ~i

The following two lemmas of Hackmack [81 ] concern again the characteristic poly-

175 nomial ~(C,n 2) of a convergent multistep m u l t i d e r i v a t i v e method f o r a d i f f e r e n t i a l system of second order; cf. (2.1.4). The f i r s t one is a modification of the t h i r d assertion in Lemma (A.2.4) f o r the case where some roots ci(n) coincide in the point n = 0. (A.2.6)

Lemma. Let Ci(n),

i = I .....

k, fulfil the assumption of Len~na (A.I.10).

Then

the associated Vandermonde matrix W(n) satisfies for -s ~ n~ ~ O, j : 1,2,

IW(nl)-Iw(n2)l

~ I + ~smaX{1,1n11-1}Inl

- n21,

IW(nl)-lw(n2)l

s 1 + <smaX{1,Bn11-1}max1~i~klci(nl ) - c i ( n 2 ) ] ,

S < ms

S = ~,

Proof. Let ~ i j be the Kronecker symbol and l e t b r i e f l y k W(nl)-IW(n2 ) = [ u i j J i , j = I. We observe that ~k . ~-I ~=lWi~Cj = 6ij

and thus can w r i t e by (A.2.5) k , u i j = ~ i j + Z~=1(wig(nl)~j(n2 )g-1 - w~g(nl)~j(nl ) g - l ) : ~ i j + [T-~.(~v(nl)

- ci(nl))-1]Z~=1(-1)~-Is~-~(nl)(cj(n21~-I

- aij{j(nl

)L-If'

~l

We now consider two cases: ( i ) I f i = j then we obtain by Vieta's root c r i t e r i u m ~ = i(_i)£.-I Sk_gtnl)~j(n2 i • )g-1 = (~i(ni)

- ~j(n2))...(~i_1(nl)

- ~j(nm))(~i+1(n I) - ~ j ( n 2 ) ) . . . ( ~ k ( n l )

- ~j(n2)).

Therefore we have in this case (A.2.7)

u i j = I [[(cv(n I) - ~j(n2))/(Cv(n I) - ~ i ( n 2 ) ) ] . ~I

As stipulated in Lemma (A.I.8) and ( A . I . 1 0 ) , the roots confluenting pairwise in n = 0 are numbered by i and i + I , i = 1,3, . . . . k.-1, 2 ~ k. ~ k. I f i > k. in (A.2.7) then ~i(n) is a simple root in [-s, 0] and we find e a s i l y that (A.2.8)

luijl

s ~slCj(n I) - ~ j ( n 2 ) l ,

i . j,

i > k..

176 I f i ~ k, then we deduce from (A.2.7)

(A.2.9) l u i j I ~ I K s l c j ( n l ) - CJ(nZ)I/ICi(nl) - Ci+1(n2)[' i = 1,3 . . . . . k.-1

L< s l E j ( n l ) (ii)

Ej(n2)]/]Ci_1(n1)

- ci(n2)I,

i

2,4,.

,k,,

i = j.

I f i = j then we have u i i = I + [I- - ! ( ~ v ( n l ) - ~ i ( n l ) ) - 1 ] L ~~k= 0 ,~- 1;, ~ - l s ik _ ~,n l )(~i(n2)~-1

_ ~i(nl)~-l).

But the elementary symmetric f u n c t i o n s s i are bounded in the i n t e r v a l finite

and i n f i n i t e

[a£

-

b~l

[ - s , 0] f o r

s, and

=

la

-

bl

~~ v- I= O a ~ - v - l v , D

I.

Thus, i f i > k, then we obtain here the bound (A.2.8) f o r l u i i - 11 and i = j ,

and i f

i ~ k, then we obtain the bound (A.2.9) f o r l u i i - 11 and i = j in a s i m i l a r way as above. An estimation of (A.2.8) and (A.2.9) by means of Lemma (A.1.10) f i n a l l y

proves

the a s s e r t i o n . (A.2.10) Lemma. Let Ci(n), i = I ..... k, fulfil the assumption of Len~na (A.I.10). Then the associated Yandermonde matrix W(n) and arbitrary Q E ~k satisfy

IW(n)-IQl ~ %(IQ[ + l~l-lmaxl¢i(O)l=~lci(O)Q~_1

- Q~]), -s ~ n2 s O, s ~ ~,

0

where Qn = ( q n - k + 2 " " ' q n ) T '

Proof. We w r i t e s ~ = 0 and s~ v = I and observe that (A.2.11)

s ~'~+I k-2 C~+I = Sk-1'

= I,.,,,k-I,

(A.2.12)

s~-1, ~

= 2,...,k, k = 2,3,...,

k-2

~-I

= Sz

k-1'

and (A.2.13)

S~,~+1 S~,~+1 = S~ k-i + k-i-1(~+1 k-i'

= I ..... k-l,

(A.2.14)

s~-1,@ s~-1,~ = s~ k-i + k-i-1(~-1 k-i'

= 2,..,,k, i : 2,,..,k , k = 2,3,...

Let ?k w* ui(Q) = ~j=1 i j ( n ) q j '

i = I . . . . . k,

177 be the i - t h element of W(n)-IQ then we obtain by (A.2.5) (A.2.15)

ui(Q) = [ ~ ( ~ ( n )

- ~i(n))-1]~:1(

-1)j-ls~_j(n)qj.

I f i > k,~ i ° e . . i f ~i(n) is a simple roots throughout I - s . 0]. then we find easily by Schwarz's i n e q u a l i t y that (A.2.16)

lui(Q) I ~ ~sIQl.

