Symposium on the Theory of Numerical Analysis: Held in Dundee/Scotland, September 15-23, 1970 (Lecture Notes in Mathematics, 193) 3540054227, 9783540054221

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

193

Symposium on the Theory of Numerical Analysis Held in Dundee/Scotland, September 15-23, 1970

Edited by John LI. Morris, University of Dundee, Dundee/Scotland

Springer-Verlag Berlin' Heidelberq . NewYork 1971

AMS Subject Classifications (1970): 65M05, 65MlO, 65M15, 65M30, 65N05, 65NlO, 65N15, 65N20,65N25

ISBN 3-540-05422-7 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-05422-7 Springer-Verlag New York· Heidelberg· Berlin 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 70-155916. Printed in Germany.

Offsetdruck: Julius Beltz, Hemsbach

Foreword This publication by Springer Verlag represents the proceedings of a series of lectures given by four eminent Numerical Analysts, namely Professors Golub, Thomee, Wachspress and Widlund, at the University of Dundee between September 15th and September 23rd, 19700 The lectures marked the beginning of the British Science Research Council's sponsored Numerical Analysis Year which is being held at the University of Dundee from September 1970 to August 1971.

The aim of this year is to promote the theory

of numerical methods and in particular to upgrade the study of Numerical Analysis in British universities and technical colleges.

This is being effected by the

arranging of lecture courses and seminars which are being held in Dundee throughout the Year.

In addition to lecture courses research conferences are being

held to allow workers in touch with modern developments in the field of Numerical Analysis to hear and discuss the most recent research work in their field.

To

achieve these aims, some thirty four Numerical Analysts of international repute are visiting the University of Dundee during the Numerical Analysis Year.

The

complete project is financed by the Science Research Council, and we acknowledge with gratitude their generous support.

The present proceedings, contain a great

deal of theoretical work which has been developed over recent years. however new results contained within the notes.

There are

In particular the lectures pre-

sented by Professor Golub represent results recently obtained by him and his coworkers.

Consequently a detailed account of the methods outlined in Professor

Golub's lectures will appear in a forthcoming issue of the Journal of the Society for Industrial and Applied Mathematics (SIAM) Numerical Analysis, published jointly by Golub, Buzbee and Nielson. In the main the lecture notes have been provided by the authors and the proceedings have been produced from these original manuscripts. is the course of lectures given by Professor Golub.

The exception

These notes were taken at

the lectures by members of the staff and research students of the Department of Mathematics, the University of Dundee.

In this context it is a pleasure to ack-

nowledge the invaluable assistance provided to the editor by Dr. A. Watson, Mr.

IV R. Wait, Mr. K. Brodlie and Mr. G. McGuire. ,

I

Finally we owe thanks to Misses Y. Nedelec and F. Duncan Secretaries in the Mathematics Department for their patient typing and retyping of the manuscripts and notes.

J. Ll. Morris Dundee, January 1971

Contents G.Golub: Direct Methods for Solving Elliptic Difference Equations • . . . . . .

1

1. Introduction . . . . . 2. Matrix Decomposition. 3. Block Cyclic Reduction 4. Applications . . . . . . 5. The Buneman Algorithm and Variants 6. Accuracy of the Buneman Algorithms 7. Non-Rectangular Regions 8. Conclusion . . . . . . . . . . . . 9. References .

2 2 6 10 12 14 15

18 18

G.Golub: Matrix Methods in Mathematical Programming 1. Introduction • . • • . . . . • . • • • 2. Linear Programming . 3. A Stable Implementation of the Simplex Algorithm 4. Iterative Refinement of the Solution 5. Householder Triangularization . 6. Projections . . . . . . . . • . . . . . . . . 7. Linear Least-Squares Problem . 8. Least-Squares Problem with Linear Constraints Bibliography • . . . . . . . ..•.

21

· 22 22 24

28 28

31 33

35 37

V.Thomee: Topics in Stability Theory for Partial Difference Operators.

41

Preface . . . • • . . . . . .. . . . . . . • . 1. Introduction .....•.... 2. Initial-Value Problems in L Constant Coefficients . 3. Difference Approximations in L to Initial-Value Problems with Constant Coefficients • . . . . . • • . • . 4. Estimates in the Maximum-Norm . . . . . . . . . 5. On the Rate of Convergence of Difference Schemes References • . . . . . . . • . . . . . . . . . . E.L.Wachspress: Iteration Parameters in the Numerical Solution of Elliptic Problems . . . • . . . . . . . . . . . . . . . 1. A Concise Review of the General Topic and Background Theory • . . . . . . . . . . . . . . . . • . . 2. Successive Overrelaxation: Theory . 3. Successive Overrelaxation: Practice . . . . • 4. Residual Polynomials: Chebyshev Extrapolation: Theory 5. Residual Polynomials: Practice . 6. Alternating-Direction-Implicit Iteration . . • . . . . 7. Parameters for the Peaceman-Rachford Variant of Adi O.Widlund: Introduction to Finite Difference Approximations to Initial Value Problems for Partial Differential Equations 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . • . • . . . . . The Form of the Partial Differential Equations . The Form of the Finite Difference Schemes An Example of Divergence. The Maximum Principle The Choice of Norms and Stability Definitions Stability, Error Bounds and a Perturbation Theorem . .

42 43 51

59

70 79

89 93

95 98

.100 .102 .103 .106 .107 • 1 11 .112 .114 .117 • 121 .124 .133

'11

7. The von Neumann Condition, Dissipative and Multistep Schemes .

•

.

.

.

•

•

•

•

•

•

•

8. Semibounded Operators . . • . • 9. Some Applications of the Energy 10. Maximum Norm Convergence for L 2 References •• • . . . . . . •

•

•

•

• •

•

•

•

• . . . . . . • Method • . . • Stable Schemes • • . . • • . .

•

.138

• .142 .145 .149 . 1 51

Direct Methods for Solving Elliptic Difference Equations

GENE GOLUB Stanford University

2

1.

Introduction General methods exist for solving elliptic partial equations of general type

in general regions.

However, it is often the case that physical problems such as

those of plasma physios give rise to several elliptic equations which require to be solved many times.

It is not uncommon that the elliptio equations whioh arise re-

duce to Poisson's equation with differing right hand side.

For this reason it is

judicious to use direct methods which take advantage of this structure and whioh thereby yield fast and accurate techniques for solving the associated linear equations. Direct methods for solving such equations are attractive since in theor,y they yield the exact solution to the difference equation, whereas commonly used methods seek to approximate the solution by iterative procedures [12].

Hockney [8] has

devised an efficient direct method which uses the reduction process.

Also Buneman

[2] recently developed an efficient direct method for solving the reduced sy stem of equations.

Since these methods offer considerable

over older teoh-

niques [5], the purpose of this paper is to present a unified mathematical development and generalization of them.

Additional generalizations are given by

George [6]. 2.

Matrix Decomposition Consider the system of equations

where M is an NxN real symmetric matrix of block tridiagonal form,

A

T

T

A

M=

(2.2)

T

T

A

The matrices A and Tare pxp symmetric matrices and we assume that AT

= TA

•

3

This situation arises in many systems.

However, other direct methods which are

applicable for more general systems are less efficient to implement in this case. Moreover the classical methods require more computer storage than the methods to be discussed here whioh will require only the storage of the

Since A and T

cOllllllute and are syDlllletrio, it is well known [1] that there exists an orthogonal matrix Q such that '1'

Q T Q :: n , and A and n are real diagonal rnatrices. A and T, and A and

The matrix Q is the set of eigenvectClt's of

n are the diagonal matrices of the p-distinct eigenvalues of

A

and T, respectively. To conform with the matrix M, we write the vectors

!t

!z x =

•

• x

Furthermore, it is quite natural to write

Ylj

x x

Y2j

2j

::

::

..

•

• x

pj

ypj

System (2.2) may be written

Tx,4 1 + Ax"'J + Tx j +1 = Y ,

j ::

2,3, ••• ,q-l ,

and

in partitioned fora,

4

Tx

"'q

-s,

From Eq. (2.3) we have A

= Q A QT

and

T

=Q 0

QT • T

Substituting A and T into Eq. (2.5) and pre-multiplying by Q we obtain +

0:lSl! :::

It

•

O:lSj_l +

+ O:lSj+l

ox:"'q_ 1+1lx "'q

=V lI..

=l j '

(j

= 2.3 •••• ,q-l)

q

where

= 1,2, ••• ,q.

j

If

and lj are partitioned as before then the ith oomponents of Eq. (2.6) may be

rewritten as

= Yil '

AiXi l +

for i

=

+ Aix i j +

Yi j

(j = 2, ••• ,q-l) ,

-

+ Aixi q = Yi q •

= 1,2•••• ,p. If we rewrite the equatd ens by reversing the rolls of i and j we may write Ai Ai f

i

• •

=

•

•

wi

wi

Ai

Xu x

i2

• • x

iq

q x q Yi l

1\

Yi2

Yi q

5

so that Eq. (2.7) is equivalent to the block diagonal system of equations, ( i = 1,2, ••• ,p)

Thus, the vecter

•

a symmetric tridiagonal system of equations that has a

constant diagonal element and a constant super- and sub- diagonal element.

After Eq.

(2.8) has been solved block by block it is possible to solve for x j = Qx.. have:

Thus we

Algorithm 1 1. Compute er determine the eigensystem of A and T.

2. Compute lj

T

(j

1,2, ••• ,q).

3. Solve

= Q lj =

(i

= 1,2, ••• ,p).

4. Compute

..

(j

1,2, ••• ,q).

It should be neted that only Q, and the since

can overwrite the lj' the

s ,.

-J

j .. l,2, ••• ,q

have to be stored,

A

Y. and the x can overwrite can overwrite the "':l.

the A

simple oaloulation will show that approximately

2p2 q +

tors are required for the algorithm when step 3 is solved

5pq arithmetic opera-

using Gaussian elimina-

tien for a tridiagonal matrix when r

are positive definite. The arithemtic operai tors are dominated by the 2pZq multiplications arising from the matrix multiplioations

ot steps 2

+ 4.

It is not easy to reduce this number unless the matrix Q ha.

special properties (as in Poisson's equation) when the fast Fourier transform can be used (see Hockney

[8]).

For our system

and the eigenValues may be written down as v .. A + 2(d eel!! iri i q+l

r .. 1,2, ••• , q

6

er that

Vi the diagonal IIlll.trix of eigenvalues of

r i and zrs

o

sJ.·n !![ • S·J.nce q+l

s

r i and r

v

have the same set of eigenvectors

Because of this decomposition, step (3) can be solved by computing

where the Z is stored for each r i'

This therefore requires of the order of 2pqa

multiplications and this approximately doubles the computing time for the algorithm. Thus performing the fast Fourier transform method in step 3 as well as steps 2 and 4 is

3.

advisable. Blook Cyclic Redootien In Section 2, we gave a method for which one had to know the eigenvalues and

eigenvectors of some matrix. system of Eq.

