Numerical Solution of Partial Differential Equations (Applied Mathematical Sciences)

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Applied Mathematical Sciences I Volume 32

Theodor Meis Ulrich Marcowitz

Numerical Solution of Partial Differential Equations

Springer-Verlag New York Heidelberg Berlin

Theodor Meis

Ulrich Marcowitz

Mathematisches Institut der Universitat zu Koln Weyertal 86-90 5000 Kiiln 41 Federal Republic of Germany

Mathematisches Institut der Universitat zu Koln Weyertal 86-90 5000 Koln 41 Federal Republic of Germany

Translated by Peter R. Wadsack, University of Wisconsin.

AMS Subject Classifications:

65MXX, 65NXX, 65P05

Library of Congress Cataloging in Publication Data Meis, Theodor. Numerical solution of partial differential equations. (Applied mathematical sciences; 32) Translation of Numerische Behandlung partieller Differentialgleichungen. Bibliography: p. Includes index. 1. Differential equations, Partial-Numerical solutions. I. Marcowitz, Ulrich, joint author. II. Title. III. Series. QA1.A647 vol. 32 [QA3741 510s [515.3'53J 80-26520

English translation of the original German edition Numerische Behandlung Partieller Differentialgleichungen published by SpringerVerlag Heidelberg © 1978. All rights reserved.

No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1981 by Springer-Verlag New York Inc. Printed in the United States of America. 9 8 7 6 5 4 3 2 1

ISBN 0-387-90550-2 Springer-Verlag New York Heidelberg Berlin ISBN 3-540-90550-2 Springer-Verlag Berlin Heidelberg New York

PREFACE This book is the result of two courses of lectures given at the University of Cologne in Germany in 1974/75.

The majority of the students were not familiar with partial differential equations and functional analysis. why Sections 1,

2,

This explains

4 and 12 contain some basic material and

results from these areas.

The three parts of the book are largely independent of each other and can be read separately.

Their topics are:

initial value problems, boundary value problems, solutions of systems of equations.

There is much emphasis on theoretical

considerations and they are discussed as thoroughly as the algorithms which are presented in full detail and together with the programs.

We believe that theoretical and practical

applications are equally important for a genuine understanding of numerical mathematics.

When writing this book, we had considerable help and many discussions with H. W. Branca, R. Esser, W. Hackbusch and H. Multhei.

H. Lehmann, B. Muller, H. J. Niemeyer,

U. Schulte and B. Thomas helped with the completion of the programs and with several numerical calculations. Springer-Verlag showed a lot of patience and understanding during the course of the production of the book.

We would like to use the occasion of this preface to express our thanks to all those who assisted in our sometimes arduous task.

Cologne, Fall 1980 Th. Meis U. Marcowitz v

CONTENTS Page

INITIAL VALUE PROBLEMS FOR HYPERBOLIC AND PARABOLIC DIFFERENTIAL EQUATIONS.

PART I.

.

.

.

.

.

.

1

.

1. 2. 3.

Properly posed initial value problems Types and characteristics Characteristic methods for first order hyperbolic .

31

4. 5. 6. 7. 8. 9.

Banach spaces Stability of difference methods Examples of stable difference methods Inhomogeneous initial value problems. Difference methods with positivity properties Fourier transforms of difference methods. Initial value problems in several space variables Extrapolation methods

40 55 73 89 97

.

.

systems

10. 11.

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207

Properly posed boundary value problems.

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Difference methods .

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BOUNDARY VALUE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

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Variational methods Hermite interpolation and its application to the Ritz method Collocation methods and boundary integral methods .

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.

SOLVING SYSTEMS OF EQUATIONS.

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.

19.

Iterative methods for solving systems of linear and nonlinear equations Overrelaxation methods for systems of linear equations Overrelaxation methods for systems of nonlinear

20. 21. 22.

Band width reduction for sparse matrices. Buneman Algorithm The Schr6der-Trottenberg reduction method

18.

1

19

119 168 192

PART III. 17.

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.

16.

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.

PART II. 12. 13. 14. 15.

.

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.

equations

APPENDICES: Appendix 0: Appendix 1: Appendix 2: Appendix 3:

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334

363

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402 417 426

444

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Introduction .

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444 447

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459

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469

Method of Massau. Total implicit difference method for solving a nonlinear parabolic differential equation Lax-Wendroff-Richtmyer method for the case of two space variables Difference methods with SOR for solving the Poisson equation on nonrectangular .

.

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484

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503

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522

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Programs for band matrices. The Buneman algorithm for solving the Poisson equation. .

Vii

.

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.

383

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.

regions

Appendix 5: Appendix 6:

334

FORTRAN PROGRAMS . .

290 317

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.

Appendix 4:

.

207 229 270

.

viii

Page BIBLIOGRAPHY

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532

INDEX .

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538

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PART I. INITIAL VALUE PROBLEMS FOR HYPERBOLIC AND PARABOLIC DIFFERENTIAL EQUATIONS

Properly posed initial value problems

1.

In this introductory chapter we will explain what is

meant by the concept of properly posed initial value problems. We start with the well-known situation for ordinary differential equations, and develop the definition with the help of explanatory examples.

This concept is an important one, for

problems which are not properly posed cannot, in general, be attacked reasonably with numerical methods. Theorem 1.1:

Let

f c C°([a,b] x IR,IR)

be a continuous func-

tion satisfying a Lipschitz condition for a constant

lf(x,z) - f(x,w)l < Ljz - wl, Then

for all

n,n e1R

x e [a,b],

z,w CIR.

there exists exactly one function

Ju'(x) = f(x,u(x)) u e Cl([a,b],IR)

L cIR:

x e [a,b]

with lu(a)

= n

and exactly one function

u e C1 ( [ a , b ] ,

IR)

with

u ' (x) = f(x,u(x)), u(a)

1

= n.

x e [a,b]

2

If

I.

L = exp(LIb-aj), then for all Iu(x)

INITIAL VALUE PROBLEMS

x c [a,b]

- u(x)I < exp(LIx-aI)In - nI < LIn -

I.

Theorem 1.1 is proved in the theory of ordinary differential equations (cf. e.g. Stoer-Bulirsch 1980, Theorem 7.1.4).

It

says that the initial value problem x c [a,b]

u'(x) = f(x,u(x)),

u (a)

=n

subject to the above conditions, has the following properties: (1)

There exists at least one solution

(2)

There exists at most one solution

(3)

The solution satisfies a Lipschitz condition with

respect to Iu(x;n)

u(x;n). u(x;n).

n:

- u(x;n)I < fin -

nI,

x c [a,b].

n,n c]R.

This motivates the following definition, which is intentionally general and which should be completed, in each concrete case, by specifying the spaces under consideration and the nature of the solution.

Definition 1.2:

An initial value problem is called property

posed (or well posed) if it satisfies the following conditions: Cl)

Existence:

The set of initial values for which the

problem has a solution is dense in the set of all initial values. (2)

Uniqueness:

For each initial value there exists

at most one solution. (3)

Continuous dependence on the initial values:

The

solution satisfies a Lipschitz condition with respect to those

1.