I f i < k, then a s u b s t i t u t i o n ui(Q) = [t-~(~ •

of (A.2.11) and (A.2.13) into (A.2.15) y i e l d s

(n) - ~ i ( n ) ) - 1 ] Z ~ =2(-1~Js ' i + 1 (n)(~i+1(n)qj-1 J ik-j

- qj)

~I

and an application lui(Q)[

of Lemma ( A . 1 . 1 0 ) ( i i )

leads to

s KsmaX{1.[nl-1}Z~=21~i+1(n)qj_1

- qjl.

But l~i+1(n)! s I hence i f Inl ~ I then (A.2.16) holds. I f Inl < I then we have because i+I s k, by Lemma (A.I.8)

l~i+1(n)qj_ I - qjl = l(~i+1(n)

- ~i+1(0))qj_1

Ks[nlIqj_11

+ l¢i+1(0)qj_1

+ ~i+1(0)qj_1

- qjl

- qjl.

Therefore we obtain in this case (A.2.17)

lui(Q) [ <= ~~s(IQl + In i-Ii¢i(0)Q~_1

0 - Qk])

I f i = k, then (A.2.16) and (A.2.17) follow using the formulas (A.2.12) and (A.2.14) instead of (A.2.11) and (A.2.13). From (A.2.16) and (A.2.17) the assertion follows with <s = ~s ~ "

A.3. A Uniform Boundedness Theorem

The main tool of the f i r s t two chapters was a theorem on the uniform boundedness of powers of Frobenius matrices whose argument varies in a closed subset R of the s t a b i l i t y region S. With respect to methods f o r d i f f e r e n t i a l systems of f i r s t order this r e s u l t has been proved by Gekeler [79 ] using the fact that in Frobenius matrices

178

the eigenvectors can be expressed e x p l i c i t e l y

by the eigenvalues. S i m i l a r r e s u l t s have

been obtained by Crouzeix [80 ] and LeRoux [79a] in d i f f e r e n t ways. In the sequel we represent an estimation of a very general form which is due to Crouzeix and Raviart [80 ]. Let A be a (m,m)-matrix and l e t ~I' . . . . ~ '

~ --< m, be the d i f f e r e n t eigenvalues

of A. Then the characteristic polynomial of A can be written in the form M.

d e t ( A - ~I) = ( - 1 ) m ~ ( ~ - xi ) i i=I and the integer mi is called the (algebraic) m u l t i p l i c i t y of the eigenvalue ~i; see e.g. Stoer and Bulirsch [80 , p. 316]. Let now D C ~ be an open domain and l e t A: D 9 ~ -~ A(~) E CmXm be a matrix-valued function. I f A(~) -= A(S- I ) exists f o r ~ = 0 then we write A(~) = A(O). We denote by spr(A) the spectral radius of the matrix A and define for R c D

m*(R) :

11

i f supsER spr(A(S)) < I

t sup~ERmax1~i~cmaxi~i(~)m~1{mi(~)}

i f supsER spr(A(S)) ~ I .

Then the f o l l o w i n g theorem is a s l i g h t modification of Crouzeix and Raviart [80 , Theorem 8.1]. (A.3.1) Uniform Boundedness Theorem. ( i ) Let D C ~ be open and l e t A: D + {mxm be a continuous matrix-valued function. (ii) Let R C D be closed in ~ and let

sup~E RSpr(A(S)) ~ I.

Then

sups E RSUPnEIN Ilnl-m*(R)A(~)nlt ~ ~R"

The proof follows Crouzeix and Raviart [80 ] and is partitioned into several steps: (A.3.2) Lemma. For every (m,m)-matrix A there exists a unitary matrix U such that

R = uHAu is

an upper-triangular matrix.

Proof. See e.g. Stoer and Bulirsch [80 , Theorem 6.4.1]. (A.3.3) Lemma. Let A: D ÷ ~mxmbe a continuous matrix-valued function in the open domain D c ~ and let a neighborhood H

A(~) have

only a single eigenvalue ~(~), ~ E D. Then there exists

of ~ and a constant ~ > 0 such that for all S E H

IIA(s)nTI ~ ~ ( I x ( s ) l n + n m - l l x ( ¢ ) l n - m + l ) ,

n = m,m+l . . . .

179 Proof. For every ( E D there e x i s t s by Lemma (A.3.2) a u n i t a r y matrix U(¢) such that the matrix R(C),

R(~) = [ r i j ( ~ ) ] im , j = I : uH(¢)A(~)U(~) is upper t r i a n g u l a r . As

Irij(¢)l --< IR(¢)I : IA(¢)I and ¢ ÷ A(¢) is continuous in ¢ there e x i s t s a neighborhood ~ f c ~ of ¢ such that r : supCEAr I r i j ( C ) I -< s u p c c x l A ( ¢ ) I < ~. Let now A* denote the matrix of the absolute values of the elements of A and l e t ]m Q = [qij i,j=1'

{10i<J qij =

else

Then the f o l l o w i n g i n e q u a l i t y holds elementwise, R ( ¢ ) * =< I ~ ( ¢ ) I z

+ rQ.

Because Qm = 0 we therefore obtain f o r n ~ m elementwise (R(~),)n < lx(~)inl . .

+ (~)ix(~)in-lrQ .

.