We now give a more direct method for solving the

(2.1).

We assume again that

A

and Tare synunetric and that

A

and T commute.

Further-

more, we assume that q = m-l and

where k is some positive integer. +

Let us rewrite Eq. (2.5b) as follows:

= lj-l

+

'

Multiplying the first and third equation by T, the second equation by -A, and addi.n& we have

Thus if j is even, the new system of equations involves x.'s with even indices. Similar equa ti ODS hold for

and ;!5m-a'

The proc e ss of reducing the equations in

7

this fashion is known as oyolic reduction.

Then Eq. (2.1) may be written as the

following equivalent system:

o •

•

o • • •

and

= lm -

k l, Sinoe m = 2 + and the new system of Eq. (3.1) involves x.'s with even indioes, the k

block dimension Itf the new system of equations is 2 -1.

Note that once Eq. (3.1) is

solved, it is easy to solve for the Zj'S with odd indices as evidenced by Eq. (3.2). We shall refer to the system of Eq. (3.2) as the eliminated equations. Also, note that Algorithm 1 may be applied to System (3.1).

Since A and T

commute, the matrix (2T2_AZ) has the same set of eigenvectors as A and T.

h(A)

= hi'

h(T)

=

Also, i f

for i = 1,2, ••• ,m-l,

Hockney [8] has advooated this procedure. Sinoe System (3.1) is block tridiagonal and of the form of Eq. (2.2), we oan apply the reduotion repeatedly until we have one block. oan stop the prooess after any step and use the method

However, as noted above, we of Seotion 2 to solve the

8

resulting equations. To define the prooedure reoursive1y, let

A(o) ::: A, T(o)

(0)

::: T;;tj

::: ;tj'

(j ::: 1,2, ••• ,m-1).

Then for r ::: a,l, •• ,k A(r+1) ::: 2(T(r»2 _ (A(r»2, T(r+1) ::: (T(r »2 , (r-1) T(r) (r-) (r-) _ A(r) y(r) • ::: lj_2 r + lj+2 r -j lj The eliminated equations at eaoh stage are the solution of the diagonal system

(r-l) A

(r-1) _ T(r-1) ( _2r-1 ::: l.j2 r _2 r- 1

)

+

j ::: 1,2, ••• ,2

k-r

After all of the k steps, we must solve the system of equations

In either oase, we must solve Eq. (3.5) to find the eliminated unknowns, just as in Eq. (3.2).

If it is done by direot solution, an ill-oonditioned system may arise.

Furthermore A ::: A( 0 ) is tridiagonal A(i) is quindiagona1 simple structure of the original system.

and so on destroying the

Alternatively polynomial factorization

retains the simple structure of A. From Eq. (3.1), we note that A(l) is a polynomial of degree 2 in A and T.

By

r induotion, it is easy to show that A(r) is a polynomial of degree 2 in the matrioes A and T, so tilat

We shall proceed iI:> determine the linear factors of P2 r(A,T).

9

Let

For t

ft 0,

we make the substitution

aft '" -2 From Eq.

(3.3),

C08

e•

we note that

P2 r+ 1 ( a , t ) = 2t

2r + 1

- (P2 r (a,t ))2

It is then easy to verify using Eqi.

(3.7)

(3.8)

•

and

(3.8),

that

and, consequently P2r(a,t) = -

(a + 2t oos

v)

.1=1 and, heme,

A(r) '" -

n

.1=1 where

= (2.1-l)v/2 r + 1 •

Thus to solve the original recursively.

it is only

to solve the factored system

For example when r = 2, we obtain A(l) '" 2T2 _ A2 = (./2 T _ A)(./2 T + A)

whence the simple tridiagonal systems (./2 T - A) ! = (./2 T + A)

=

!

are used to solve the system '"

1. •

We call this method the oyclic odd-even reduction and faotorization (CORF) algorithm.

10

Example 1.

Poisson' 8 Equation with Diriohlet Boundary Conditt ons.

It is instructive to apply the results of Seotion 3 to the s:>lution of the

finite-difference approximation to Poisson's equation on a reotangle, R, with specified boundary values. u

xx

Consider the equation

= f(x,y) for (x,y)€R,

+ U

ss

= g(x,y) for (x,y)E8R •

u(x,y)

(Here 8R indio ates the boundary of R.)

We assume that the reader is familiar

with

the general teohnique of impesing a mesh of d.iscrete points onto R and approximating The equation u v

i-l,j

= fi,j

-2v

(llx

xx + uyy

= f(x,y)

.+v i+l,j

is approximated at (xi'YJ') by v

+

i,j-l

- 2v

i'1 (liy 2

+ v

i,J+l

i , n-l, 1 , j , m-l) ,

(1

with appropriate values taken on the boundary

and

Then v i j is an approximation to u(xi'Yj)' and fi,j

= f(Xi'Yj)'

gi,j

= g(xi,y j).

Hereafter, we assume that

When u(x,y) is specified on the boundary, we have the Dirichlet boundary condition.

For simplicity, we shall assume hereafter that tsx = liy.

-4

1

1

-4

A:

0

1

• 1

0

1

and T

= In_I'

-4

(n-l) x (n-l)

Then

11

The matrix I

1 indicates the identity matrix of order (n-l). A and Tare symmetrio nand commute, and, thus the results of Sections 2 and 3 are applicable. In addition, sinoe A is tridiagonal, the use of the factorization (3.10) is greatly simplified.

The nine-point difference formula for the same Poisson's equation can be treated similarly when -20

4

4 -20

o 4

A =

4

I

I

4

o

I

T=

0

4 4

-20

I

0 I

(n-l )x(,n-l)

4

Example II The method can also be used for Poisson's equation in reotangular regions under natural boundary conditions provided one use s

au u(x + Ax,Y) - u(x ax= 2h

Ax,Y)

and similarly :; , at the boundarie s , Example III Poisson's equation in a rectangle with doubly periodio boundary conditions is an additional example when the algorithm can be applied. Example IV The method can be extended successfully to three dimensions for Poisson's equation. Fer all the above examples the eigensystems are known and the fast Fourier transform can be applied.. Example V The equation of the ferm (a(x)u) xx +

yy + u(x,y)

= q(x,y)

on a rectangular region oan be solved by the CaRF algorithm provided the eigensystem is oaloulated sinoe this is not generally known.

12

The oounterparts in oylindrioal polar oo-ordinates oan also be solved using CORF on the raotangleSin the appropriate co-ordinates.

5.

The Buneman algorithm and variants In this seotion. we shall desoribe in detail the Buneman algorithm [2]

variation of it.

and a

The difference between the Buneman algorithm and the CORF algo-

rithm lies in the way the right hand side is calculated at each stage of the reduotion.

Henoeforth. we shall assume that in the system of Eqs (2.5) T .. I • the p

identity matrix of order p. Again oonsider the system of equations as given by Eqs. (2.5) with q

= 2k+l-1.

After one stage of cyclic reduction. we have

for j = 2.4••••• q-l with side of Eq.

(5.1) (1)

lj

where A(l) ..

Xo .. x

-

1 = 0 , the null veotor. -

Note that the ri8ht hand

may be written as .. lj-l + lj+l

(2I _ A2)

_

Av

_

-j{,j -

A(l) A-'

lj + lj-l + lj+l - 2A-'lj

•

P

Let us define

(These are easily oaloulated sinoe A is a tridiagonal matrix.)

After r reductions, we have by Eq. (r+l)

lj

«r) =

(r»

lj_2 r + lj+2 r

Then

(3.4) A(r)

(r)

-

Let us write in a fashion similar to Eq. (5.3) v(r) K.j

Substituting Eq.

2I p

=

A(r) nCr)

(5.5)

A(r+l) from Eq.

"'j

+ n(r) "j.

into Eq.

(3.4),

(5.4)

and making use of the identit,y (A(r»2 ..

we have the following relationships:

13

(r+l) (r) (r) 2 (r+l) . E

d 'X >

J.

inversion formula




'x J

Further, for u

J u(f)

\\ u \\

e

=

t:f

J

and aa a oonsequenoe of the latter, the set

L

00

:?X·""?· . J J" \

we have Fourier's

d"1

\I\A\\)

eo

/\."'"

transforms in

•

of functions in

1

with Fourier

is dense in L2

In the sequel we shall oonsider N-vector valued funotions u(X)::(U1(X), ••• It is clearly natural to define u(x) E holds for each oomponent u ' j = 1, ••• ,N. j

to N-vectors e.g.

and for Nl

Lo....

etc. by demanding that this

Single bars will denote norms with respect

53

and double bars will indicate norms with respeot to L2 , so that for the N-veotor

II u \\ For later use we need the following Lemma 1

Let

Jr1-

Then

Be a dense subset of La and let a( 1 ) be a oontinuous NxN matrix.

110. V II

\IV\\

Let u(x, t) be an N-vector-function defined for x initial-value problem 0

=

>

0 and a C such that

)

'" We then have that i f (1) is parabolic in Petrovskii's sense, the corresponding

initial-value problem is correctly posed in L2 •

For by Lemma 2 we have for

T, \ et-p

-s

(1:

which is clearly bounded.

C

l\ +

l+\1\M)N-1

(6)

In particular, the heat equation "d'2\J\

C>

X

".L.

olearly falls into this oategory. Solutions of parabo1io

are smooth for t

>

0; we have

Theorem 2 Assume that (1) is parabolic in Petrovskil's sense.

Then for t

0,

0(

D E( t)v E L2 for any

cJ.

and for ars T '> 0 and any 0\ there is a C such that \ ... \

\\ tl00

let [t] be its integral part.

\e"r(t.

We have.

D:l

e c

\ -s

which proves the lemma. Combining Lemmas 3 and 4 we have at once: Theorem 4

If (1). (2) is correctly posed in L2 then there is a oonstant

such

that

(n) satisfies the oonditions of Lemma 3. suoh that

On the other hand i f there is a constant

satisfies at least one of the conditions of Lemma

oorrectly posed in L2.

¥

3. then (1), (2). is

59

One oommonly used criterion iSI Theorem 5 only if

Let p(;t) be a normal matrix.

(10)

Then

(1), (2)

is correotly posed i f and

holds. Sinoe p(

By Theorem 3 we only have to prove the suffioienoy.

can find a unitary

is diagonal.

uCl )

is normal we

such that

Henoe

whioh proves the result. For later use we statel Theorem 6

If

(1), (2)

posed in L2 then

is

tive oonstant Ct and C2 and for each

1E

(10) holds

and there are posi-

d R a positive definite hermitian matrix

such tmt

(12) and

(13 ) f!:2.2!