3

initial values for which the problem is solvable.

a

We next consider a series of examples of initial value problems for partial differential equations, and examine whether or not these problems are properly posed. interest will be in classical solutions.

Our primary

These are charac-

terized by the following properties: (1)

a region

The differential equation need only be satisfied in G, i.e., in an open, connected set.

The solution

must be as often continuously differentiable in G

as the or-

der of the differential equation demands. (2)

subset

r

Initial or boundary conditions are imposed on a of the boundary of

tinuously differentiable on

G.

G U r

The solution must be conas many times as the order

of the initial or boundary conditions demand.

If only func-

tional values are given, the solution need only be continuous on

G u r.

Example 1.3: T > 0.

Let

e C1(IIt,IR)

be a bounded function and let

Then one of the simplest initial value problems is

uy(x,y) = 0 u(x,0) _ fi(x)

x e IR,

y e (0,T).

Obviously the only solution to this problem is

u(x,y) = fi(x).

Therefore we have

I1

u(

=1k -all,,,

so the problem is properly posed. This initial value problem can be solved "forwards" as

above, and also "backwards" since the same relationships exist


I.

4

y e [-T,0].

for

However, this situation is by no means typi-

cal for partial differential equations.

o

An apparently minor modification of the above differential equation changes its properties completely: Example 1.4:

Let

and

0

T

be chosen as in Example 1.3.

Then the problem

ux(x,Y) = 0

xe]R, ye (0,T)

u(x,0) _ 4(x)

is solvable only if

is constant.

4

In this exceptional case

there are nevertheless infinitely many solutions

4(x) +

for functions

The problem

c

CI((0,T),RR)

with

therefore is not properly posed.

:y(0) = 0.

(y)

o

The following example contains the two previous ones as special cases and in addition leads heuristically to the concept of characteristics, which is so important in partial differential equations. Example 1.5:

A, B, T e]R

Let

with

4 e C1(]RJR)

be a bounded function and let We consider the prob-

A2 + B2 > 0, T > 0.

lem

Aux(x,y) = Buy(x,y)

u(x,0) _ 4(x) For

x e ]R,

y C (0,T) .

B = 0, the problem is not properly posed (cf. Example

1.4).

Assume

B # 0.

(xc(t), yc(t)) = (c

we have for

t e (0,T):

On the lines

-

Bt,t),

t eIR

a parameter

1.


S

Bx(xc(t),Yc(t)) + uy(xc(t),Yc(t)) = 0.

d-t-u(xc(t),Yc(t))

This implies that u(xc(t),Yc(t))

=_

4'(c)

= 4'(xc(t) + Byc(t)),

t

e [0,T).

The problem thus has a uniquely determined solution

u(x,Y) _ (x + g Y) The family of lines

and is properly posed.

constant (c

Bx + Ay = Bc =

the family parameter) are called the characteris-

tics of the differential equation.

They play a distinguished

role in the theory and for more general differential equations consist of curved lines.

In the example under consideration

we see that the problem is properly posed if the initial values do not lie on a characteristic (cf. Example 1.6).

Further we

note that the discontinuities in the higher derivatives of propagate along the characteristics.

4'

o

In the following example we consider systems of partial

differential equations for the first time, and discover that we can have characteristics of differing directions. Example 1.6:

Initial value problem for systems of partial dif-

ferential equations. a2 +

S2

Let

n c]N, T > 0, and

Define

> 0.

G = {(x,y) c IR 2

r = { (x, y) and suppose

a,B aIR with

4'

A c MAT(n,n)R)

c iR

c C1(r,IItn)

2

ax + By c (0,T) } ax + By = 0} bounded, q c C1(G,IItn) , and

real diagonalizable.

solution of the problem

We would like to find a

6


I.

uy(x,Y) = Aux(x,Y) + q(x,y),

(x,Y) e G,

u(x,Y) _ 4(x,Y),

(x,Y) e r.

The given system of differential equations may be uncoupled by means of a regular transformation matrix A

to the diagonal form

u = Sv, q = Sr, and

which takes

S

S-1AS = A = diag(Ai).

Letting

= Sip, we obtain the equivalent system

vy(x,y) = Avx(x,Y) + r(x,y),

(x,y) e G,

v(x,Y) = V(x,Y),

(x,y) e r.

Analogously to Example 1.5, we examine the ith equation on the lines

x + Aiy = c

with the parametric representation

(xc(t),Yc(t)) = (c-ait,t),

t c 1R.

From the differential equation it follows that for the ith component

vi

of

v, and for all

t e 1R

a(c-ait) +

with

St e (0,T) : av.

d

av.

dt vi(xc(t),Yc(t)) = -aiax (xc(t),Yc(t)) +

ay(xc(t),Yc(t))

= ri(xc(t),Yc(t)) therefore t

vi(xc(t),Yc(t)) = ni + I0 ri(xc(T),Yc(T))dT, ni c 3R

arbitrary.

When considering the initial conditions, we have three possible cases: Case 1:

aai - a # 0.

The two lines

have exactly one intersection point

r

and

x + Xiy = c

1.


c,Yc)

=

7

ac

-9c (aai 8

and

ni = ni(c) =

i(XC,Yc)

1vc -

ri(xc(T),Yc(T))dT.

0

Thus for all

vi

we obtain the following representation: Y

vi(x,Y) = ni(c) + j0 ri(c-aiy,Y)dy

c = x + aiy One can check by substitution that

v i

is actually a solution

of the ith equation of the uncoupled system. Case 2:

aa .-$ = 0 i

x + aiy = c

and

c = 0.

are now identical.

The two lines

and

r

The ith equation of the un-

coupled system can be solved only if coincidentally dt i(xc(t),Yc(t)) = ri(xc(t),y (t)),

a(c- A t) + St C (0,T).

In this exceptional case there are, however, infinitely many solutions. Case 3:

aai -

x + aiy = c

B = 0

and

c # 0.

The two lines

now have no point of intersection.

r

and

The ith equa-

tion of the uncoupled system has infinitely many solutions. As in Example 1.5, the family of lines (c

x + aiy = c

the family parameter) are called the characteristics of

the system of partial differential equations.

characteristics coincide with the line

r

If none of the

on which the ini-

tial values are given, then we have case 1 for all

i, and

8


I.

v1 (x, Y)

u(x,y) = Sv(x,y) = S vn(x,Y)

is the uniquely determined solution of the original problem. To check the Lipschitz condition, let

0 E Cl(r,IR")

and

Then it follows that

= Spy.

)II

Ilu(

v(, ;ip)II,,

=

Thus we have shown that the problem is properly posed exactly when

is not a characteristic.

r

u = min{aili = l(l)n}

Let

From the representation of at the fixed point

u

u

v = max{aili = l(l)n}.

v

we see that the solution

and

(x0,y0)

ally only on the values of

and

depends on

(x,y)

q

and addition-

on the line segment con-

necting the points I -S(xo+1Y0) l

ap-Is

a(xO+uy0)

l

-1611-,7-75

J

and

( -B(xo+vyo) l av- 13

a(xo+vyo) av-

l

}

This segment is called the domain of dependence of the point (x0,y0).