+

+ ( n )i~(~)in-m+1(rQ)m-1 m-1

But IA(¢)nl = IR(¢)nl ~ l(R(~)n)*l ~ I(R(¢)*) n] by the Theorem of Perron and Frobenius ( c f . e.g. Varga [62 ] IA(¢)nl ~ Ix(E) n + ( ~ ) [ x ( ¢ ) i n - l e

hence, w r i t i n g o = I r Q l ,

+ . . . + (m_1)ln~(¢)in-m+lem-1

:< I x ( ~ ) n + ~r,mmaXo~j~m_2{sup ~ E # l ~ ( ~ ) l J } n m - l l x ( ~ ) l n-m+1 which y i e l d s the assertion by the c o n t i n u i t y of ~ in ~ and the norm equivalence theorem, Notice that ~ = ~r,m f o r ~ E Y~rn R i f sup~ E RIX(~)l

~ I.

(A.3.4) Lemma. Let A: D + {mxm be a continuous matrix-valued function in the open domain D C ~ and let A(~) have ~ different eigenvalues, ~1(~)s..., X~(~)s I < ~ ~ ms E D. Then there exists a neighborhood/Vof ~ and a continuous matrix-valued function

H : Y ~ ÷ Cmxm such that for all ~ E H

180

H(¢)-IA(~)H(~)

:

where the matrices Ai(~) have different eigenvalues and Ai(~) has the single eigenvalue k i ( ~ ) ,

i = I , . . . . ~.

Proof. For i = I , . . . , ~ ,

l e t Bi be an open disk with center ~i(C) such t h a t Bi n Bj =

f o r i ~ j . Then there e x i s t s a neighborhood

Y~I of ~ such t h a t f o r ~ E J~I every Bi con-

t a i n s e x a c t l y mi eigenvalues of A(~) where mi denotes the m u l t i p l i c i t y value ~ i ( ~ ) .

of the eigen-

We define the p r o j e c t i o n s

Pi(~)

=

~I

[ (sl - A(~))-Ids,

V ~ E Y~I, i : I . . . . . ~,

aBi where aBi denotes the positively oriented boundary of Bi , By Kato [66 , Section 2.5.3] the matrix-valued functions ~ + Pi(~) are continuous in ~. Let further

Zi(~) = { P i ( ~ ) z ,

z E cm}

be the ranges of Pi(¢). Then, for ~ E/~I , Zi(~) is a linear subspace of dimension mi which is invariant with respect to A(~), and

Cm : ZI(~) ® Z2(~) e . . .

m Z (¢);

cf. Kato [66 , Section 1.5.4]. We choose a basis of Zi(~), zi(~),.."

z '

i (~), mi

i :

I,...,~.

and w r i t e (A.3.5)

k = 1,...,m i , i = I , . . . , ~ ,

i Zk(¢) = pi(¢)z~(5) '

then the functions ~ ÷ z~(~) are continuous in ~ and hence there exists a neighborhood Y ~ c ~I of ~ such that the vectors (A.3.5) are linear independent in 14/'. Now the matrix with the columns z~(~),

H(¢) = [ z l ( ¢ ) , . . . , z

ImI(~) . . . . .

z ~ ( ¢ ) , . . . , Z m ~ (¢)]

has the desired properties for ~ £ ~ . Proof of the Theorem. By Lemma (A.3.4) there exists for every ~ E D a neighborhood Y~;(~) and a constant < such that for a l l n E IN with n z m*(~)

181

(A.3.6)

IA(~)n[ ~ ~(spr(A(~)) n + n m * ( 5 ) - I s p r ( A ( ~ ) ) n-m*(5)+1)

But R is closed in ~ hence there e x i s t s a f i n i t e

v ~ E Y~/(C) ~ D.

number of open sets J~j, j = 1 , . . . , J ,

which cover R and have the property t h a t (A.3.6) holds f o r a l l ~ E v~jj. Because spr(A(~)) s I f o r a l l supc c ~ j

~ E R we have

~ R IA(c)nl ~ <j(1 + nm*(R)-1) ~ <~n m*(R)-IJ

hence

SUpsER IA(~)nl ~ max1~jsjsup~EIV~nR IA(~)nl s mRnm*(R)-1 which is the desired r e s u l t .

A.4. Examples to Chapters I and IV

In t h i s section we give some examples of m u l t i s t e p m u l t i d e r i v a t i v e methods f o r differential

systems of f i r s t

order:

( i ) The general c o n s i s t e n t s i n g l e step method w i t h one d e r i v a t i v e has the characteristic

polynomial =(~,n) : (1 - ~n)c - (1 + (1 - ~ ) n ) ,

c f . Section 4.2. ( i i ) The general s i n g l e step method w i t h two d e r i v a t i v e s has the c h a r a c t e r i s t i c nomial

(A.4.1)

poly-

x(~,n) = (~01 + n~11 + n2~21 )~ + (SO0 + n~10 + n2~20 )"

a) For the maximum a t t a i n a b l e order p = 4 we obtain a uniquely determined method w i t h the polynomial ~(c,n) = (12 - 6n + n2)~ - (12 + 6n + n2). This method has the s t a b i l i t y x 4 = 1/720 ( c f .

region S = {n E C, Ren ~ O} u {~} and the e r r o r constant

(1.3.6)).

b) The methods (A.4.1) of order p = 3 have the c h a r a c t e r i s t i c

polynomial

=(~,n) = (12 - 12(I + y)n + (4 + 6y)n2)~ + (-12 + 12yn + (2 + 6y)q2).

182

They are A - s t a b l e i f f -

2/3, i . e . ,

~ ~ - I / 2 and have the e r r o r constant ×3 = ( I + 2 y ) / 2 4 . For y =

f o r m21 = O, the r e s u l t i n g method is not A o - s t a b l e ( c f . D e f i n i t i o n

(I.6.1)).

The c o n d i t i o n o2(n) = (12 - 12(I + y)~ + (4 + 6y)n 2) : leads to y = ± ~ / 3 .