By Theorem 4 there is a oonstant

oondition (iv) of Lemma 3 with C = Ct.

II

such that the family.r

Thus for each

!

f:

in (11) satisfies

d R there is a positive

defini te H(,t ) such that

RQ. ( But by

3.

(12)

n-)' I))

UCr)( PL

this implies

0

(13).

Differenoe approximations in L2 to initi.al-value problems 'Iii th constant ooefficients Consider again the initial-value problem

(1)

uh.)o) .., v(x)

(2)

60

For the approximate solution of (1), (2) we oonsider explioit differenoe operators of the form

=

where h is a small positive parameter,

•••

with

integer,

are

NxN matrioes whioh are polynomials in h, and the summation is over a finite set of

We introduoe the symbol of the operator

::: whioh is periodio with period 2JJ /h in transform of

j and notioe that for v

j ,

is

Assume that the initial-value problem (1), (2) is oorreotly posed. to ohoose

the Fourier

We then want

so that it approximates the solution operator E(k) when k is a positive

parameter tied to h by the relation = oonstant;

we aotually want to approximate u(x,nk) = E(nk)v = E(k)nv by shall emphasise the dependence on k rather than h and write To aocomplish this, we shall assume that E

k

following; definition.

We say that

In the future we as in Lecture 1.

satisfies the condition in the

is oonsistent with (1) i f for any solution of

(0"':>

(1) in

\...

-'

If O(k) oan be replaoed by kC(hl"), we say that

is acourate of order

r.

Clearly

any oonsistent soheme is aoourate of order at least 1. We can express oonsistency and aocuraoy in terms of the symbol (of. [35]): Lemma 1

E" The

is oonsistent with (1) if and only if

The operator

n

-=

url

K

+

61

The proof of

(3),

say, oonsists in proving like in the special case in Leoture

1 that oonsistenoy is equivalent to a number of algebraic conditions for the coeffi-

cients, whioh turn out to be equivalent to the analytio functions exp(kP(h- 1 Ek(h- l

)

having the same coefficients for h

j

Zol..

))

and

up to a oertain order.

Using Lemma 1 i t is easy to dedoo e that if \: is oonsistent with (1) in the present sense then we also have consistency in the sense of Lecture 1. G

For the set

of genuine solutions in the previous definition we can for instance take the ones

eo.

/'""

oorrespondinc to v E:

From Lax's equivalence theorem it is clear that we want

to discuss the stability of operators Theorem 1

of the form described.

We have

1he operator \: is stable i f and only if for any T :> 0,

\ c We notioe that

E1c( t)n

E

:>

Rc!

)

is the symbol of

It follows in the same way as

in Lecture 2 that

1\

whioh pr-ovea

l\

\\

)

" the theorem.

We now turn to the algebraic oharaoterization of stability.

We first prove

For any NxN matrix A we denote byf (A)

the neoessity of the von NeUIIB.nn condd.td.on,

its spectral radius, the maximum of the moduli of the eigenvalues of A. Theorem 2

Z If \: is stable in L

f' We have for nk

,

there exists a oonstant '(

such that

'ZE:

'-

(4)

1,

and so

\"\-

K )

It is easy to prove by counter-examples that (4) is not sufficient for Necessary and sufficient conditions for stability have been given by Kreiss (18) and Buchanan [5]; we quote here Kreiss' result. concentrated in the following Lemma.

The main content in Kreiss' theorem is

Here we have introduced

notation: For H hermitian and positive definite, we introduce

the following

62

l \-\

:=

\ \k \

1-\

1 t\ \ H

IA , "')

IAu.\'-I

s« ?

-=.

Y"l. ,

\ \.1\ Ii

Reoall again that for hermitian matrices, A

B means (Au,u)

Lemma 2

Then the following four oonditions are

Let

be a family of NxN matrices.

(Bu,u).

equivalent.

(i)

sup

(ii) sup

t \ f\ "" \ ..; A ) t (\ \- \')\R (A ; z) l ;

(iii) For A .)(e,\J, 'J) where

e,. ':.'I then

65

l€.

2

\J)

\J

-z l

J3 'N -' \N)

":: \

\J ,;z. 'W \

-'

]'A and henoe with w

? )v,

=

\ £1 \/) \

\\J\.

\w\ -\

or

.

C\'"2.\ - \) \W\

)

which proves the result. One can aotually prove that

\Anl

s 2,

A E.

F

This result oan be used to prove the stability of certain generalizations of the Lax-Wendroff operator to two dimensions (see [24]). Consider again the symmetric hyperbolio system (6) and a differenoe operator of the form (7), oonsistent with (6).

Then

) =

is independent of h.

We say with Kreiss that E is dissipative or order \) ('\) even) if there is a k such that

\-

g,> 0

-v

\- N, y" -N leH + ( 1'\ l\

matrix A with spectral radius

\1\(\

I

(II

Proof of Theorem 10

.

.l .

By consistency we have

E (h -i "$ ) _ "I

+ 0 (K T \ '1 \K)

0.

K

We therefore have for n

\ t

;

)

T, with

«-x. '7> -

...

P (t ? ("1 + IN))) € ,

t

)

=

'«-t l·1j)

I'll

/ N-\

-=-

J

cl

•

\""\

+M

C e- 0,

any

0( ,

is a:>nsistent with (12) and

e

and any v G.

we have for nk

= t,

as the operator approximating Do( is again rather arbit-

One can indeed show that the same result as in Theorems 4 and 5 hold for any

differenoe operators oonsistent with D "'. Theorems 3, 4 and 5 generalize to variable coeffioients.

5.

On the rate of convergenoe of differenoe schemes

In this leoture we shall work in a slightly more general setting than before. Let LP

= LP(Rd ) ,

p

1




J

p=

aD

oD ,

-'

\y

Sf."

Consider also the Sobolev spaoes


•

and i t follows that

Yf.>

S' One can prove that B C BS2 if S, p p .

s+6 B p

s s c;: WeB •

>

S2, and tm t f or integer s and

E. > 0

.

The maan property of these spaces that we will need is P P then the following interpolation property: Assume that 1 P (J). m is a natural number, and

8

is

a real number with 0 < s < m.

Then there is a constant C such

that any bounded linear operator A in W with p ""

1\

\' C\

P\ \J \ \

l

p

\\\Jllv-I. \>

C'1 \\ \J \\

we have

,-'U

II R V \t w\>

eel

Theorem 2 and (7) with A Theorem 3 posed in

f' . nk

W p

=

c

vJ;

\I V

II

prove immediately the following result:

Assume that the initial-value problem (2), (3) is strongly correctly and that R

Then for 0

is stable in

0

consistent with

(2),

and accurate of order

there is a constant C such that for any v EB

T,

II (1::1'. -,=If\\0

with order of accuracy

83

and let v =

¢ E;

where

e;

and

"'Iv

is the Heavyside function (0).

By above

we have in this case

For dissipative operators

stronger results have been obtained in Apelkrans [1],

and Brenner and Thom;e [4], where also the spreading of discontinuities is discussed. It is natural to ask i f for a parabolic system, the smoothing property of the solution operator can be used to reduce the regularit,y demands on the initial data in Theorems 2 and

3.

This is indeed the case.

Before we state the result we give

the following result, which follows easily from properties of fundamental solutions. v

Theorem 4 1

Assume that (1) is parabolic in Petrovskii I s sense.

p

eo , any. m

>, g

Then for axry p with

and T > 0 there is a constant C such that

C -t

--

\\

_

\L\ '1'1 \> l

0

,"*

-;S

T

J

We can now state and prove the result about the rate of convergence in the parabolic case. Theorem 5

Assume that (2) is parabolic in Petrovskii's sense and that E is stable k

in W oonsistent with (2) and accurate of order p' S

there is a constant C such that for v E B , nk P

fA •

Then for any e

For details, see [27].

v G B; where

t" D

we obtain the same result as i f

formulate this result in terms of the Banach space

va EM

were stable in

e.

We shall

of functions v with support in

[-M,O] such that

is finite. Va

Bj,(

Using the 1 2 convergence result, Sobo1ev's inequality, and the fact that

is continuously embedded in B 2

one can prove the following result [37].

88

Theorem 10

Consider a L2 stable operator

positive M and s

i P:

+1 and for

.

\\ E"" - Ell\K)\I \\ K

for the equation (5).

T,

e

This result also holds when

C \\ depends upon x,

Then for given

. ))

II

u

M

89

REFERENCES [1]

M.Y.T. Apelkrans.

On difference schemes for hyperbolic equations with dis-

continuous initial values. Math. Compo 22 (1968), 525-539. [2]

Ph. Bremer.

The Cauchy problem for symmetric hyperbolic systems in Lp• Math. Scand. 19 (1966), 27-37.

Ph. Bremer and V. Thome'e. difference schemes.

,-

[5]

p

Math. Scand ,

Ph. Bremer and V. Thomee. schemes.

Stability and convergence rates in L for certain To appear.

Estimates near discontinuities for some difference

To appear.

M.L. Buchanan.

A necessary and sufficient condition for stability of diffe-

rence schemes for intial-value problems.

J.Soc.lndust.Appl.Math. 11

(1963), 919-935. " Uber die partiellen Differenzen-

R. Courant, K. Friedrichs and H. Lewy.

gleichungen der mathematischen Physik. Math. Arm. 100 (1928), 32-74. A. Friedman.

Partial differential equations of parabolic type.

Prentice-Hall. Englewood Cliffs, New Jersey, 1964.

[8]

K. Friedrichs.

Symmetric hyperbolic linear differential equations.

Comm. Pure Appl. Math. 7 (1954), 345-392. I.M. Gelfand and G.E. Schilow.

Vera11gemeinerte Funktionen III.

Deutscher Verlag der Wissenschaften, Berlin, 1964.

[10]

S.K. GodunoT and V.S. Ryabenkii. schemes.

[:a]

Introduction to the theory of difference

Interscience. New York, 1964.

G.W. Hedstrom.

The

of the Lax-Wendroff method.

Nume r, Math. 7 (1965), 73-77. G.W. Hedstrom.

Norms of powers of absolutely convergent Fourier series.

Michigan Math. J. 13 (1966), 393-416. G.W. Hedstrom.

The rate of convergence of some difference schemes.

SIAM J. Numer. Anal. 5 (1968), 363-406. G.W. Hedstrom.

The rate of convergence of difference schemes with constant

coefficients. BIT 9 (1969), 1-17. F. John.