On the other hand, if

q(x,y)

is known only on the

segment connecting the points I l

-$a

u-'

as

and J

(_i.ab 1 av-0 '

av-S }

then the solution can be computed only in the triangle

ax + By > 0,

x + uY > a,

x + vy < b.

This triangle is called the domain of determinancy of this segment.

These concepts are clarified further by the

1.


example in Figure 1.7.

Figure 1.7:

9

o

Characteristics

and domains of determinancy

and dependence.

The uncoupling of a system of differential equations as used in the preceding example is also helpful in investigating other problems.

10


I.

Initial boundary value problem for systems of

Example 1.8:

partial differential equations.

r= DG, q c C1(G,1R )

n E]N, G =

Let

0,4 E C1([0,m), ]Rn)

and

real diagonalizable.

A E MAT(n,n,]R)

(0,co)2 ,

bounded, and

We seek a solution to the

problem uy(x,Y) = Aux(x,Y) + q(x,y),

u(x,0) _ 4(x),

x>0

u(0,Y) _ ky),

Y > 0

with the compatibility conditions A4''(0) + q(0,0).

(0) _ (0)

ct'(0)

=

With the notation used there, it

follows that for the ith component with

and

The system of differential equations can be

uncoupled as in Example 1.6.

t EIR

(x,Y) e G

t > 0

and

vi

v, and for all

of

c - X.t > 0: i

t

vi(xc(t),Yc(t)) = ni +

ri(xc(T),Yc(T))dT,

f 0

e 1R

arbitrary.

There are three possible cases for the initial boundary condition:

Case 1:

ai < 0.

The characteristic

x + Aiy = c

one intersection point with the boundary the values of

0

or

4, ni

With the aid of

r.

and therefore

has exactly

vi

is uniquely

determined. Case 2: for

ai > 0.

The characteristic

x + Xiy = c

c > 0, one intersection point with the positive

and one intersection point with the positive case

then has,

ni

y-axis.

x-axis In this

is overdetermined in general, and the ith equation

of the uncoupled system is not solvable.

1.

11


ai = 0.

Case 3:

the solution

For

c > 0,

vi(xc(t),yc(t))

ith component of

S-14

for

is uniquely determined, but

i

converges continuously to the c

only in exceptional cases.

0

The problem is properly posed if and only if all eigenvalues A

of

are negative, i.e., when only the first case occurs. The following example, the wave equation, is an impor-

tant special case of Example 1.6. (Example 1.9:

C1(II2, ]R)

Wave Equation.

With

.

Let

a = 0, 0 = 1, A =

T > 0

l0

and 1],

q

01,02 e =_

0

and the use

of componentwise notation, Example 1.6 becomes aul/ay = au2/ax au2/ay = aul/ax

x e ]R,

y C (0,T) .

u1(x,0) = Yx) u2(x,0) = 42(x) ((

We have

Xl = 1, a2 =

1, S =

11

Ii

_lJ

and

S-1

equation

becomes

Thus the solution of the wave ul(x,Y)

= S i1(x+Y)

u2(x,y)

*2(x-Y)

_ S

2(Yx+Y) + 2(x+Y)) Z(01(x-Y)

- 42(x-Y))

Yx+Y)+Yx+Y)+Ol(x_y)-02(x-Y)

x CIR,

1

yC

[0,T).

2 1(x+y)+02(x+y)-q1(x-y)+02(x-y) The wave equation can also be written as a partial differential equation of second order equation with respect to pect to

y

by differentiating the first and the second equation with res-

x, and then subtracting one equation from the other.

The initial value for

u2

becomes an initial value for au1/ay

D


I.

12

With the change of nota-

with the aid of the first equation. tion of

for

u

u

and

u1

yy

- u

xx

41, we obtain

for

4)

= 0

u(x,0) = ¢(x)

y e (O,T).

x e ]R,

uy(x,O) = Yx) = Ve(x) Yet another form of the wave equation arises from the coordinate transformation

y = b + a, x = C

-

a.

The differential

operators are correspondingly transformed into a

a

a

a

a

a

'9-y

ax

It then follows that the given differential equation can also be written as

(ay + ax) (ay

2 u(_-a,_+a)

aCaa

-

=

ax) u(x,y) = 0,

a2 u C a)

= 0.

DCao

In order for the wave equation problem in this form to be properly posed, it is necessary that the initial values not be specified for

g ° constant and

lines are the characteristics.

a =_ constant, since these o

Another important type of partial differential equation is exemplified by the heat equation, which is presented in the following examples. Example 1.10: For

Initial value problem for the heat equation.

T > 0, a > 0, and

the following problem:

q,(p e C°(R,IR)

we seek a solution to

1.


ut(x,t) = auxx(x,t) - q(x)

u(x,0)

fi(x)

13

t e (0,T).

x e 1R,

However this problem is not uniquely solvable and therefore also not properly posed (cf. Hellwig 1977).

This

shortcoming can be overcome by imposing conditions on 0

q

and

The additional

and by restricting the concept of a solution.

conditions lq(x)lexp(-Ixl)

and

lu(x,t)lexp(-1x1)

(k(x)lexp(-1x1)

x e]R

and

C°(JR,IR)

and

bounded for

determine linear subspaces of

0(x)

(lull

x

=

One can show that the problem is

t c [0,T)

For the norm we use

C2(IR x (0,T) ,]R) fl C°(IR x [0,T),IR). 11011 = sup exp x eIR

bounded,

u(x t)

sup

xe R, te[O,T)

exp

x

now properly posed.

The

solution of the homogeneous equation vt(x,t) = avxx(x,t), v(x,O) = fi(x),

x cIR,

1p(x)1 exp(-Ixl)

t e (0,T)

bounded,

p E C°(IR,IR)

can be derived using Fourier transforms (cf. §9): +2 -1/2 (4nat) exp ( (x )4(T)dt for

t e (0,T)

v(x,t) = for

fi(x)

t = 0.

To obtain the solution of the inhomogeneous problem, we first consider the equation awxx(x,t)

Obviously

- q(x) = 0.

14

I.

fx r

w(x,t) = w(x) = 1

a J0

is a particular solution. that

lw(x)lexp(-Ixl)


q(T)dtd

J

0

A straightforward computation shows

is bounded.

Now let

solution of the homogeneous equation with Then

u(x,t) = v(x,t) + w(x)

be the

v(x,t)

V(x) = fi(x)

- w(x).

is the solution of the original

inhomogeneous heat equation.

It is apparent from the integral representation of the solution that there is no finite domain of dependence.

There-

forethis problem has no practical significance, in contrast to the following initial boundary value problem. Example 1.11: equation.

Let

a

Initial boundary value problem for the heat T > 0, a > 0; q,c c C°([a,b], It)

c C°([0,T], Si).

and

We seek a solution of the problem

ut(x,t) = auxx(x,t)

- q(x),

u(x,0) _ ct(x),

x c

(a,b),

x c

[a,b]

t e

(0,T)

u(a,t) = Vpa(t), u(b,t) = iyb(t),t c [0,T). Since there are two conditions on each of

u(a,0)

id

u(b,0), we also need the following compatibility conditions

0(a) _ $a(0),

0(b) = * b(0).