For ¥ = 8 / 3

12(I - an) 2

the r e s u l t i n g method is A - s t a b l e w i t h - £ S (Cala-

han's method, c f . e . g . F r i e d [79 ] ) . c) The methods ( A . 4 . 1 ) of o r d e r p = 2 have the c h a r a c t e r i s t i c (A.4.2)

polynomial

~ ( c , n ) = (2 - 2(1+y)n + (1+2y-26)n2)C + (-2 + 2yn + an2).

They are A - s t a b l e i f f

¥ ~ - I / 2 and 0 ~ a s (2y + I ) / 4 and have the e r r o r constant

x2 = (I + 3y - 6 n ) / 6 .

In p a r t i c u l a r ,

we o b t a i n f o r y = - I / 2 and 6 = 0 the t r a p e z o i d a l

r u l e and f o r ¥ = - I / 2 and ~ = I a method due to J e l t s c h [78a] which i s s t a b l e on the imaginary a x i s but not A - s t a b l e . The s t a b i l i t y

region o f t h i s method i s given in

figure (A.4.7). (ii)

Backward d i f f e r e n t i a t i o n

teristic

methods are l i n e a r m u l t i s t e p methods w i t h the charac-

polynomial k i ~ ( c , n ) = ~i=Omi c - n~C k

( c f . e . g . Lambert [73 , po 2 4 2 ] ) . The c o e f f i c i e n t s

of the k - s t e p methods o f o r d e r p = k

are given f o r k = I . . . . ,6 in t a b l e ( A . 4 . 3 ) ° The r e s p e c t i v e r o o t locus curves and the m o d i f i e d r o o t locus curves are given in f i g u r e ( A . 4 . 8 ) and ( A . 4 . 1 2 ) . (A.4.3) Table o f Backward D i f f e r e n t i a t i o n

(iii)

ml

Methods.

k

B

~0

m2

m3

I

I

-I

2

2

I

-4

3

3

6

-2

9

-18

11

m4

m5

m6

I

4

12

3

-16

36

-48

25

5

60

-12

75

-200

300

-300

137

6

60

10

-72

225

-400

450

-360

147

The methods o f Cryer {73 ] are l i n e a r k - s t e p methods o f o r d e r p = k w i t h the

characteristic

~(c,n)

polynomial

= p(C) - n(c + d) k.

183 For a given d these methods are defined in a unique way by Lemma (1.1.12) because of the prescribed order. For d = - I + 2 / ( I + 2 k+1) and k = I , . . . , 7 and the modified r o o t locus curves are given in f i g u r e (A.4.9)

the r o o t locus curves and (A.4.13).

( i v ) The methods of Enright [74a,b] are n o n l i n e a r k-step methods with the c h a r a c t e r i s t i c polynomial ~(c,n)

Ck

k-1

=

k

i

_ n~i=OBi ~

2 -

q ~

k •

The k+2 free parameters can be chosen such that the order is p = k+2. The c o e f f i c i e n t s of the r e s u l t i n g methods are given f o r k = I . . . . ,7 in table (A.4.5) and the associated root locus curves are given in f i g u r e (A.4.10). I f we choose ok(n) = (I - ~n) 2 as in Calahan's method, i . e . ,

(A,4.4)

y = - (Bk/2) 2,

then the remaining k+1 f r e e parameters can be chosen such t h a t the order is k+1. However, in t h i s case the r e s u l t i n g methods are no longer uniquely determined. The methods of order p = k+1 with (A.4.4) proposed by Enright are given in t a b l e (A.4.6) and the associated r o o t locus curves are given in f i g u r e (A,4.11), These curves consist of two segments f o r k ~ 4 but the methods remain A(m)-stable at l e a s t up to k = 6. (A.4,5) Table of E n r i ~ h t ' s Methods I. Y I

3

2

4

3

5

4

6

5

7

6

8

7

9

7

9

80

81

-I

-I

1

-I

-19

T~

7 i

T~

82

83

84

85

2

5

29

-I

19

307

TC

~

TC~

-3

-17

I

-41

47

3133

-863

41

-529

373

-1271

2837

317731

-275

-731

179

-5771

-13823

12079

-33953

8563

-35453

86791

157513

-133643

8-/

1758023

86

8131 -2797

247021 1147051

184

(A.4.6) Table of E n r i g h t ' s Methods I I . k

1

Z

60

3

v"6-

3

4

/5"

I

2

Z

Z

2

-5 -g

4

Z

4Z + -g

Z

-257

137

-1103

6Z + TZT

5

3853 T~

66

65

v~T75"7~

12

T

-2083057

~

125Z -20297

~

5Z

(A.4.7) Figure. S t a b i l i t y (2 + n - 2n2).

-

-12449

500Z

4Z

T6~

24 25"

Z

16Z

12Z

-

+T4~

3Z

I-[

-

+ T~

2416169 T

10Z

1231883 7

-3661 - T~

-261979

-I~

54Z

+ TTT

16Z

T2T

~C~TTC

6

27Z

TZT TT0~

Z +

6

64

I-[

7T~T

1057

5

63

2

2

4

B2

81

T~

+ 7"5"

2889973

-5534137

120

T ~

7T0~

137

250Z

+II~

250Z

-T~

5Z

+B7-%

124541 -2887799

587501

-10783

7 ~

T ~

-7~

25Z

+T~

20Z

-T~

region of the method defined by x(~,n) = (2 - n - 2n2)~ -

40 Tg Z

185

(A.4.8) Figure. Root locus curves of backward d i f f e r e n t i a t i o n methods, 21,00

7.

k=

0

-7FO0

=4

0

7,00 ILl. O0 REALTEIL

21.00

28,00

(A.4.9) Figure. Root locus curves of Cryer's methods.