On integration of parabolic equations by difference methods. Pure App1. Math. 5 (1952), 155-211.

H.O. Kreiss.

"

II

Uber Matrizen die beschrankte Ha.Ibgr'uppea erzeugen.

Math. Scand. 7 (1959), 71-80.

90

[17]

H.O. Kreiss. Uber die Losung des Cauchy problems fur lineare partielle Differentialgleichungen mit Hilfe von Differenzengleichungen. Acta Math. 101 (1959), 179-199.

[18]

H.O. Kreiss.

fUr

&ber die

Differenzengleiehungen

die partielle Differentialgleichungen approximieren. BIT 2(1962), 153-181. [19]

H.O. Kreiss. 109-128.

Uber sachgemasse Cauchyprobleme.

Math. Scand. 13 (1963),

[20]

H.O. Kreiss.

On difference approximations of dissipative type for hyper-

bolic differential equations. Comm, Pure Appl. Math. 17(1964), 335-353. [21]

H.O. Kreiss, V. Thom6'e and O.B. Widlund. Smoothing of initial data and rates of convergence for parabolic difference equations. Comn. Pure AppI , Math.

[22]

P.D. Lax and R.D.

Survey of the stability of linear finite

difference equations. [23]

To appear.

P.D. Lax and B. Wendroff.

Conm , Pure Appl. Math. 9 (1956), 267-293. Systems of conservation laws.

Comni,

Pure App l.,

Math. 13 (1960), 217-237. [24]

P.D. Lax and B. Wendroff. Difference schemes for hyperbolic equations with high order or accuracy. Comn. Pure Appl. Math. 17 (1964), 381-398.

[25]

J.

Besov spaces in theory of approximation.

Ann. Math. Pure Appl. 85 (1970), 93-184. [26]

G.G. O'Brien, M.A. Hyman and S. Kaplan.

A study of the numerical solution

of partial differential equations. J. Math. and Phys. 29(1951), 223-251. [27]

J. Peetre and V. Thomee.

value problema. R.D.

On the rate of convergence for disorete initialMath. Scand. 21 (1967), 159-176.

and K.W. Morton.

problems. V.S.

Difference methods for initial-value

2nd ed., Intersoience, New York, 1967. and A.F. Fillipow.

Uber

die Stabilitat von Differenzen-

gleichungen. Deutscher Verlag der Wissenschaften, Berlin, 1960. S.I. Serdjukova.

A study of stability of explicit schemes with constant , , real coefficients. Z. Vycisl. Mat. i Mat. Fiz. 3 (1963), 365-370.

S.I. Serdjukova.

On the stability in C of linear difference schemes with

,

,

constant real coefficients. Z. Vycisl. Mat i Mat. Fiz. 6(1966), 477-486. W.G. Strang. Polynomial approximation of Bernstein type. Trans. Amer. Math. Soc. 105 (1962), 525-535 • .-

V. Thomee. Stability of difference schemes in the maximum-norm. J. Differential Equations 1 (1965), 273-292.

91 ,-

V. Thomee.

On maximum-norm stable difference operators. Numerical Solution

of Partial Differential Equations (Pro c; Sympos. Univ. Maryland, 1965), pp. 125-151.

[35]

v.

[36]

V.

Academic Press. New York.

Parabolic difference operators. Math. Scand, 19 (1966), Tl-107.

Thome-e.

Stability theory for partial difference operators.

SIAM Rev. 11 (1969), 152-195.

[37]

V. Thomee.

On the rate of convergence of difference schemes for hyperbolio

equations.

Numerical Solution of Partial Differential Equations. (Proo.

Sympos. Univ. Maryland, 1970).

[38]

O.B. Widlund.

To appear.

On the stability of parabolic difference schemes.

Math. Compo 19 (1965), 1-13. [39]

O.B. Widlund. norm.

[40]

Stability of parabolic difference schemes in the maximum-

Numer. Math. 8 (1966), 186-202.

O.B. Widlund.

On the rate of convergence for parabolic difference schemes,

II. Comm. Pure Appl. Math. 23 (1970), 79-96.

Iteration Parameters in the Numerical Solution of Elliptic Problems EUGENE L. WACHSPRESS General Blectric Company Schenectady, New York

94

These notes are intended to serve

all a

guide to a deeper stud,y of material

presented in a series of lecturell delivered in September, 1970 at the University of Dundee as a part of the special one year sympollium on The Theory of Numerical Analysis.

Subject

Lecture 1

A Concise Review of the General Topic and Background Theory

2

Successive Overrelaxation: Theory

3

Successive Overrelaxation: Practice

4

Residual Polynomials and Chebyshev Extrapolation: Theory

5

Residual Polynomials: Practice

6

Alternating-Direction-Implicit Iteration: Theory

7

Parameters for the Peaceman-Rachford Variant of ADI

Referenoe text: "Iterative Solution of Elliptio Systems," by Wachspress (Prentice

Hall, 1966).

95

1.

A CONCISE REVIEW OF THE lrENERAL TOPIC AND BACKGROUND THEORY We are concerned with iterative approximation to the vector

which satisfies

the system of linear equations (1)

Ax where:

1s

is a known m-vector

A is a given nonsingular mxm matrix. and is an m-vector which is to be found. An approximation

to

is acceptable when

II

II

II

where E is some prescribed error bound and

(2)

< E.

II

II • II

a designated norm.

We shall first oategorize various iteration procedures. cribe a measure of efficiency or rate of convergence.

We shall then des-

Having done this. we will

indioate a rather general approach to demonstrating convergence for a wide class of methods for iterative solution of linear systems.

Finally. an example of each of

three major classes of linear iteration procedures will be given. It is convenient to restrict matrix A in (1) to be real. symmetric. and positive definite.

(Our definition of positive definite is such that a real matrix

which is p.d. must be symmetric.)

Although less restrictive conditions are subject

to analysis. the discussion is greatly simplified in this manner. A STATIONARY LINEAR ITERATION procedure is characterized by: a known "trial" vector T

This procedure is convergent iff In order that

Defining the error vector!.n =.!n !.n -+

1s.

-+

= A- 1 1s for any

n = 1.2 •••••

and

1s.

be a stationary point. we require:

= T .!

'!n

+ R

+ R

+

o.

(4)

and subtracting (4) from (3). we get

= T ,!n-l = Tn ,!o

Q for arbitrary ,!o iff Tn

1s • •

96

The spectral Tadius of T is r(T) :: max/gi(T)/ where the ii are ei,envalues of T.

Thus, r(T) must be less than unity for oonvergence.

so that (I-T)A- 1

for any

R

::

From (4),

(3) oan be written

R, and

(6)

:: T !.n-l +

on the right hand side of (6), we would have no need for

If we oould oompute

Thus, T must be such that the right hand side of (6) does not require

iteration.

oomputation of A-I or B! ::

::

To olarify this, suppose we can solve the system

for ! where B approximates A in some sense. B !n .. (B-A) !n-l +

We m8¥ attempt to iterate by:

, or

(I_B- 1A) !n-t +

!n Here, T :: I_B- 1A and (I_T)A- 1

..

B- t :: R.

We note that B is a "iood" approximation to A for this iteration when the speotral radius of (I_B- 1 A) is much less than unity. We will now derive a oondition sufficient to assure convergence whioh has application to many iteration techniques. A is positive definite and henoe has a unique positive-definite square root, 1

1

Thus, (I_B-tA) is similar to

1

whose spectral radius is bounded

above by the square root of the largest eigenvalue of (I_AiBT-1Ai) The

::

(7)

I _ AiB-t(B+BT_A)BT-1Ai •

T condition sufficient for oonvergenoe is that B + B - A be positive definite.

For then we can define K ::

T

1

T

1

1

and M ::

1

and rewrite (7) as

T

MM :: I-KK ; we note here that K is nonsingular, being the produot of nonsingular matrices.

Thus

0

r(I_B- 1A) < 1 i f B+B

T

T)

g(W ) :: g(I_KK

T

T) :: 1 - g(KK < I, and we have proved that

- A is positive definite.

To oonvince ourselves that this is

not a neoessary oondition, we need only find one oounterexample: A

T.

and B :: 2

::

:]

is not positive definite.

with r(T) ::

i

- 18

yield

[ 39

while B+B

T

- A

97

liT II ;. r( T) II s II T lin. II.!.o II;' r n II

If we choo se our norm so that

II,!n II:: II • n::

log E 10i r

•

Tn.!.o

and i f

II

= Q. then .!.O =

II.!.n II II !oIl

so that

and

< E for

We note that when r is close to unity the number of iterations

required to satisfy a prescribed oonvergence criterion varies as l/(l-r). A PARTIALLY STATIONARY LINEAR

prooedure is one for whioh the approximation

is obtained as a linear combination of vectors which could be obtained by a stationary procedure.

(j :: 1.2••••• n) be iterates generated by

Let

(6).

Then the n-th

iterate of a partially stationary procedure based upon this stationary iteration would be If we define !n = 1.n -:I. :: :Ln -

then

!n = Pn(T) .!.O , where Pn(T) is a polynomial of degree n in T. normalized to unity when T=I.

(I is the identity matrix of order the same as T.)

Converienoe is established by showing that for a given Pn and T: 1

r :: limit n ...

[sup IPn(gi)

ex>

In

J

< 1

where the ii are the eigenvalues of matrix T. A NONSTATIONARY LINEAR i terati ve prooedure is one for whioh the iteration matrix is a funotion of parameters whioh may change from iteration to iteration: +

spectral radius of

n

n T

j=1

j

•

(I

n

A- 1 !£ . If r n is the n T) j

- j=1

then the asymptotic convergence rate is limit n ...

00

rn

1/n

When the T all oommute, analysis is quite similar to that applied to partially j

stationary schemes.

When the T do not oommute convergenoe theory is often less

J

definitive. In examining relative merits of iteration procedures. we endeavor first to

establish oonvergenoe for a range of iteration parameters. seoond to determine the speotral radius as a function of these parameters. and third to ascertain how theBe parameters may be chosen to minimise this speotral radius. arises in the analysis of eaoh iterative procedure. will be considered in subsequent lectures.

Thus. a minimax problem

Three commonly used techniques

98

These are: I. Successive Overrelaxation II. Chebyshev Extrapolation (partially stationary) III. Alternating-Direction-Implicit Iteration Each of these is well documented in the literature and these notes are intended only as an introduction to the subject rather than a detailed analysis. 2.

SUCCESSIVE OVERRELAXATION:

THEORY

We may "improve" the value of a componen t of the vector! by computing as a new value during iteration n that number which yields satisfaction of the p-th equation with values from iteration n-l substituted in the equation for the remaining components of !: k

x

p

j

from iteration n-l for j n for j

An iteration consists in improvement of all components of!.