We know from the literature that the problem is properly posed (cf. e.g. Petrovsky 1954, §38).

It can be solved, for example,

by a Laplace transform (reducing it to an ordinary differential equation) or by difference methods. on

¢(Q

s c

[0,t].

for all

E c

[a,b]

u(x,t)

and on a(s),Ys)

is dependent for all

1.


15

In contrast to Example 1.3, the problem at hand is not properly posed if one attempts to solve it "backwards" [for t c(-T,0)]-heat conduction processes are not reversible.

For the problem

then either is not solvable or the solution does not depend continuously on the initial values.

This state of affairs is

best explained with the following special example:

a = 0, b = it, a = 1, q = 0, a =

0,

lI,b

= 0, w c IN, y c IR,

(x;Y,w) = y sin wx. ut(x,t;Y,w) = uxx(x,t;Y,w),

x e (O,n),

u(x,O;Y,w) _ 0(x;Y,w),

x c

u(O,t;Y,w) = u(ir,t;Y,w)

t e

(-T,0)

[0,n]

t c ('T,0]

0,

One obtains the solution u(x,t;y,w) = y exp(-w2t)sin wx. For the norms we have Ilu(-,-;Y,w)

- u(',';O,w)II = y exp(w2T) Y.

The ratio of the norms grows with

beyond all bounds.

w

there can be no valid Lipschitz condition with respect to dependence on the initial values. Example 1.12:

o

Initial boundary value problem of the third

kind for the heat equation.

Nonlinear heat equation.

T,a > 0; q, a C°([a,b],]R); 0a,Ya,Rb2Yb > 0; sb+Yb > 0;

c C°([0,T],]R).

Let

0a+Ya > 0;

We consider the problem

Thus

16

I.


ut(x,t) = auxx(x,t) - q(x),

x e (a,b),

u(x,0) = O(x),

x e [a,b]

t e (0,T)

0au(a,t) - yaux(a,t) = *a(t) t e

abu(b,t) + ybux(b,t) =

(0,T).

b(t)

Compatibility conditions:

Bab(a)

*a(0)

-

Bb0 (b) + YO' (b) _ *b (0) The boundary conditions imposed here are of great practical They are called boundary values of the third

significance. kind.

The special cases

Ya = Yb = 0

and

Ba = Bb = 0

are

called boundary values of the first and second kinds, respectively.

One can show that the problem is properly posed.

The methods of solution are Laplace transforms or finite differences, as in Example 1.11.

The nonlinear heat equation ut = [a(u)ux]x - q(x,u)

a(z) > e > 0 ,

z

e lR

is frequently rewritten in practice as follows: strongly monotone function

f:]R +]R

rzz

f(z) = f

0

and set

w(x,t) = f(u(x,t)). It follows that

wx = a(u)ux wt = a(u)ut wt = a(u)[wxx - q(x,u)]

With the notation

by

define a

1.


&(Z) =

17

a(f-1(z)) q(x,f-1(Z))

q(x,z) =

one obtains a new differential equation:

wt = a(w)[wxx - Q(x,w)) All steady state solutions

(wt = 0) satisfy the simple equa-

tion

wxx = q(x,w). Example 1.13:

o

Parabolic differential equation in the sense

of Petrovski. Let T > 0, q c 1N, a C IR bounded-.

0 C Cq(]R, IR)

and

We seek a bounded solution for the problem

ut(x,t) = a(ax)qu(x,t)

x c IR, t c (0,T) .

u(x,0) = fi(x) q

odd or

(-1)4/2a < 0.

Special cases of this problem are given in Example 1.5 (for B # 0) and in Example 1.10.

The parabolic equations in the

sense of Petrovski are hyperbolic equations for

q = 1

are parabolic equations in the ordinary sense for (see §2).

For larger

q = 2

q, the properties of the problem

resemble those in the case lem is properly posed.

and

q = 2.

One can show that the prob-

The solution methods are difference

methods or Fourier transforms even if

q >

(cf.

2

§9).

The above equation has physical significance even when a

is complex.

For example, letting

a = 4nm

and

q =

2

yields the Schrodinger equation for the motion of a free particle of mass q

m

(h

is Planck's constant).

must be modified for complex

The condition on

a, to become

I.

18


Re(aiq) < 0.

Example 1.14: T > 0

o

Cauchy-Riemann differential equations. C°(R,IR)

and

bounded functions.

Let

We consider

the problem uy(x,Y) = -vx(x,Y)

x E IR, y E (0,T)

vy(x,Y) = ux(x,Y) u(x,0) = c(x),

v(x,0) = (x).

These two differential equations of first order are frequently combined into one differential equation of second order: uxx + uyy = 0.

This equation is called the potential equation and is the most studied partial differential equation.

The Cauchy-Riemann

differential equations are not a special case of the hyperbolic system of first order of Example 1.6 since the matrix A =

li

0,

is not real diagonalizable.

elliptic type (see §2).

Rather they are of

Although the initial value problem

at hand is uniquely solvable for many special cases, there is no continuous dependence on the initial values. Example:

y,w EIR, 4(x) = y sin wx, *(x) = 0.

As a solu-

tion of the Cauchy-Riemann differential equations one obtains: u(x,y) = y sin(wx) cosh(wy) v(x,y) = y cos(wx) sinh(wy). With

w = (u,v)

and

X =

(4,ip)

I I w ljW= y cosh(wT) IIXIIW = Y.

this yields:

2.

Types and characteristics

19

Thus the solution cannot satisfy a Lipschitz condition with respect to the initial values, and the problem is not properly posed.

This property carries over unchanged to the equivalent

initial value problem for the potential equation. In practice, only boundary value problems are consid-

ered for elliptic differential equations, since the solution does not depend continuously on the initial values.

2.

o


Since initial and boundary value problems in partial differential equations are not always properly posed, it is worthwhile to divide differential equations into various types. One speaks of hyperbolic, elliptic, and parabolic differential equations.

Of primary interest are initial value problems for

hyperbolic equations, boundary value problems for elliptic equations, and initial boundary value problems for parabolic equations.

Typical examples for the three classes of equa-

tions are the wave equation (hyperbolic, see Example 1.9), the potential equation (elliptic, see Example 1.14), and the heat equation (parabolic, see Examples 1.11, 1.12).

In addi-

tion, the concept of the characteristic proves to be fundamental for an understanding of the properties of partial differential equations.

In keeping with our textbook approach, we will consider primarily the case of two independent variables in this and the following chapters.

We consider first scalar equations of

second order, and follow with a discussion of systems of first order.

In the general case of

m

independent variables we

restrict ourselves to a few practically important types, since

20


I.

a more complete classification would require the consideration In particular, for the case

of too many special cases. m > 2

there exist simple equations for which none of the

above mentioned problems is properly posed. Definition 2.1:

with

C°(G xIR3,]R)

for all

Let

be a region in

G

a(x,Y,z)2

(x,y) E G, Z EIR3.