.300 ~--

=2

.200 ..__1 LLI I.,--

~n L~

"

i00

CIZ Z £0 CIZ

-. oo -.iGo

°o

.100

.200 REALTEIL

.300

(A.4.10) Figure. Root locus turves of Enright's methods I .

15.00

.OC

~[fie

0

0

6 O0 REflLTEIL

18o00

.~00

.500

• 6'00

186

(A.4.11) Figure. Root locus curves of Enright's methods I I ,

15.00-

IoS

\

\

W F-W (3Z Z CO

(!

~.00 REALTEIL

-~.00

]2

O0

20.00

(A.4.12) Figure. Modified root locus curves of backward d i f f e r e n t i a t i o n

28.00-

k=6

21.00-

,

lu,. O0-

Iu.J

-7.~0

0

7.'00

ILI,'O0 21.'00 28.'00

REALTETL

methods.

187

(A,4.13) Figures. Modified root locus curves of Cryer's methods with enlargements. 600500LtOO-

k:3

soo~ 200100

-. ~oo -o

k: :4

Ski5

o

o 6'00

100

-.200

.0100Z

-.0'I00

RE~

O0

-.0100

.0100-

.0100-

\

-.0'I00

i

RE~E TE I L .OiO0

-.0100

-.0100

~

0

-,OtO0

\\

. oioo

188 A.5. Examples of NystrDm Methods

A method of Nystr~m type (2.4.9)

cf.

is described completely by i t s i t e r a t i o n

G(n) = - o(n) - I r~o(n)

~(n)]

Lx0(n)

×~(n)]

matrix

( 2 . 4 . 1 1 ) . To keep the notations simple we w r i t e

~ J , d~(n) = Z~=O6jn ~ J', ~o(n) = Zj=O~jn

~

j

~ ~

~

j

×o(n) = Zj=oYjn , x~(n) = ~j=o~jn j , ~(n) = Zj=Imjn . Then we have m0 = I by the s t i p u l a t i o n

( 2 . 4 . 2 ) , and the method is c o n s i s t e n t by

(2.4.10)

iff

(A.5.1)

~0 = ~0 : ~1 : ~0 = - 1, ?0 = O, ~1 + ~1 : - 1/2.

In the sequel, s ~ 0 denotes the l e f t end p o i n t of the r e l e v a n t s t a b i l i t y

inter-

val I - s , O] c S. I f the method has not maximum a t t a i n a b l e order then s depends on the free parameters. As a r u l e , t h i s dependance is very s e n s i t i v e to small m o d i f i c a t i o n s of the data. The presented special examples stem from the attempt to f i n d methods w i t h large s t a b i l i t y

i n t e r v a l s by numerical experiments.

Let us f i r s t

consider the general Nystr~m type method w i t h ~ = 2. By Lemma

( 2 . 4 . 5 ) , t h i s method has order p i f fulfilled

for u = 3,...,p+I: = 3:61

u = 4:~2 (A.5.2)

(A.5.1) holds and the f o l l o w i n g c o n d i t i o n s are

= 5

+ ~1 + 1/6 = O,

~1 + ml + I / 2

+ ~1/2 + m2 + 1/24 = O,

~2 + ~1 + 1/6 = O;

~2 + ~1/6 + ~2 + 1/120 = O,

u = 6 : ~ 1 / 1 2 + m2 + 1/360 = O, = 7:mi/20

+ m2 + 1/840 = O,

~2 + ml/2 + m2 + 1/24 = O; ~1/6 + ~2 + 1/120 = O; ~1/12 + ~2 + 1/360 = O.

(A.5.3) General method w i t h ~ = I:

1 + ( e(n) : (1 + ~n) - I

+ ~o)n ½ q

= O;

1 + (~+

~o)n]

1 + (½ + ~o)nJ'

189

This method has order p = 2 and no s t a b i l i t y

interval

[ - s , 0] f o r a l l m E IR.

(A.5.4) ( i ) F u l l i m p l i c i t method with c = 2 and order p = 4: mI , m2 EIR, I+

(~+ml)n +

I+

+

ml

2

G(n) : (1+mln+m2n2) -I

(ii) (iii)

Maximum order p = 5: ml = - 1/15, m2 = 1/360; s = 0. Order p = 4, m2 = 0: s = 8 f o r ml = 0.0416.

( i v ) Order p = 4, ml = m2 = 0, i . e . e x p l i c i t method: s = 7.06. (v) Order p = 4, ml = m, m2 = m2/4' i . e . o(n) complete square: s = 8.18 f o r m = 0.0445.