"Relaxation" or "simultaneous relaxation".

p

p.

We may call this

On an array oomputer with many arithmetic

units working in parallel, it would be possible to improve all components simultaneously. We may, alternatively, use new neighbour values as soon as they are computed. Then

a

x pj jn

k

P

p

= 1,2, ••• ,m.

Components are now improved in some order, and we call this "successive relaxation". Although this latter approach is often better than simultaneous relaxation, a more significant gain in efficiency is usually achieved by extrapolation.

If we denote

the unextrapolated result of successive relaxation at point p by x*pn , then before proceeding to the next point during the n-th iteration, we compute for a prescribed extrapolation plrameter w: x

pn

= x

pn-1

+

w(x* - x ) pn pn-1

(8)

Numerical solution of elliptic type difference equations is accomplished with w in (:+,2) and since w is greater than unity this is called "successive overrelaxation." Among factors responsible for extensive literature on SOR are its simplioity,

99

wide applicability, and firm theoretical foundation. D.M. Young's analysis provided a basis for efficient utilization of SOR.

He

introduced the concept of a "consistent ordering" which is related to what is now lmown as Youni's Property A.

The equations of a system having this property may be

ordered as follows: An index s(p) is assigned to unknown x

p

associated with the p-th equation.

Then for every nonzero a ., the ordering s(p) is said to be consistent if PJ

S(j)

s(p)-l for j < p

a(j)

s(p)+l for j > p.

Components of .! are improved in order of increasing s.

Points (equations) with a

common s-value which are coupled to one another directly must be updated simultaneously.

Many systems arising in practice may be consistently ordered without a need

for simultaneous improvement even though several components may have common a-values. Five-point difference stars are an example. Young established an intimate relationship between eigensolutions of the simultaneous relaxation and consistently ordered SOR iteration matrices derived therefrom.

If g is an eigenvalue of the SR matrix, then there is a correspondin£

eigenvalue h of the SOR matrix satisfying: h

=

2

(w - 1)

+

The optimum extrapolatien parameter w is obtained by solving the minimax b problem: = maximum /h(w,g)/

/g/ ,

< 1

= minimum H(w,G). w

The solution to this minimax problem (whioh is a rather simple minimax problem) is

=

2

1 + ./l_G

2

(10 )

and

The remarkable gain in efficiency of SOR over SR is evidenced for the case G=l-r for r « 1 by oomparing the relative number of iterations required by the two methods for an error reductions of E: n SR = -In E/r while n

SOR =

-In E

2.fir

(n )

100

The potenoy of the oonvergenoe theorem given in the first lecture is illustrated by applying it to the SOR iteration matrix. T

A = D- R - R

1et the coeffioient matrix be

where D is diagonal (and positive since A is positive-definite by

hYpothesis) and R is strictly lower triangular.

If SOR is applied with the natural

ordering (successive updating of oomponents 1,2, ••• ) then the SOR iteration matrix is This may be written in the alternative form

1

=I

- (Q - R)-t A.

Thus, B in equation (7) is equal to

w

Q w

R in this oase, and

B + BT _ A = 2D/w _ R _ RT _ D + R + RT = (2/w - 1) D. The speotral radius of 1 is less than unity when 2/w - 1 is greater than zero, or when w is in (0,2). It oan also be shown by this approaoh that SOR oonverges even when the ordering and the extrapolation parameter are ohanged eaoh iteration. When one digs deeper into the theory, one finds that the SOR iteration matrix with optimum parameter and oonsistent ordering does not have a diagonal Jordan form. The resulting eigenveotor deficienoy has an adverse effeot on convergenoe. 3.

SUCCESSIVE OVERRELAXATION:

PRACTICE

For effio:ie nt implementation of SOR, one must choose an appropria te ordering and estimate the optimum parameter w One must also ohoose a strategy consistent b' with the charaoteristios of the computer for whioh the iteration program is designed. This latter point is sometimes overlooked.

One illustration is that on the CDc-6600

there is a staok feature whioh leads to a gain in speed by a faotor of ten in the basio arithmetio when one programs the "inner arithmetio Lcop" in maohine language, taking full advantage of the staok feature rather than relying on FORTRAN.

Another

consideration is relative effioienoy of getting data in and out of fast memory and of oomputation onoe the data is in memory. be

On some maohines several iterations oan

performed in the time it takes to read the data in and out of memory.

The method

of oonourrent iterations enables one to perform several iterations with one pass over the equations.

This is partioularly important when solving large problems where all

101

the data oannot be oontained in memory. Periodio boundary oonditions present minor diffioulties.

Line relaxation is

benefioia1, and the proper ohoioe of lines enables retention of oonsistent ordering. (I often think in terms of problems arising from disoretization of partial differential equations to obtain five­point or seven­point differenoe stars.) A major oonsideration in any event is choioe of w.

As the spectral radius of

SR approaches unity, it becomes increasingly more important to choose a good w. Several elaborate techniques have been described for estimating the extrapolation parameter.

However, a reasonably effective procedure which I have had success with

for many years does not require any sophistioated additional programming. One starts with a parameter, wo, chosen deliberately smaller than w and iterb a.tes until an asymptotic convergence rate is established. This convergence rate is measured by oomparing suocessive changes in the vector:

- 2£n­1 II II2£n­. ­2£n­zll For Wo


1. The Pj and qj approach asymptotic values of order magnitude unity all b approaches unity, roundoff error is no longer a serious problem, and we need no longer deoide in advance upon the number of iterationll. There are other polynomial extrapolation prooedures in ooumon use.

One family

of iteration sohemes based on Lanczos' work involves oomputation of the Pj and qj from oertain inner products of the x

-j

and x" veotors. -j

These include the steepest

desoent and conjugate gradient methods. 5.

RESIDUAL POLYNOMIALS: PRACTICE

The eigenvalue interval (a,b) upon whioh the extrapolation parameters are based may be estimated from observed convergence with assumed bounds.

Oscillatory

error behavior indicates that the assumed lower bound is too large while a uniformly signed error indicates the upper bound is too low (for the case where the assumed bounds lie within the true bounds).

By noting the sign of the error and the rate of

change, one can update the eigenvalue interval. Although it is possible to start a new parameter cyole after eaoh updating, one may use the asymptotio values for Pj and qj iumediately.

This has proven to be

qUite satisfaotory for many problems. Systems which satisfy Young's Property A should be treated by the method whioh reduoes the arithmetio by a factor of two. erence text.)

(See Pp. 155-6 in the ref-

104

The Chebyshev and SOR methods are oomparable in efficiency.

The Chebyshev

method does often yield a greater reduction in error norm for a given number of iterations, but other factors often outweigh this.

Computer and problem oharacter-

istics often dictate which approach is better. We lIBy examine more precisely means by which the interval [a,b] may be estimated.

Although I have had no oocasion to use the procedure which will now be

described, thus making this discussion more "theory" than "practice", the means by whioh parameters are updated is one of the more practical aspects of iteration and thus falls appropriately in this section. When Chebyshev extrapolation is based upon an assumed eigenvalue interval [a',b'] which oontains the true interval [a,b], the asymptotic convergenoe rate is

r -lim t ..

C t

00

where Il

e

(J:.) Il

e

::

I

+.fl=7e

lIE

When [a,b] is not contained in[a',b'], error oomponents assooiated with eigenvalues outside [a,b] () [a' ,b'] will eventually predominate. either a' " a or b'

If it is known that

b , we seek only one bound and the procedure is analogous to

that already desoribed for SORe

We shall describe a more sophisticated approach for

estimating both a and b. Mter sufficiently maDY iterations, s , we suppose that suocessive iterates satisfy and define for k :: 1,2,3,

t t + Al !.1(S) + Az !.z(s) , I

It is easily mown that

.!1 -

t = 1,2,3, •••

105

We may ascertain values for CI and (3 which minimize II !. (CI,(3) 11 2

.,!2) (!.J.!.J) - (!1.!.J) (,!2.!.J)

=

CIa

(30 =

and

(.!.e.!.J) - (,!, •.!,3)_ (,!2•.!.e) (,!2.,!2) (!.J.!.J) - (!.a•.!,3)a

We may for example, compute

CIa

and (30 after ever,y ten iterations

(8 = 10,20,30, ••• ) until values are obtained whioh do not change appreoiably with 8. Having determined Clo and (30 we may e stlmate A, and All by using the relationship :

1

'i:, Thus

+

ao +

=

1

'i:a

= ao

JClg 2(30

4Bo

and

=

ag - Jal - 4/3g 2(30

are the esti.Da ted values for A. Referring to the Chebyshev polynomials with eigenvalue Z of the basic iteration (SR) matrix, we have cosh ,(2Z-(a'+b')] oosh [( s+t-l ) ooshb'-a' Now let x = 2Z - (a'+b') b'-a' Then

A ;, r (x +

JT:l.).

and we have

The estimates for a and b are obtained by substitution in the above equation:

and

b =

i

[(a'+D')

a =

i

[(a'+b') +

2

(!:

+ h)] r

if

A,

(!:

+ Aa)] r

if

Aa,,-r

A,

2

Aa

r,

This analysill ill intended primarily as a guide to further study of methods for estimating extrapolation parameters. speoific problems.

Numerical pro cedures should be molded to suit

We have indioated how a oomparison of observed oonvergenoe with

theoretical oonvergence provides a means for

updating parameters.

106

6.

ALTERNATINlO-DlRECTION-IMPLICIT ITERATION

A family of non-stationary procedures i8 generated by splitting A into the sum of two matrices so that a two step procedure of the following type is formed: A = H + V

= -(V - wl):f j_1 +

(H +

(V

+ ZjI):fj =

-(H -

+ j =

1,2, •••

The matHces H and V are ohosen so that this iteration is numerioally convenient.

For example, five-point differenoe

equations can be handled when H

inoludes horizontal (along lines of oonstant y-value) ooupling while V includes all vertical (along lines of constant x-value) ooupling.

Then (23) involves a correo-

tion of eaoh horizontal line treated as a block followed by oorreotion of eaoh vertioal line.

When H and V are both positive-definite and all the w and Zj are j

equal to a single positive oonstant, say w, convergenoe is easily demonstrated:

= Tn

where T

= (V+wI)-t(H-wI)(H+wI)-t(V-wI).

T is similar to T' = K(H) • K(V) where K(X) .. (X-wI)(X+wI)-t has a speotral norm less than unity for a:ny positive-definite X.

Henoe, the spectral radius of T' is

less than unity. Convergenoe is not assured without further oonditions when wj and with j.

Zj

vary

theorem asserts that by using a large enough number of parameters

in a monotonically nonincreasing sequenoe (wi thin the interval of the eigenvalues of H and V) one oan obtain a spectral radius less than unity.