IR2

b(x,Y,z)2

+

and a,b,c,f c c(x,Y,z)2 > 0

+

The equation

a(x,Y,P)uxx + 2b(x,Y,P)uxy + c(x,Y,P)uyy + f(x,Y,P) = 0 with p(x,y) = (u,ux,uy) differential equation. au

xx

is called a quasi-linear second order The quantity + 2bu

xy

+ cu

yy

is called the principal part of the differential equation.

The description quasiZinear is chosen because the derivatives of highest order only occur linearly.

The differential equa-

tion is called semilinear when the coefficients of the principal part are independent of

p, and

a, b, and f

c

has the

special form f(x,y,p) = d(x,Y,u)ux + e(x,Y,u)uy + g(x,y,u) with functions

d,e,g a C°(G xlR,IR).

A semilinear differential

equation is called linear when the functions independent of

u, and

g

d

and

e

are

has the special form

g(x,y,u) = r(x,y)u + s(x,y)

with functions

r,s E C°(G,IR).

A linear equation is called

a differential equation with constant coefficients when the functions

a, b, c, d, e, r, and

s

are all constant.

a


2.

21

In order to define the various types of second order partial differential equations we need several concepts originating in algebra.

A real polynomial (real) form of degree

x cIRm

and all

quadratic form.

P(x) = P(xl,...,xm)

k

t cIR.

It may also be represented in matrix form

Without loss of generality A

holds for all

A form of degree two is called a

P(x) = xTAx,

Then

P(tx) = tkP(x)

if

is called a

A E MAT(m,m, IR),

x E IRm.

A may be assumed to be symmetric.

is uniquely determined by

P

and vice versa.

The

usual concepts of symmetric matrices positive definite

:

all eigenvalues of A greater than zero

negative definite

:

all eigenvalues of A less than zero

definite

positive definite or negative definite

positive semidefinite

all eigenvalues of A greater than or equal to zero

negative semidefinite

:

all eigenvalues of A less than or equal to zero

semidefinite

:

positive semidefinite or negative semidefinite

indefinite

:

not semidefinite

thus carry over immediately to quadratic forms. Definition 2.2:

To the differential equation of Definition

2.1 assign the quadratic form P(E,n) = a(x,Y,P)E2 + 2b(x,Y,P)En + c(x,Y,P)n2 Then the type of the differential equation with respect to a

I.

22

u c C2(G,]R)

fixed function


and a fixed point

(x,y) E G

is

determined by the properties of the associated quadratic form:

Type of d.e.

Properties of

P(t,n)

hyperbolic

indefinite (i.e. ac-b2 < 0)

elliptic

definite (i.e. ac-b2 > 0)

parabolic

semidefinite, but not definite (i.e. ac-b2 = 0)

The differential equation is called hyperbolic (elliptic, parabolic) in all of

with respect to a fixed function, if, with

G

respect to this function, it is hyperbolic (elliptic, parabolic) for all points

(x,y)

c G.

o

The above division of differential equations into various types depends only on the principal part. equations, the type at a fixed point with respect to all functions efficients

a, b, and

For semilinear

(x,y) c G

is the same

u c C2(G,]R); for constant co-

c, the type does not depend on the

point, either.

In many investigations it is not sufficient that the differential equation be hyperbolic or elliptic with respect to a function in all of the region.

In such cases one fre-

quently restricts oneself to uniformly hyperbolic or uniformly eZZiptic differential equations.

By this one means equations

for which the coefficients

a, b, and

dent of

and for which in addition

c G, z EIR3

(x,y)

ac - b

2

< -Y < 0

c

are bounded indepen-

(unif. hyperbolic)

2

ac - b

>

Y > 0

(unif. elliptic)

(x,y) c G, z d R 3

2.


where

23

y = const.

Linear second order differential equations with constant coefficients can, by a linear change of coordinates, always be reduced to a form in which the principal part coincides with one of the three normal forms uxx - uyy

hyperbolic normal form

uxx + uyy

elliptic normal form

uxx

parabolic normal form.

Even for more general linear and semilinear equations one can often find a coordinate transformation which achieves similar The type of a differential equation is not changed

results.

by such transformations whenever these are invertible and twice differentiable in both directions.

For the definition of characteristics we will need several concepts about curves. and let

I

with

a,b

c

a

G

A mapping

IR.

if

be a region in

G

¢

E

image of the curve

for all

t c

I.

is called the tangent to the curve the set

at the point

is called a

C1(I,G)

41(t)2 + 4Z(t)2 > 0

(¢i(t),$2(t))

1R2

(a,b), (a,-), (--,b), or

be one of the intervals

smooth curve in The vector

Let

0(I)

is called, the

0.

Definition 2.3:

We consider the differential equation in

Definition 2.1.

A vector

a = (B10B2) c ]R 2,

B # (0,0)

called a characteristic direction at the fixed point with respect to the fixed function

u E C2(G,IR)

is

(x,y)

if it is

true that

a(x,Y,P)BZ - 2b(x,Y,P)$1S2 + c(x,Y,P)B2 = 0.

E G

I.

24

The image of a smooth curve

in

4


is called a characteris-

G

tic of the differential equation with respect to the tangents

for aZZ

u

whenever

are characteristic directions for

(4i(t),421(t))

the differential equation at the points respect to

u

(41(t),42(t))

This means that

t e I.

0

with

is a solu-

tion of the ordinary differential equation a(41,42,P(41,42))42(t)2 -2b(01,02,P(01,02))01(t)0z(t) +c(01,02,P(01,(P2))01(t)2

The condition

(2.4)

0

t E I.

=

aa2 - 2b$162 + c02 = 0

can also be put in the

form S

when a # 0.

1

c # 0.

=

(b2-ac)1/2

b ±

S2

(2.5)

c

An analogous rearrangement is possible when

This implies that a hyperbolic (parabolic, elliptic)

differential equation has two (one, no) linearly independent characteristic direction(s) at every point. Examples: The wave equation characteristic directions

The heat equation istic direction The curve

uyy(x,y) - uxx(x,y) = 0 (1,1)

and

(1,-1)

ut(x,t) = auxx(x,t) - q(x) (1,0)

0

has

at every point. has character-

at every point.

is not sufficiently determined by the

differential equation (2.4).

Thus one can impose the normaliza-

tion condition

1(t)2 + 4 (t)2 = 1, Subject to the additional condition

t e

I.

(2.6)

a,b,c a C(G 1 x]R,IR), 3

2.


25

it can be shown that the initial value problem for arbitrary to c

for the ordinary differential equations (2.4), (2.6)

I

has exactly two (one) solution(s) with distinct support, if it is the case that the corresponding partial differential equation is hyperbolic (parabolic) in all of

G

with respect to

u.

In the hyperbolic case it follows that there are exactly two characteristics through every point

(x,y) e G, while in the

parabolic case it follows that every point

(x,y) c G

initial point of exactly one characteristic.

is the

The equation has

no characteristic when it is of elliptic type. The differential equation of the characteristic can be simplified when

c(x,y,p(x,y)) # 0

for all

(x,y) c G.

In

lieu of (2.6) we can then impose the normalization condition 42(t) = 1.

dy

.