( v i ) Order p = 3, ~I = ~' ~2 = ~2/4' ~2 = ~2 = 0: s = - f o r ~ z 1.92. In the remaining p a r t of t h i s section we consider e x p l i c i t methods f o r the i n i t i a l (A.5.5)

y"

= f(t,y),

Runge-Kutta-Nystr~m

value problem t > 0, y(0) = Y0' y ' ( 0 )

with conservative d i f f e r e n t i a l

= y~

equation. As i t was emphasized several times, these

methods coincide f o r m a l l y with Nystr~m type methods f o r the problem ( 2 . 4 . 8 ) , (A.5.6)

y''

= A2y + c ( t ) ,

t > 0, y(0) = Y0' y ' ( 0 ) = y~,

with exception of the treatment of the time-dependent r i g h t side c ( t ) . Recalling the notations f o r Runge-Kutta methods introduced in Section 5.1,

t n , i = (n + T i ) A t , 0 ~ Ti ~ I , Vn, i = V ( t n , i ) ,

fn,i(v)

= f(tn,i,v),

i = I .... ,r,

a Runge-Kutta-Nystr~m method f o r (A.5.5) reads

(A.5.7)

2 r Vn, i = v n + ~iAtWn + At Z j = 1 ~ i j f n , j ( V n , j ) ,

(A.5.8)

2r Vn+ I = v n + Atwn + At Z j = i B j f n , j ( V n , j )

(A.5.9)

zXtWn+1 = Atw n + A t 2 Z ; = I Y j f n , j ( V n , j ) ,

We use the same n o t a t i o n s as in Section 5.1 but w r i t e

i = 1,...,r,

n = 0,I,...

190 a = (31 . . . . . ~r )T, b = (BI . . . . . Br )T' q = (YI . . . . . Yr )T" Then we obtain in complete analogy to (5.1.9) f o r the l i n e a r problem (A.5.6) with time-varying matrix A(t) 2 the computational device

Vn+I : (I + At2bTA_~2(Z- At2p~A2)-IZ)Vn + (I + At2bTA_~2(~- ~t2p_~A2)-la)AtWn + At2bT(~ - at2p_~A2)-Ic_n AtWn+ I : At2qTA~2(Z- At2p_~A2)-Izvn + ( I + ~t2qTA_~2(Z-At2p_~A2)-la)AtWn

+ At2qT(~ - At2p_~A2)-~c~q, n : 0,1 . . . .

In p a r t i c u l a r , we f i n d f o r the t e s t equation y "

= x2y w r i t i n g Vn = (v n, AtWn)T and

n = At2~2 Fgl(n) G(q) = Lg3 (n)

Vn+ 1 = G(n)V n,

g2(n) 1 g4 (n)]

with the f o l l o w i n g r a t i o n a l functions in n g1(n) = I + qbT(l - n p ) - I z , g2(n) = I + nbT(l - n p ) - l a , (A.5.10) g3(n) = ~qT(l - n p ) - I z , g4(n) = I + qqT(l - n p ) - l a . A Runge-Kutta-Nystr~m method is described by the vectors a, b, and q, and by the matrix P, i . e . ,

by the matrix

(q,b,a,P) =

[!i r

i 11 :it1 •

Br

~r

~rl

-

E.

~rr]

The method is emplieit i f P is a lower t r i a n g u l a r matrix with zeros in the diagonal. As in Runge-Kutta methods, the order of consistence can have d i f f e r e n t values f o r the general nonlinear problem (A.5.5) and the l i n e a r problem with constant matrix (A.5.6). In the f o l l o w i n g examples, p denotes the order with respect to (A.5.5) and with respect to the corresponding Nystr~m type method, i . e . ,

with respect to (A.5.6). p is

said to be maximum - with respect to (A.5o5) and (A.5.6) - i f i t is the maximum order f o r a l l explicit methods with the same stage number r. (A.5.11) r = ~ = I , p = I (max.), S = [ - 4 , = (y1,~.l,T1,0) = ( 1 , 1 / 2 , 1 / 2 , 0 ) .

0], c f . Hairer [77 ]:

191

(A.5.12) r = ~ = 2, p = 2 (max.), c f . H a i r e r [77 ] ; s = 3.67 I ==

I -

4(3~2-3~+1)

:

0

4(3~2-3e+1)

3(1-2~) =

(1-2m)(1-3~)

2-3~

2(3~2-3~+1)

4(3~-3~+1)

4(3~-3~+1)

3(1-2~)

f o r ~ = 0.265:

Oolo<

9(1-2~) 2

(A.5.13) r = JL = 3, p = 3 (max. f o r ( A . 5 . 5 ) ) , cf. H a i r e r [77 ] ; s = 6.69 v m E IR: ½-~ ~ =

(~-

½

~ ) ( I +/3~ I /3 ~_-~, 7~-~

1,1

~3~

½+

~_-~j

0 I

/3

~--~

0

I

0

0 ,VmEIR.

I -

0

Hairer [77 ] ;

0

0°

0

6(I-2~) 2

4(6~2-6m+I)

E =

0

7~i~ l~lj

(A.5.14) r = L = 3, p = 3 (max. f o r ( A . 5 . 5 ) ) , c f .

6(I-2~) 2

0

6(I-2~) 2

2(6~2-6~+I)

I

(I-4~)(1-2~)

6(I-2~) 2

~

8(6~2_6~+I)

I

I-~

6(I-2~) 2

0

2~(I-2~)

0 , 0<~ < I.

½(1-2~)(I-4m)

This method has order p = 4 with respect to (A.5.6) f o r m = (5 + ~ ) / 2 0 ; f o r ~ = (5 + ~ ) / 2 0

and s = 15.33 f o r ~ = (5 - ~ ) / 2 0 .

(A.5.15) r = c = 5, p = 5, cf. Albrecht [55 ] ; s = 9.24:

71907190 o 24/90 I/4

J32/90

o

o

1/32

0

o

0

o

0

= : 112/90

6/90

I/2

-I/24

I/6

0

0

132/90

8/90

3/4

3/32

I/8

1/16

0

0

I

0

3/7

-I/14

I/7

L

7/90

0

6(I-2~) 2

il .

s = 6.47

192

(A,5,16) Figure, Root locus curves of method ( A , 5 , 4 ) ( i i i ) , i

i

15.0 ~ oc uJ o: z

i

~

i

t

i

t

i

1

-4 .1o

10.0

.05

s.0 0

-i

o

U, -s.o ~-1o.o

4

~ -,05 ~-.10

-15.0 -20.0

_1

-.15 I I I I / -15.0 - I 0 . 0 - 5 . 0 0 5.0 REALTEIL

(A.5.17)

i

.15

20.0

I I0.0

I 15.0

81

-

.0

Figure.