Repetitive appli-

oation of such a parameter oyole is oonvergent. The theory for this prooedure is not as useful in application as theory for SOR and polynomial extrapolation. The analysis of the model problem, wherein H and V oommute, is quite elegant. Optimum parameters are found by solving the minimax problem: t

g(x,!!,!.) ..

n

w

j

- x

107

= maximum I

H

a " x " b

(26)

c " Y " d H

0

0

,!.

= minimum

H

The existenoe of a unique solution to this minimax problem was reoently established in Il\Y thesis (see IFIP '68 pro oeedings for a oonoise summary).

Several

means are available for choosing nearly optimum parameters. It is interesting to review the literature on this problem and note how the theory has been developed during the past fifteen years.

An analytic solution for

o 0 !: and!. found by W.B. Jordan culminated the searoh for optimum parameters.

theles., this minimax problem was actually solved about 100 years earlier (as observed by J. Todd)!

Jordan first devised a bilinear transformation of variablesto

reduoe the problem to an analogous one with identical eigenvalue intervals for both variables.

My thesis oould then be used to establish that the set of wj is identi-

oal to the Zj (except for order) for the transformed problem.

7.

PARAMETERS FOR THE PEACEMAN­RACHFORD VARIANT OF ADI

The optimum parameters are obtained by theory involving modular transformations of elliptio functions.

Numerioal evaluation turns out to be quite easy.

An appro­

ximat10n valid over a wide range is:

rj -

2t

•

j = 1.2••••• t.

To illustrate the mathematical eleganoe of analysis of convergence rates of (23). we will derive the equations for generating parameters a

o j

which solve the

minimax problem:

(28) H = maximum

I

a " x "b

I

H

a

0

minimum H

Multiplying numerator and demoninator of each faotor in the product on the right

108

hand side of (28) by ab /ajx. we obtain

,(x,!.) =

!!!._!H. x a,; ab ab -+x &j

II tTl j=1

As x varies from a to b, ab/x varies from b to a. o

as the set a j

(30)

Henoe the set

ab

0

/a j

is the same

by virtue of the uniqueness of the parameters for e:ny given eigenCombining the faotors with &j and ab/a j , we get

value interval. (a (a

j

- x)(&b/a j - x)

j

+ x)(&b/a j + x)

(Xl + ab) - (&.1 + ab/a,,)x

=

(Xl

=

(x + ab/x) _ (a.1 +

+ ab) + (a

j

+ ab/aj)x

(x + ab/x) + (a + ab/a j) j

a' _ x'

Ig(x.!.) I --

.1 ;r-:;:x, j=1

(31)

j

where (ab)i = a' , x' , b' = (a+b)/2. Continuing in this fashion. we sucoessively reduoe the number of faotors in the produot until we arrive at the one parameter problem:

We may work baokwards to obtain a parameter "tree" by suooessive solution of quadratios:

= a;s) + ./(a;s»1 _ (a{s»1 = (a(S»2 /

109

Although (27) looks a lot simpler, this teohnique was developed before the elliptic function solution was known.

There is an intimate connection between this

process and Landen transformations for evaluation of elliptic functions.

Introduction to Finite TIifference Approximations to Initial Value Problems for Partial TIifferential Equations OLOF WITILUNTI New York University

This work was in part supported by the U.S. Atomic Energy Commission, Contract AT(30-1)-1480 at the Courant Institute of Mathematical Sciences, New York University

112

1. Introduction The study of partial differential equations and methods for their exact and approximate solution is a most important part of applied mathematics. mathematical physics and numerical analysis.

One of the reasons for this is of course that very

many mathematical models of continuum physics have the form of partial differential equations.

We can mention problems of heat transfer. diffusion. wave motion and

elasticity in this context.

This field of study also seems to provide a virtually

inexhaustable source of researoh problems of widely varying difficulty.

If in

particular we oonsider finite difference approximations for initial value problems we find a rapidly growing body of knowledge and a theory which frequently is as sophisticated as the theoryfhr partial differential equations.

The work in this

field. as in all of numerical analysis. has of course been greatly influenced by the development of the electronic oomputers but also very muoh by the recent progress in the development of mathematical tools for problems in the theory of partial differential equations and other parts of mathematical analysis. Much of this progress has centered around the development of sophistioated Fourier techniques.

A typical question is the extension of a result for equations

with oonstant ooefficients. to problems with variable coefficients.

In the oonstant

ooeffioient case exponential functions are eigenfunctions and such a problem oan therefore. via a Fourier­Laplaoe transform, be turned into a, frequently quite difficult. algebraic one.

Much recent work in the theory of finite difference

schemes, including much of that of the author has been greatly influenced by this development.

These techniques are usually referred to as the Fourier method and will

be the topic of several of lectures will be different.

lectures here in Dundee.

The emphasis of these

We­will concentrate on explaining what is known as the

energy method after a discussion of the proper ohoice of norm, stability definition eto.

We will also try to make some effort in relating the mathematics to the under-

lying physics and attempt to explain difference approximations.

a philosophy of constructing classes of useful

113

We have deoided to use as simple technioal tools as possible, frequently oonoentrating on simple model problems, to illustrate our points.

Some generality will

undoubtedly be lost but it will hopefully make things easier to understand and simplify the notations.

A oonsiderable amount of time will be spent on analysing

the differential equations we are approximating.

Experienoe has shown that this is

the most convenient way to teach and work with the material.

The properties of the

differential equation are almost always easier to study and a preliminary analysis of the differential equations can frequently be translated into finite differenoe form.

This is partioularly useful when it comes to choosing proper boundary oondi-

tions for our difference schemes. The objeotive of our study is essentially to develop error bounds for finite difference schemes; methods to tell useful from less usefUl schemes and to give guidelines as to how reliable olasses of schemes can be found.

On the simplest

level finite difference methods are generated by replacing derivatives by divided differences, just as in the definition of a derivative, discretizing coefficient functions and data by evaluating them at particular points or as averagas over small neighbourhoods.

As we will see there are many choioes involved in such discretiza-

tion processes and the quality of the approximate solutions can vary most The finite differenoe appr­oech has some definite advantages as well as disadvantages.

Thus the most one can hope, using a finite difference scheme, is to be able

to get a computer program which for any given set of data will give an accurate answer at a reasonable oost.

The detailed structure of the mapping which transforms

the data into the solution will of course in general be muoh too complicated to understand.

Thus the olassioal approach giving closed form solutions to differential

equations frequently gives much more information about the influence on the solution of changes in data or the model.

The same is true perhaps to a somewhat lesser

extent, of methods of applied mathematics such as asymptotic and series expansions. However finite difference sohemes and the closely related finite element methods have proved most useful in many problems where exact or asymptotic solutions are unknown or prohibitively expensive as a computational tool.

114

The main referenoe in this field is a book by is a seoond edition of a book by

It

[1957] whioh, in its theoretical part, is

based to a great extent on work by Lax and inf1uenoed by the work of Kreiss.

and Morton [1967J.

The new edition is heavily

A second part of the book disousses many speoifio

app1ioations of finite differenoe schemes to problems of continuum physics.

There

is also a survey article by Kreiss and the author [1967J, with few proofs, based on lectures by Kreiss which still awaits publication by Springer Verlag.

It may still

be available from the Computer Soience Department in Uppsa1a, Sweden.

Also to be

mentioned is a olassioal paper by Courant, Friedrichs and Lewy [1928J which has appeared in English translation together with three survey artioles containing useful bibliographies [1967]. very much worth a study. [1969J.

Another

olassical paper, by John [1952J, is also

Among recent survey articles we mention one by Thomee

That paper essentially discusses the Fourier method.

2. The form of the partial differential equations We will oonsider partial differential equations of the form,

where u is a veotor valued function of x and t.

The variable x

= (X1'.'.'Xs ), S

varies in a region n which is the whole or part of the real Euo1idian space R

•

S

When n is all of R we speak of a pure initial or Cauohy problem; in the opposite case we have a mixed initial boundary value problem.

The differential operator P is

defined by

11

a

x

where 1111

=

Ll1

i

s s

and the rna trioes A (x, t) have suffioiently smooth elements.

11

The

degree of the highest derivative present, m, is called the order of the equation. If we let the ooeffioients depend on u and the derivatives of u as well we say that the problem is nonlinear.

115

We will restrict our attention almost exclusively to linear problems and to the approximate calculation of classical solutions, i.e. solutions u(x,t) which are smooth enough to satisfy our equation in the obvious sense. In order to turn our problem into one with a possible unique solution we initial values u(x,O) = f(x).

It is thus quite obvious that for the heat equation

a specification of the temperature distribution at some given time is necessary in order to single out one solution.

Frequently when 0 is not the whole space we have

to provide boundary conditions on at least part of the boundary

ao

of

o.

Sometimes

we also have extra conditions such as in the case of the Navier-Stokes equation where conservation of mass requires the solution to be divergence free.

The

boundary conditions, the form of which might vary between different parts of the boundary, have the form of linear (or nonlinear) relations between the different components of the solution and their derivatives.

The essential mathematical

questions are of course, whether it is possible to find a unique continuation of the data and to stuqy the properties of such a solution. Like in the case of systems of ordinary different ial equations, we can hope to assure the existence of at least a local solution by providing initial data, etc. It is however, often not immediately clear how many boundary conditions should be supplied.

Clearly the addition of a linearly independent boundary condition in a

situation where we already have a unique solution will introduce a contradiction which in general leads to a

nonexistence of a solution.

Similarly, the removal of

boundary condition in the same situation will in general lead to a loss of

uniqueness.

Existence is clearly necessary for the problem to make sense, similarly

to require uniqueness is just to ask for a

deterministio mathematical model.

The

correct number of boundary conditions as well as their form is often suggested by physical arguments (or, §3). We now mention some simple examples which will be used in illustrating the theory.

The equation

116

is the simplest possible system of first order, a class of problems of great importance.

We have already mentioned the heat equation

The equation

" is a simple model for equations of Sohrodinger type.

It has several features in

common with

which arises in simplified time dependent elasticity theory.

Finally we list the

wave equation with one and two space variables:

The last three equations do not have the form we have considered until now being second order in t. variables.

This can however be easily remedied by the introduction of new

Thus let v =i\U and w

=

will be equivalent to the equation provided.

Then

=

Initial conditions have to be

We first note that u and 8

both have to be given like in the case of tu ordinary differential equations of second order. This is also clear from the analogy with a mechanical system with finitely many degrees of freedom.

The initial

conditions for w can be formed by taking a second derivative of u(x,O)o The wave equation, in two space variables, can be transformed into the required form in several ways.