=

With (2.5) it follows from (2.4) that

o (c UV,Y,P )) ±Vp,Y

b(* ,

A(,P,Y,p) = The image set

,

a('V,Y,p)c(1V,Y,p)

x = i(y)

is a characteristic.

simplification is possible for

An analogous

a(x,y,p(x,y)) # 0.

Finally we consider the special case where c(x,y,p(x,y)) = 0

at a point.

characteristic direction. plies that

(0,1)

This implies that

Thus also

(1,0)

a(x,y,p(x,y)) = 0

is a characteristic direction.

is a im-

Since it is

possible for hyperbolic equations to have two linearly independent characteristic directions, both cases can occur simultaneously.

Indeed, with an affine coordinate transformation

one can arrange things so that the characteristic directions at a given point with respect to a fixed function are in arbitrary positions with respect to the coordinate system, so

I.

26


the above representation is possible.

We next consider the type classification and definition of characteristics for systems of first order partial differSince parabolic differential equations

ential equations.

arise in practice almost exclusively as second order equations, we restrict ourselves to the definition of hyperbolic and elliptic. Let

Definition 2.7:

h e C°(G x 1Rn, IRn)

n elN, G

and

a region in

1R2,

A e C°(G x IRn, MAT(n,n, IR)).

The

equation uy - A(x,y,u)ux + h(x,y,u) = 0 is called a quasiZinear ferential equations.

system of first order partial dif-

The quantity uy - Aux

is called the principal part of the system.

The system is

called semiZinear if

u.

A

does not depend on

system is called linear when

A semilinear

has the special form

h

h(x,y,u) = B(x,y)u + q(x,y) with functions

B a C°(G,MAT(n,n,1R)), q c C°(G,IR').

A

linear system is called a system with constant coefficients A, B, and

if the functions Definition 2.8:

are all constant.

q

o

The system of differential equations in Defini-

tion 2.7 is called hyperbolic (elliptic) with respect to a fixed function if the matrix

u e C1(G,1Rn)

A(x,y,u(x,y))

real eigenvectors. G

and a fixed point has

n

(x,y) e G

(no) linearly independent

It is called hyperbolic (elliptic) in all

with respect to a fixed function if at every point

(x,y) e G

2.


27

it is hyperbolic (elliptic) with respect to this function. We consider the system of differential equa-

Definition 2.9:

A vector

tions in Definition 2.7. 0 # (0,0)

u C C1(G,IRn)

(x,y) e G

cIR2

of the matrix

An equivalent condition is in

respect to

01

y eIR

and an eigenvalue

A(x,y,u(x,y))

+ag2 = 0.

such that

The image of a smooth

is called a characteristic of the system with

G

u

with respect to a fixed function

if there exists a

A = A(x,y,u(x,y))

4>

a = (a1,a2 )

is called a characteristic direction of the system

at a fixed point

curve

a

if for aZZ

t e

I

the tangents

are characteristic directions of the system with respect to u

at the points

($1(t),02(t)).

This means that

0

is a

solution of the following ordinary differential equation: fi(t) + A(01,4>2,u(4>1,4>2))42(t) = 0,

t e I.

M

(2.10)

From the above definition and the additional normalization condition

S2 = 1

it follows at once that a hyperbolic

system has as many different characteristic directions at a point

(x,y) e G

eigenvalues. direction.

as the matrix

A(x,y,u(x,y))

has different

In an elliptic system there is no characteristic The differential equation (2.10) can be simplified

since we may impose the additional normalization condition 01(t) = 1.

We obtain 1P, (y) + A(p,Y,u(*,Y)) = 0.

The image set of the straight lines

x = ip(y)

is a characteristic.

Consequently

y e constant are never tangents of a

28


I.

characteristic in this system; this is in contrast to the previously considered second order equations. Examples:

uy = Aux + q

The system

from Example 1.6 is of hyperbolic

A

diagonalizable matrix

with the presumed real

The characteristics were already given in that example.

type.

(0

The Cauchy-Riemann differential equations

uy

1

01 0

ux

of Example 1.14 are a linear first order elliptic system with constant coefficients.

If all the coefficients are not explicitly dependent on

u, every quasilinear second order differential equation

can be transformed to a

first order system.

2 x 2

This trans-

formation does not change the type of the differential equaThus, we consider the following differential equation:

tion.

a(x,Y,ux'uy)uxx + 2b(x,Y,ux,uy)uxy + c(x,Y,ux,uY)uYY + Setting

v = (ux,uy)

f(x,Y,ux,uy)

yields the system av

av l

av 2

2

a(x,Y,v)x + 2b(x,Y,v)ax + c(x,Y,v)aY

If

3v2

av1

TX_

dy

c(x,y,z) # 0

solve for

v

for all

(x,y) c G

and all

y 0

1

vy = -a/c

-2b/cj

The coefficient matrix has eigenvalues 2 =

A 1 ,

= 0.

c

(-b ±Vb

.

+ f(x,y,v) = 0

z

eIR2

we can

2.

29


The corresponding eigenvectors are

When

(l,A1,2 ).

Al = X2 Thus the type of this first

there is only one eigenvector.

order system, like the type of the second order differential equation, depends only on the sign of

b2

-

ac.

We next divide partial differential equations with

m

independent variables into types, restricting ourselves to the cases of greatest practical importance. Definition 2.11:

Let

m c 3N, G

a region in

aik, f C C°(G XIRm+1,IR) (i,k = 1(1)m). a/ax. = a..

IRm

and

We use the notation

The equation m aik(x,P(x))aiaku(x) + f(x,P(x)) = 0

E

i,k=l

with

p(x) = (u(x),alu(x),...,amu(x))

second order differential equation. ity, we may assume the matrix

is called a quasiZinear Without loss of general-

A = (aid)

to be symmetric.

Then the type of the differential equation with respect to a fixed function

u c C2(G,IR)

and a fixed point

x c G

is

determined by the following table:

Type of d.e.

Properties of

hyperbolic

All eigenvalues of A(x,p(x)) are different from zero. Exactly m - 1 eigenvalues have the same

elliptic

All eigenvalues of A(x,p(x)) are different from zero and all have the same sign.

parabolic

Exactly one eigenvalue of A(x,p(x)) is equal to zero. All the remaining ones have the same sign.

A(x,p(x))

sin.

a


I.

30

Definition 2.12:

h e C°(G x ]Rm, IRn)

v = 1(1)m-1.

Let

m,n eIN, G

and

a region in

IRm,

Au a C°(G x ]Rn,MAT(n,n, ]R))

for

The system m-1

a

u(x) m

A (x,u(x)) a u(x) + h(x,u(x)) = 0 u=1

u

u

is called a quasiZinear first order hyperbolic system if there exists a

with

C e C1(G xIR", MAT(n,n,]R))

x e G, z cIRn.

regular for all

(1)

C(x,z)

(2)

C(x,z)-lAu(x,z)C(x,z) z eIRn, u = 1(1)m-l.

symmetric for all

x c

f

o

The concepts of principal part, semiZinear, constant coefficients, and the type with respect to a fixed function in all of G

are defined analogously to Definitions 2.1 and 2.7.