I

-01.0

i

Root l o c u s c u r v e s

15o

o f method ( A . 5 . 4 ) ( i v ) ,

ioo -4

I

I

I

-ql.O -21.0 RERLTEIL

l

l

r

J

5.0

,=, (z z

0

-5.0

-I0,0

-15.0 I -25.0

t -20.0

I -15,0

I -I0.0

_510

i 5 0

10 I.

0

1 .0

RERLTEIL

(A,5.18) Figure. Root locus curves of method ( A . 5 . 4 ) ( v ) . L

I

~

i

l

i

i

i

i

i

1

I

I

I

i

i

1

I

I

I

I

l

l

1or

15.0

10.0 5.0 ~z ~

~

.15 F

q

20.0

-05 t

o~-

0 -5.0

~-1o.o

t

- , 0 5

-15.0

-.10

t

-20.0 i

I

-20.0

I

I

-10.0

(A,5,19) Figure,

8,0

Root locus curves of

6.0

method (A.5,4)(vi) for m = 1,92,

u w

-

15 -9I. 0 I

t

0 RERLTEIL

10.0

-7.0

i

~

-21,0

0

i

-5.0 -3.0 RERLTEIL

i

E

2.0 ~i0 RERLTEIL

610

I

I

- I .0

1.0

i

]

u~.O 2.0

a: z

0

~ -2,0 -u.,O -6.0 -8,0 i

-~,0

r

I

810

101,0

-8.0

-6.0

-~.0

I -25.0

I -20.0

I -15.0

I I -10.0 -5.0 REALTEIL 5.0

I

i

J I0.0

i

-5,0

-25.0

-20.0

-15.0

~-lO,O

0

5.o

I0.0

15.0

20.0

25.0

I

-5.0

I

-I0.0

-15,0


I

b

REALTEIL

51o

t

10.0

i

15i. 0

i

o

5.o

~:

-J

I -IL~.O

i

I -12,0

i

I -i0.0

i

-8 I. 0

i

6] .0

RERLTEIL

-

~

_i. 0

i

-2.0

~

I 0

i

i 2 0

i

-15.0

-I0.0

-5,0

o

5.0

I0.0

15.0

I -30.0

I -25.0

I

-20.0

I

I

-i0.0

RERLTEIL

-15.0

I

-5.0

0

i

I

7

51o

i

(A.5.23) Figure. Root locus curvesofmethod (A.5.14),m=(5-¢~)/20.

-15.0

-10,0

:E -5.0

m

I0.0

15.0

(A.5.21) Figure. Root locus curves of method (A.5,13).

d

(A.5.22) Figure. Root locus curves of method(A.5.14),m= ( 5 + 4 ) / 2 0 .

c~ a:

~ z

J

0

2.0

W-2.o

z

W

w

d

~.0

6.0

8.0

(A.5.20) Figure. Root locus curves of method (A.5.12).

194

( A . 5 . 2 4 ) Figure. Root locus curves of method ( A . 5 . 1 5 ) . i

i

i

,

i

i

i

i

i

BO.O 20.0

.25 .20 .15 .iO

J

io.o

~

.os

o

z

a:

0

z -.o5 -i0.0

e:-.

10

:~-.ls

-20.0

4

-.20 -.25

-30.[J I

1 0

-20.0

I

I 20.0 RERLTEIL

I

LlO I. 0

I

J

I -I0.0

I _81.0

1

~ I I I _21 0 1 -6.0 -u,.o REIqLTEIL

I

A.6. The (2,2)-Method for Systems of Second Order

The general 2-step method ( 2 . 1 . 4 ) with ~ = 2 and order p ~ 4 has the characterist i c polynomial

(A.6.1)

#(~,n2) = ( 2 _ 2 ~ + I )

- n2(~ 2+ ( I - 2 m ) ~ + )

+ q4(e~2+ ( ~ - 2 e - 1/12)~+e).

For instance, i f we choose ~ = I / 2 and e = I / 4 then the resulting method of order 4 has the s t a b i l i t y interval [ - = ,

O] and is also strongly D-stable in [ - = ,

0]. The

s t a b i l i t y region of this method is plotted in Figure (A.6.3). Notice that the curved l i n e and the entire real l i n e with exception of the unplotted part together make S = 8S. The plotted imaginary axis does not belong to S. For e = ( 3 0 w - I ) / 3 6 0 Wmin = ( - 5 - 2

the method has order p ~ 6. Let

I/T5)/60,

mopt = ( - 5 + 2

I~)/60

then we have for mmin ~ w ~ mopt

(A.6.2)

[-s(~),

O] C S,

s(~) = 12/(I - 12w).