We will mention two of these because of their importance in

the following diSCUSSion.

Let us first just introduce the new variable v = 8

tu

and

rewrite the wave equation as

As we will see later on this is not a convenient form and we will instead introduce three new variables

117

u:s =

ay u

which gives the equation the form

Initial conditions for this new

are provided as above.

In order to illustrate our discussion on the number of boundary conditions we consider u(x,O)

= t'(x )

•

If f has a continuous derivative then f(x+t) will be a solution of the equation for x

0, t

0 and it can be shown that this solution is unique.

of the origin only.

The boundary consists

If we introduce a boundary condition at this point say u(O,t) =

get), a given function, we will most likely get a contradiction and thus no solution. The situation is quite different if 0 atu

= -axu

and 0 = (x; x

01).

= (x;

x

01

(or what is essentially the same,

The solution is still f(x+t) for x + t

0,

0

t

but in order to determine it uniquely for other points on the left halfline a boundary condition is required.

It can be given in the form u(O,t)

once continuously differentiable. g(x+t).

The solution for 0

= get),

x + t

f(O)

t, t

= g(O),

get)

0 will be

Thus different get) will give different solutions and the specification of

a boundary condition is necessary for uniqueness. us that no jump occurs across the line x + t

The condition f(O)

= g(O)

assures

= o.

3. The form of the finite difference schemes. We begin by introducing some notations. defined on lattices of mesh points meshes:

= {x; Xi = nih, n

of the fineness of our mesh.

i

We will be dealing with functions

For simplicity we will consider uniform

= 0, ± 1, ± 2•••• J. The mesh parameter h is a measure We also discretize in the t-direction:

= !t; t

= nk,

118

n = O,1,2, ••• l.

When we study the convergence of our difference schemes we will

let both hand k go to zero.

It is then convenient to introduce a relationship

between the timestep k and the meshwidth h of the form k 0, k(O) = O.

decreasing when h m, k = Ah m

= the

= k(h),

k(h)

Often this relationship is given in the form

order of the differential equation and A a positive constant.

The divided differences, which replace the derivatives, can be written in terms of translation operators T . defined by h

where

6

i

is the unit vector in the direction of the positive Xi-axis.

Forward,

backward and central divided differences are now defined by D+ i 0, independent of h, such that (1/e)llfll p, h"

l l lrl l l p, h '

eillfill p, h

for all l' E L h.

p,

It is easy to verify that a scheme strongly stable with respect to some norm is stable.

The strong stability reflects an effort to control the growth of the solution

on a local level.

Note that we have already established that certain finite diffe-

rence approximations to the heat equation are strongly stable with respect to the maximum norm.

Our proof that the L

2

norm of any solution of a symmetric first order

system has a limited growth rate gives hope that certain difference schemes for such problems will turn out to be strongly stable with respect to the L

2

,h

norm.

In many cases we will however be forced to choose a norm different from 11 011 in order to assure strong stability. refer to Kreiss For a norm with

[1962] and

p,

h

For a discussion of this difficult subject we

and Morton [1967].

stable scheme, the coefficients of which do not depend on n, there exists respect to which the scheme is strongly stable.

This can be shown by

the following trick which the author learned from Vidar Thomee. The fact that the coefficients of the difference schemes do not depend on time makes Eh(nk,n,k) a function of n-n, only.

We can therefore write it as Eh(nk-n,k)o

Introduce IlItlll

p,

h::: sup Ile-alk E. (lk)fll

rso

-h

It is easy to show that this is a norm,

It

cllellp, h

h.

It is equiValent to

stability and a choice of 1 ::: 0,

llellp, h ' llltlil p, h

p,

"

11.11 p, h

because by

136

Our difference soheme is olearly strongly stable beoause,

II IEh(k)flIIp,h = f:E ..

e

ak

lIe-

alk

=

lIe-

a lk

Eh«1+1 )k)fllp,h ..

Illrlll p,h·

We oould oonsider using a weaker stability definition.

A oloser study gives tie

following analogue of the Hada.ma.rd condition. Definition

A finite differenoe scheme is weakly stable in L if there exist p

constants a, C and p such that

A theory based on this definition would however suffer from the same weakness

as one based on the Hadamard definition of well posedness. of. Kreiss

[1962] or Richtmwer and Morton [1967].

For a detailed discussion

It is also clear that, in genera4

we will stay closer to the laws of physics if we chao se to work with the stronger stability definit ions. This far we have only dealt with homogenous problems. genous case is however quite simple.

Going over to the iOOomo-

We demonstrate this for an explicit scheme

u(x,t+k) = u(x,t) + kQou(x,t) + kF(t). u(x,O) = rex) • Using the solution operator we get n

u(x,nk) =

F«v-1 )k) •

f(x) + k

v=1 If the soheme is stable in L

P

we get

1IU(nk) II h" C(exp(ank) Ilfll p,

n

h + k

x [max] IIF(t)11 h). te O,nk

v=

Now if a:

ft 0

P,

137

Notice that this is really essentially a matter of computing compounded interests on an

original capital IIfll and periodic savings IIF(t

)lI.

The formalism is known

as Duhamel's principle. Its most common application is to error bounds for finite difference schemes. Put the solution of the differential equation into the difference scheme.

As was

pointed out before we will then get an extra inhomogenous term, the local truneaticn error, of the fora kT(X,nk,h).

Introduoe the error which is the difference between

the approximate and the exact solution.

Subtract the two difference equations.

The error will then satisfy the same differenoe equation with the truncation error as an inhomogenous term.

It is easy to see from our estimate that we have conver-

gence in L for 0 , t , T i f p

max IIT(nk,h)11 h goes uniformly to zero with the

O,nk,T

p,

lIlesh size and that we have a rate of convergence hr if T(nk,h) = O(hr) uniformly. We recall that

T

can be computed using just Taylor series.

We now turn to the theorem on perturbation mentioned in the previous section. For simplicity we give a proof only for time independent coefficients. Theorem

Consider a finite difference scheme u(x,(n+1 )k) = Q u(x,nk)

stable in L , statisfying p

Ilu(nk)11 p, h" C exp(ank) Ilu(o)11 p, h. Let Q' be an operator, uniformly bounded in L h. p,

Then solutions of

vex, (n+1 )k) = (Q+kQ') v(x,nk) satisfy Ilv(nk)1Ip, h ' C exp(13nk) Ilv(o)1Ip, h with

f'\=lX+Ce-akIlQ'11

p,

h "

Consider the 2 v terms in the development of (e- Ilk Q + e-

Ilk

kQ')v. (;)Of

of these consist of j factors ke-IlkQ' and there at most (j+1) factors of the form (e- Ilk Q)p, P some natural numbers. j+ The norm of such. a. term is by our assumption bounded by k j C 1 IIQ' II j e -akj.

138

Thull

II(e-ak

Q+

ke-

ak

Q,)vlI , c

t (;)

(kCe-

ak

IIQ'II)j

=

j=O

czk where y = C e -

IIQ'II •

Thus we have verified a finite differenoe version of our perturbation theoremo A differential equation theorem can now be derived by taking, for any particular differential equation, a stable difference scheme, applying the theorem just proved and taking a limit by letting h and k go to zero. We have not shown that a stable scheme always can be found for any well posed differential equation but that is in fact the case.

Also notice that there is

never any chance to derive stronger inequalities for a finite difference scheme tb8n those which hold for the corresponding differential equation.

The most we can hope

ill to get exaot analogues of what is true in the oontinuous case. We leave it to the reader to give another proof of our latest theorem using the norm

111.111 p, h

which we constructed in our proof that any stable scheme is

strongly stable at least With respect to one norm.

7. The von Neumann condition, dissipative and multistep schemes. In this section we will use Fourier series to derive stability conditions and also introduce a number of useful schemes for hyperbolic equations. We first consider the periodio problem atu

= axu,

t

0, u(x,O)

= f(x) =

If' f(x) is sufficiently smooth it can be developed in a convergent Fourier series f(x) =

(Xv e

i vx

v=-oo

The solution of the equation takes the form +00

u(x,t) =

(Xv eiv(x+t)

v=-oo

139

The Euler differenc e scheme v(x,t+k) = v(x,t) + kDo v(x,t), can be studied in the same way.

where

v(x,O)

f(x)

Its solution is

).. = k/h.

In contrast to the differential equation case the amplitude of Fourier components will growo

v

f

O.

).. = 1

This is so becuase 11 + i)" sin v hi =

J1

+

)..2

sin2 V h > 1 for

From our discussion of the Courant­Friedrichs­Lewy condition we know that would be ideal and we see that choosing ).. equal to a constant will lead to

very rapidly growing high frequency components.

It is easy to show that for

very smooth initial value functions this very strong amplification of high components will lead to arbitrary large and wildly divergent approximate solutions. The amplification of the lowest frequency modes is however not very large. This might lead us to the following idea.

Replace the initial value function by a

fixed partial sum of its Fourier serie s.

If the initial data is sufficiently smooih

we can do this changing the values of the initial value function and the solution by an arbitrarily small amount.

If we use the new initial value for the finite

difference scheme we can see, from the explicit solution formula, that the discrete solution will converge to the correct one when h goes to zeroo

In fact the same

argument shows that we could proceed with a ).. larger than 1 because for any constant).. (1 + i)" sin

vh)n will converge to e

i vkn

for any fixed value of v.

This approaoh however suffers from the same weakness as a theory for differential equations based on analytic functions only.

In fact a finite Fourier series

represents an analytio function and much of Hadamard's criticism of the CauchyKowaleski type theory carries over to the finite difference case. To see that something is drastically wrong with our argument above we urge the reader to carry out a few steps with the Euler method using ).. value function which is

€

= 10

at one mesh point and zero elsewhere.

and an initial

The rapid growth

of this special solution will assure us of a totally unacceptable growth of round off errors.

One could say that the error of measurements which played an important

140

part in Hadamard's argument are replaced by the round off errors.

From the error

bound in section 6 we see that we will not be seriously affected by round off errors if a difference scheme is stable.

This is thus a reason. perhaps the most impor-

tant one. why we insist on using only stable schemes for computations. There is a simple remedy for the lack of stability of the Euler scheme namely the addition of a so oalled dissipation term.

The term corresponds to a finite

difference approximation of yet another term in the Taylor expansion of u(x,t+k) with respect to t.

This way we get the 1ax­Wendroff scheme.

v(x.t+k) = v(x.t) + kDrv(x.t) + The coefficient of e

i vx

v(x.a)

D_D+v(x,t)

f(x) •

is now amplified by a factor in each step.

It is elementary to verify that this factor is less than or equal to one in absolute value for A

"1.

This clearly ensures strong 1

2

stability.