The

hyperbolic type of Definition 2.12 coincides with that of Definition 2.8 in the special case of

m = 2.

So far we have considered exclusively real solutions of differential equations with real coefficients. and boundary conditions were similarly real.

The initial

At least insofar

as linear differential equations are concerned, our investigations in subsequent chapters will often consider complex solutions of differential equations with real coefficients and complex initial or boundary conditions.

This has the effect

of substantially simplifying the formulation of the theory. It does not create an entirely new situation since we can always split the considerations into real and imaginary parts.

3.

3.

Characteristic methods for hyperbolic systems

31

Characteristic methods for first order hyperbolic systems Let

be a simply connected region and consider

G C I R 2

the quasilinear hyperbolic system

uy = A(x,y,u)ux + g(x,y,u).

A E C1(G x IRn, MAT (n,n, IR)) , g E C1(G x IRn, IRn) , and

Here

is an arbitrary but fixed solution of the sys-

u e C1(G,IRn)

For the balance of this chapter we also assume that

tem.

always has

A(x,y,z)

x, y, z, and

that

au(x,y)z),

different real eigenvalues

n

Their absolute value shall be bounded independently

u = 1(1)n. of

(3.1)

p < v

The eigenvalues are to be subscripted so

u.

implies

A

V

< Xv.

If the eigenvalues of a matrix are different, they are infinitely differentiable functions of the matrix elements.

When multiple eigenvalues occur, this is not necessarily the case.

Our above assumption thus guarantees that the are continuously differentiable (single-valued)

au(x,y,u(x,y))

functions on

There are always

G.

real eigenvectors. to

1.

linearly independent

Their Euclidean length can be normalized

They are then uniquely determined by the eigenvalue up

to a factor of G xJRn

n

+1

or

In the simply connected region

-1.

the factor can be chosen so that the eigenvectors are

continuously differentiable functions on exists an

E e Cl(G xJRf, MAT(n,n,IR))

G xlRn.

Thus there

with the following

properties: is always regular.

(1)

E(x,y,z)

(2)

The columns of

(3)

E(x,y,z)-1A(x,y,z)E(x,y,z) = diag(au(x,y,z))

Naturally, -E

E

are vectors of length

has the same properties as

E.

Let

1.

32

I.


From (3.1) one obtains the first

D(x,y,z) = diag(a)i (x,y,z)).

normal form E-Iuy For

E, D, and

=

E-1

DE-Iux

+

(3.2)

g.

we suppressed the arguments

g

(x,y,u(x,y)).

A componentwise notation clarifies the character of this normal form.

Let

g = (gv)

u = (uv),

E-I = (euv),

This implies

v l

n

au

n

euv y

n

au

v

ax

u v£leu

+

V

Ll euvgv,

u = 1(1)n

(3.3)

or

n

n

a

euvgv,

euv[ay - au axaluv

p = 1(1)n.

(3.4)

= v=1

Each equation contains only one differential operator, 8y

Au 8x' which is a directional derivative in a characteris-

tic direction (cf. Example 1.6).

However, this does not mean

that the system is uncoupled, for in general depend on all the components of

euv, au, and

gv

u.

For a linear differential equation, it is now natural to substitute

v(x,Y) = E(x,Y)-lu(x,Y) This leads to the second normal form 1

vy = Dvx + (aay

D aax ) Ev + E - Ig.

(3.5)

-

In componentwise notation, these equations become av

av

n +

ayu = u -1

(buv) _

E

buvvv + gu,

u = l(l)n

v=1

aay

-1

D aax )E,

(gu) = E-Ig.

(3.6)

33


3.

The original hyperbolic system and the normal forms obviously have the same characteristics.

They may be represented para-

metrically as follows:

x = 0(t) ,

y=t p = 1(1)n.

0,

V(t) + For each characteristic, p

Since the

is fixed.

(3.7)

are con-

au

tinuously differentiable, they satisfy local Lipschitz condiIt can be shown that there are exactly

tions.

same

is exactly one.

there

p

Two characteristics for the

No characteristic of

cannot intersect each other.

u

our system touches the

x-axis.

different

G; thus for each choice

characteristics through each point of of

n

Each characteristic cuts the

x-axis at most once.

We will now restrict ourselves to the special case n = 2.

In this case there are particularly simple numerical

methods for handling initial value problems. characteristic methods.

tial values on the set

They are called

For simplicity's sake we specify iniWe presuppose

r = {(x,y) c Gly = 0).

that:

is a nonempty open interval on the x-axis;

(1)

r

(2)

Every characteristic through a point of sects

G

inter-

r.

The second condition can always be satisfied by reducing the size of

G.

It now follows from the theory that the course of depends only on the initial values on

u

in all of

G

is the domain of determinancy of

two points of

G

r.

r.

Let

and

Q1

The characteristics through

Q1

and

then bound the domain of determinancy of the interval

r.

be

Q2 Q2

QlQ2'


1.

34

x-axis, one can

Since every characteristic intersects the

choose the abscissa of the intersection point

(s,0)

parameter for the characteristics, in addition to

V.

as a

A char-

acteristic is uniquely determined when these two are specified.

From (3.7) one obtains the parametric representation

_u(s,t) + at

(3.8)

0

= s,

E r,

s

u = 1,2.

11

The solutions are continuously differentiable.

are two characteristics through each point and

two abscissas, sl = pl(x,y) It is true for all

(x,y).

s =

Thus

pl

and

Since there

(x,y), there are

s2 = p2(x,y), for each point

t

that

p(0u(s,t),t)

s

u = 1,2.

e r,

are solutions of the initial value problem

p2

ap

ap

(x,y) e G, u = 1,2

ayu(x,y) = au(x,y,u(x,y))aXL(x,y), pu(x,0) = x,

x e r.

(3.9)

To prove this statement one must first show that the initial value problems (3.9) are uniquely solvable and that the solutions are continuously differentiable.

For these solutions it

is obviously true that pu(0u(s,0),0) = pu(s,0) = s,

On the other hand, the functions on

e r, u = 1,2.

s

t, since their derivatives with respect to ap

30

ax 7 t-

aP

ay

aP

u ax

With the aid of the projections

do not depend

pu(0u(s,t),t)

app

u ax pl

t

are zero:

= 0.

and

p2

one arrives at

3.

Characteristic methods for bvperbolic systems

Figure 3.10.

The domain of determinancy of the interval in the (x,y) plane and in characteristic coordinates (a,r).

35

PQ


I.

36

a new coordinate system in

G, called a characteristic coordi-

nate system (cf. Figure 3.10): a = 12-[P2(x'Y) + P1(x,Y)]

T =

12

[P2(x,Y)

- Pl(x,Y)] On

By previous remarks, the transformation is one-to-one.

r

one has a=x and T=y=0. The characteristic methods determine approximations for u, x, and

at the lattice points with characteristic co-

y

ordinates {(o,T)

Here

h

I

a = kh, T= th

with

k,Q E ZZ).

is a sufficiently small positive constant.