For mmin < w < mopt the entire s t a b i l i t y region S = ~S consists of the interval (A.6.2) and a nearly straight line which intersects (A.6.2) v e r t i c a l l y . For ~ = Wmin and m = mopt this vertical l i n e degenerates to a point being indicated in Figure (A.6.4). For = - I/6, - I/2,

I / 4 , O, 4/100 the s t a b i l i t y region is plotted in this figure. The

marks on the real l i n e denote from right to l e f t the corresponding l e f t end points -s(m) of the interval (A,6.2). For ~ ~ [~min' mopt] the set S ~ IR consists of two disconnected interval hence the method cannot possibly have the s t a b i l i t y interval (-=,

O] for order p ~ 6. For these values of m we find that [ - s ( w ) ,

O] c S for

195

s(~)

= 6115(1-4~)

- V~(720J+120m-7)1/2]/[13-120m]

hence s(m) ÷ 0 f o r m + - ~ and s(m) ÷ 6 f o r m ÷ + ~. For m = 11/252 = 0.0436... and e = 13/15120 the method given by (A.6.1) has order p = 8. Since mmin < 11/252 < mopt = 0.0457... the s t a b i l i t y form as in Figure (A.6.4) with S n IR ~ [ - 25.2, 0]. stability [-26.6,

(A.6.3)

region has here the same

In comparison with t h i s the l a r g e s t

i n t e r v a l o f a method given by (A.6.1) o f order 6 w a s [ - S ( m o p t ) , 0] 0].

Figure.

14.oo 12,00!0.008.00 B.O0 4.00 2.00 z

~

o

~

-2.0Q-~.00

-6,00 -8,00-10.00-

-12.00 -14.00

-20',00

-30.00

(A.6.4)

io. O0

-30'.00 REELL

Figure.

3~ 3> Z

3>

0-1/24

t ~o

4,/10(

.

,

.

,

.

.

iv,

.

.

,

.

.

,

.

~

.

.

,

.

L

.

,

.

.

,

.

,

.

.

,

,

,

,

,

2,°It

~

,

,

b

,

, Z ,

i

,

,

References

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Glossary of Symbols

i f f := i f and only i f ; LN set of positive integers, IRm set of real m-tuples, ~m set of complex m-tuples, N0 = IN v { 0 } , (a,

IR+ = { x £ JR, x > 0 } ,

b] = { x E i R ,

~ = ~ v

{~};

a < x ~ b},

X ({) Y = {x + y, x C X, y E Y when X n ¥ = ~}; i n t e r i o r of the set S C ~, ~ closed hull of S, aS boundary of S, ~ \ S

complement of

S in C; e = a/~t, T: y ( t ) ÷ y ( t + A t )

s h i f t operator; Ren (or Re(n)) real part of n E ~, Imn imaginary part of n C ~, arg(re i¢) = ¢, r > O, 0 S ¢ < 27, [n] largest integer not greater than n EIR, sgn(n) sign of n E IR;

deg(~(n)) degree of the polynomial ~(~); lrxlI a r b i t r a r y vector norm, llXIlp = (Z~=11x IP) I / p , I s p ~ ~, Ix I = llx, 2, x E cm; iiAii = maxllXll=111Axll, Sp(A) set of the eigenvalues of A, spr(A) spectral radius of A, det(A) determinant of A, Re(A) = (A + AH)/2, A (m,m)-matrix;

the superscript T stands for 'transpose', the superscript H stands for 'conjugate transpose'; P ~ Q <=> xH(p - Q)x m 0 v x E {m <=> Re(P - Q) p o s i t i v semidefinit, P,Q (m,m)-matrices; cP(IR;Rm) set of p-times continuously d i f f e r e n t i a b l e functions f : IR ÷IRm, 111fllln = maxo~t~nAtlf(t) I, f : JR ÷JRm, n t I N , At > O; For symbols used only in Chapter Vl see Section 6.1.

Subject Index

A-function 98 algebraic function 157 c h a r a c t e r i s t i c polynomial 2, 35, 56, 66, 158 c r i t i c a l value 16 Daniel-Moore conjecture 31 degree of a Runge-Kutta method 118 d i s c r e t i z a t i o n error 2, 35, 46, 56, 60, 117 error constant 4 Frobenius matrix 6 Galerkin approximation 147 Galerkin procedure 143 growth parameter 12, 39 Hurwitz matrix 24 method A-stable, A(~)-stable 23 A(O)-stable, Ao-stable 23 asymptotically A(~)-stable 26 backward d i f f e r e n t i a t i o n 182 consistent 2, 35, 46, 56, 61, 117 convergent 12, 37 G-stable 96 I -stable 30 l~ap-frog 32 multistep m u l t i d e r i v a t i v e I , 35 of l i n e s 142 r-stage 115 s t i f f l y stable 29 strongly D-stable 29, 43, 58, 63 symmetric 40 zero-stable 12, 37 method o f , Calahan 182 , Cowell 68 • Cryer 182 , Enright 183 , Euler 92 , Houbolt 70 Milne-Simpson 9 , Numerov 68 , Nystr~m type 60, 62 Runge-Kutta 115 Runge-Kutta-Nystr~m 189 , St~rmer 68 midpoint rule 92 M~bius transformation 24 modal analysis 16 m u l t i p l i e r 106 order, of a method 2, 35, 46, 56, 61, 117 , of an i n t e g r a t i o n formula 118 ortogonal damping 16, 55 Pad~ approximant 20, 139 p e r i o d i c i t y i n t e r v a l 40 Puiseux diagram 165 r a m i f i c a t i o n index 161 region, of s t a b i l i t y 7, 37, 47, 57, 63 - , of ~-exponential s t a b i l i t y 9 - , of r e l a t i v e s t a b i l i t y 28 Ritz projection 143 -

-

,

,

,

root, unimodular 7 , p r i n c i p a l 12, 38 • essential 12 root locus curve 11 , modified 112 Routh-Hurwitz c r i t e r i o n 24 z-transformation 67 Sobolev norm 142 Sobolev space 142 spectral condition 8 trapezoidal rule 92 Tschebyscheff polynomial 32 Vandermonde matrix 173