From this, convergence

follows as well as a relative insensitivity to round off errors. We have now developed a simple technical tool which allows us to decide the qualities of all the sohemes suggested for the heat equation.

The results will be

revealed shortly. The dissipation of the 1ax­Wendroff scheme aots to damp out the higher frequenqy modes of the solution and this is frequently just as well because they must contain rather serious phase errors.

For sufficiently smooth solutions, which means quickly

decreasing Fourier coefficients with increasing v, these modes playa ver,yunimportant part in the representation of the solution.

Similar considerations

make very much sense in cases when we do not have oonstant coefficients.

Heuristi-

cally we can argue that the variability of the coeffioients will make various Fourier modes interact in a way which is very hard to analyse. expect serious phase errors for high modes.

However we can

Not only will these oomponents be in

error but they will interaot with other oomponents in a totally erroneous way. such a case it seems advisable to damp out such modes in the discrete model.

In

141

Sometimes model.

we however are quite anxious to have an energy preserving discrete

This is for instance the case when we have to calculate over long periods of

time and with only very weak forcing functions. the energy for the case BtU

= aXu is

One simple scheme which preserves

the leap frog scheme also mown as the mid-point

rule when it is used for ordinary differential equations, v(x,t+k) = v(x,t-k) + 2kD. v(x,t) • Another one is the Crank-Nicolson scheme (1 - k/2 Do) v(x,t+k)

= (1

+ k/2 Do) v(x,t) •

Fourier analysis shows that the amplification per step for the Crank-Nicolson scheme is (1 + i h/2 sin vh)/(1 - i A/2 sin vh)

and thus that the amplitude is preserved.

=

For the leap frog scheme we look for solutions of the form v(x,nk) get, by the solution of a quadratic equation, the two roots

J1 -

AZ sin2 vh which, for A

'lli"l

1, both lie on the unit circle.

,i It 1• VX e and

= iA sin vh .:t The multistep

character is reflected in the existence of two independent solutions. A similar analysis for the backward scheme shows that

11

_ i).1 sin vh l

1 which

again implies strong stability. We now turn to aFourier analysis of the schemes suggested for the heat equation. Using). for k/h2 we find that the amplification factor for Euler's method is 1 - A4 sin2 (vh/2), thus it is stable for).

For the Crank-Nicolson scheme:

(1 - 2A sin2 (vh/2»/(1 -+ 2A sin2 (vh/2». unconditionally stabl.e ,

For the backward

scheme 1/(1 + 4A sin2 (vh/2», unconditionally stable. For the mid-point rule an 'y iv ivx \J Ansatz of the form GL e leads to -4A sin2(vh/2) .:t J1 + (4). sin2(vh/2»2 which shows that the scheme is unstable for any constant Ao

In a similar way we

could also show that the Dufort-Frankel scheme is unconditionally stable. Another interesting method, of fourth order accuracy in time, is Milne's method (1 -

J kQ) v(x,t+k) = (1

where Q stands for Do or D+D••

+

31 kQ)

v(x,t-k) +

34 kQv(x,t)

•

The roots of the corresponding quadratic equation

2 'Aq .:t..,1I + 3 1( 1 . hi are ( 3 Aq)2) / (1 - 3 'Aq) Where q stands for 1. S1n vh or -4 sin2v 2

0

142

Thus the method

preserves energy for the hyperbolic case, provided " ,

./5,

and is

violently unstable for the parabolic case. The instability of the Milne and mid-point methods is closely related to the well known weak stability of these methods when applied to ordinary differenia1 equations. Cf. Dahlquist [1956], [1963].

In fact parabolic equations are very stiff

equations and weakly stable schemes are therefore quite useless. We are now well prepared for the following definition. Definition

Consider a linear finite difference scheme Un+t

=

QUn

and define its symbol by = exp( i
0 in

We will see by an example that this is a most

natural restriction. We first show that Ln '" Ln_l•

Rewrite the equation as

un+1 - un-, = kQ 1 (un+1 + un_I) + 2kQoun and take the scalar product with un+1 + un_I.

Then

The first term on the right hand side is less than or equal to zero because of one of our assumptions.

Rearranging and adding lIunll2 on both sides we get Ln '" Ln_, •

To show that Ln is positive and equivalent to the natural L norm we start by 2 observing that

Therefore

148

To see that kllQol1 10 1 - /) is a natural condition consider the case Qo = Do and

= 00

This Q. has, as is easily verified, an L norm equal to 1/h. Thus the 2 restriction just means k/h '1 - &, essentially the Courant-Friedrichs-Lewy condition.

Q,

This is a natural condition in terms of Qo alone because in the case Q, = 0 the method is explicit o For a more general discussion and a comparison of the growth rates of the exaot and approximate solutions we refer to Johansson and Kreiss [1963J. Sohemes of Dufort-Frankel type can be discussed in very much the same wayo We will now show that the se ideas can be used to design stable and efficient schemes, so called alternating direction implicit schemes, for certain two dimensWnal equations.

We suppose that our problem has the form

and that the operators and the boundary conditions are such that P, and Pz are semibounded.

For simplicity we assume that

Re(u, P.u) , 0 ,

j = 1,2,

J

and that we have finite difference approximations Q. to P., j J

ReCu, Q.u) , 0 , J

j

J

1,2, such that

= 1,2.

We will consider the following two schemes

and

These schemes are particularly convenient if Q, and Qz are one dimensional finite difference operators.

In that case we only have to invert one dimensional operators

of the form (r - akQi) and this frequently leads to considerable savings.

This

becomes clear if we compare the work involved in solving a two dimensional heat equation, using an alternating direction implicit method with Q, = Qz =

and the application of the standard backward or Crank-Nicolson scheme

with Q = D_xD+ x +

The former approach only involves solutions of linear

systems of tridiagonal type while the other, in general, requires more work.

149

The L stability of the first scheme is very simple to prove. 2 i

1,2, both have inverses the

L

2

Thus I - k Qi'

norms of which are bounded by 1.

stability of the other scheme is more involved.

The proof of the

Let

and

Then

(1 + k/2Q, )zn or Yn+, - Zn

= k/2

Q,(Yn+l + zn)

0

Forming the inner product with Yn+l + zn, just as in the proof of the stability of the Crank-NicolsGn method, we get

Now

= Ilun+,11

2

- k/2 Re(Q2Un+"

Un+l) +

Ir/4

IIQ2Un+,11

2

and

Therefore, because Re(Q2u, u) , 0,

It is easy to see that this implies L stability if kQ2 is a bounded operator. 2

If

kQ 2 is not bounded we instead get stability with respect to amxonger norm, a result which serves our purpose equally well. We refer to an interesting paper by Strang [1968] for the construction of other stable accurate methods, based on one dimensional operators o

10. Maximum norm convergence for L2 stable schemes In this section we will explain a result by Strang [1960] solutions of L

2

stable schemes of a certain accuracy converge

shows that in maximum norm with

the same rate of convergence as in L provided the solution of the differential 2

equation is sufficiently smooth.

150

1et

u,,+1 = QUn ,

Uo (x) = f(x)

be a finite difrerence approximation to a linear problem, U

well posed in 1

(x,O)

= f(x)

,

2•

To simplify matters we assume that the two problems are periodic. assume that we have an 1

2

stable scheme.

We also

It is known that if f is a sufficiently

smooth function the solution will also be quite smooth.

We now attempt to establish

the existence of an asymptotic error expansion of the error.

where we choose r as the rate of convergence

in 120

Make the Ansatz

If we substitute this

expression into the difference equation we find that the appropriate choice for er, er+l' are solutions of equation of the fora

e.(O)=O J

where 1

j

are differential operators which appear in the formal expansion of the

truncation error.

The solutionsof a finite number of these equations are under our

assumptions quite smooth. To end our discussion we have to verify that en,N(x,h) = u,,(x) - u(x,nk) -

(x,nk) l=r

is O(h r) in the maximum norm for some finite N, i.e. that hre error term.

r

is indeed the leading

This is done by a slight modification of the error estina te of §6 0

r N1 derive a difference equation for an,N and find that its L2 norm is 0(h + + ) . assumption we have a periodic problem. fore bounded by hspace dimensions.

s/2

times its 1

2,

We By

The maximUlll norm of a mesh function is there-

h norm over a period, where s is the number of

This concludes our proof.

151

We remark that an almost identical argument shows that we can relax our stab:ilii;y requirements and require only weak stability (Cf. §6) and still get the same results for sufficiently smooth solutions. REFERENCES Brenner, P.; 1966, Math. Scand., V.19, 27-37. Courant, R., Friedrichs, K. and Lewy, H.; 1928, Math. Annal., V.100, 32-74 also; 1967, IBM J. of Research and Development, V.11, 213-247. Dahlquist, G.; 1956, Math. Scand., V.4, 33-53. Dahlquist, G.; 1963, Proc. Sympos. Appl. Math., V.15, 147-158. Friedman, A.; 1964, Partial Differential Equations of Parabolic Type. Prentice-Hall. Garabedian, P.; 1964, Partial Differential Equations.

Wiley.

Gelfand, I.M.; Shi1ov, G.E.; 1967, G€neralized Functions, V.3 Academic Press. Hadamard, J.; 1921, Lectures on equations. Johansson,

problem in linear partial differential

Yale University Press. Kreiss, H.O.; 1963, BIT, V.3, 97-107.

John, F.; 1952, Comm. Pure. Appl. Math., V.5, 155-211. Kreiss, H.O. ; 1959, Math. Scand., V.7 71-80. Kreiss, R.O. ; 1962, BIT, V.2, 153-181 Kreiss, H.O. ; 1963, Math. Scand., V.13, 109-128. Kreiss, H.O

Q ;

1963, Numer-, Math. , V.5, 27-77.

Kreiss, H.O., Widlund, 0.; 1967, Report, Computer Science Department, Uppsala, Lax, P.D.; 1957, Duke Math. J., V.24 Lax, P.D.; 1963, Lectures on hyperbolic partial differential equations, Stanford University (lecture notes). Littman, W.; 1963, J. Math. Mech., V.12, 55-68. R.D.; 1957, Difference methods for initial-value problems. Wiley Interscience. R.D., Morton, K.W.; 1967, Difference methods for initial-value problems Q 2nd Edition Wiley Interscience.

152

Strang, W.G.; 1960,

Duke

Math. J., V.27, 221-231.

Strang, W.G.; 1966, J. Diff. Eq., V.2, 107-114. Strang, W.G., 1968, SIAM J. Numer. Anal., V.5, 506-617. /

Thomee,

v.;

1962, J. SIAM, V.I0, 229-245.

Thom:e, V.; 1969, SIAM Review, V.ll, 152-195.

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