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 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 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


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 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 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 __ ~. 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{- 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
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 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 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- 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 < . . . . __0, (4.3.21) yields

At2[~n-la eTA,,(C )e _ eTnA,(Cn)en]

n ->- T :>

or, if A"(t)

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 =

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 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

=

~

0

such that

< : 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 + O. Then for n = 1,2,..., 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 _- 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

~ 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

~ 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 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 _- 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 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 ~,


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 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) [ 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) ~