The simp-

lest method of characteristics is called Massau's method, which we will now describe in more detail. Let

Q0, Q1, and

Q2, in order, denote the points with

coordinates

a

r= t hh

k h,

a = (k-l)h,

T = (t-1)h

a = (k+l)h,

T = (L-1)h.

Massau's method uses the values of Q2

to compute the values at

for

T = 0

Q0.

u, x, and

y

at

Q1

and

Since the initial values

are known, a stepwise computation will yield the

values at the levels

T = h, T = 2h, etc.

Here one can ob-

viously restrict oneself to the part of the lattice with even or

k + 1 at

Q0

and

k + 1

Q1, as is

odd.

a + T

We note that at

Q0

and

a - T

is the same

Q2.

Therefore, Q0

and

Q1

lie on the characteristic

pl(x,y) = (k-l)h

and

Q2

lie on the characteristic

p2(x,y)

=

(k+l)h

and (cf.

Q0

3.


Figure 3.11).

37

In this coordinate system the characteristics

are thus the straight lines with slope

and

+1

-1.

th

(2-1)h (k-l)h kh (k+l)h

Figure 3.11.

Layers in Massau's method.

The numerical method begins with the first normal form (3.4)

and the differential equations (3.8) for and

E-1

72Q0'

01

and

are regarded as constant on the intervals

02.

A, A,

7

and

Their values are fixed at Q1 and Q2, respectively.

The derivatives along the characteristics are approximated by the simplest difference quotients. 0,1,2

We use superscripts

to denote the approximations for and

E-1 = (e,1)

g = (g v)

j

=

u, x, y, At Au,

at the points

Q0, Q1, Q2.

Then

we have

xo -x v = 1,2; j = 1,2 o- uj

yo

=

(ay

Aj aX)uv(x3,y3,u1).

y In detail, the computation runs as follows:

38


I.

(1)

Determine

AJ, (E-1)J, A

Determine

x°

for

j

= 1,2

and

u = 1,2. (2)

from the system of equa-

y°

and

tions x°-xJ + A J = 0, yo-yJ

j

= 1,2

J

or (x°-xl) + all (y°-yl) = 0 (x°-xl) + a22(y°-yl) = (x2-xl) + A22(y2-y l). (3)

Determine

tions

ui

u0-uI

2

from the system of equa-

u2

and 2

v=lejv y°-y) = v=l eI

t1,2

j

gV ,

=

or eI (u°-u l) + e1 (uo-u1) 2 2 11 1 1 12

_

(yo-y1)(eI119

e21(u1-u1) + e22(u2-u

=

e21L(y°-y2)g1

1 + 1

+

eI 12gI) 2 u1

-

u1]

2)

+

e22L(yo-y2)g2

+

2

u2

-

u2]

The rewriting of the systems of equations in (2) and (3) is done for reasons of rounding error stability. When

is sufficiently small, the matrices in both

h

systems of equations are regular.

For when

h

is suffici-

ently small, we have 1

1

2

A2 + Al

2

and 1

1

1

1

ell

e12

ell

e12

2

2

1

1

e21

e22

e21

e2

1

= regular matrix.

Massau's method sometimes converges in cases where value problem has no continuous solution.

the initial

As a rule, it is

3.


easily seen numerically that then the same pair

(x,y)

39

such a case has occurred, for

occurs for different pairs

Then there is no single-valued mapping

(x,y)

-

(a,T).

(a,T).

The

accuracy of Massau's method for hyperbolic systems is comparable to that of the Euler method for ordinary differential equations.

But there also exist numerous characteristic

methods of substantially greater accuracy.

The extrapolation

methods (see Busch-Esser-Hackbusch-Herrmann 1975) have proven themselves particularly useful.

For nondifferentiable initial

values, implicit characteristic methods with extrapolation are also commendable.

All these methods differ from Massau's

method in their use of higher order difference quotients.

All

in all one can say that under the conditions formulated above--

two variables, systems of two equations, A

has distinct real

eigenvalues--the characteristic methods are probably the most productive.

There also exist generalizations for

more gen-

erally posed problems; unfortunately, they are much more complicated and much less useful.

For that reason we want to

conclude our treatment of characteristic methods at this point and turn to other methods, known as difference methods on rectangular lattices.

The theory of normal forms may be found in Perron (1928), and the convergence proof for characteristic methods in Sauer (1958).

A FORTRAN program may be found in Appendix I.

40

4.

I.


Banach spaces

There are many cases in which initial value problems for linear partial differential equations can be reduced to initial value problems for ordinary differential equations.

However, in such cases the ordinary differential equations are for maps of a real interval into an appropriate Banach space of non-finite dimension.

One result of this reformulation of

the problem is that it is easier to make precise the concept of a properly posed initial value problem, as discussed in Chapter 1.

Lax-Richtmyer theory concerns itself with stability

and convergence criteria for difference methods.

As it starts

with the reformulated problems, a knowledge of these "Banach space methods" is absolutely essential for an understanding of the proofs.

The situation is different for practical applica-

tions of difference methods.

For then one almost always begins

with the original formulation as an initial value problem for a hyperbolic or parabolic differential equation.

Elliptic

equations do not play a role here, since the corresponding initial value problems are not properly posed.

In this section are defined the basic concepts of Banach space, linear operator, differentiability and integral in a Banach space, etc.

Also presented are several important

theorems which are necessary for the development of Banach space methods.

Definition 4.1:

Let

B

be a vector space over a field

is called a complex (real) Banach space whenever the following holds:

1K = t

(DC = IR) .

1.

B - [0,co),

In

B

B

there is a distinguished function

called a norm, with the following properties:

Banach spaces

4.

41

(a)

Ilall = 0 a= 0

aeB

(b)

Ilaall = JXI hall

A

(c)

IIa+bll

The space

2.

ology induced by of elements of that e

Ilan

{an}

11-11;

is complete with respect to the topi.e., every Cauchy sequence

converges to an element

B

amll< E.

element

a

a

{a

in

B.

n } neIIV Recall

is called a Cauchy sequence if for every positive

there exists an integer -

a,b a B.

IIaII * Ilbil B

e IK, a e B

in

n

The sequence

0

such that {an}

if the sequence

B

n,m > n

0

implies

is said to converge to the {Ila

-

anll} converges to 0. 0

Every Banach space thus consists of a vector space together with a defined norm.

Thus two Banach spaces with the

same underlying vector space are distinct if the norms are different.

In particular it is worth noting that an infinite

dimensional vector space which is complete with respect to one norm by no means need have this property with respect to any other norm.

In the following, we will speak simply of Banach spaces insofar as it is clear from context whether or not we are dealing with complex or real spaces.

Since later developments

will make heavy use of Fourier transforms, we will almost exclusively consider complex Banach spaces.

The vector space n becomes a Banach space with

Example 4.2:

either of the two norms

llxIl = m,x

Ix.I,

Ilxll = (

The same is true for any other norm on 1960, Ch. V).

o

4n

(cf. Dieudonne

42


I.

The set of all maps

Example 4.3:

x :2Z +

n

for which the

infinite series n

j=--

k=l

xk(j)x)

converges, becomes a vector space over

with the usual

definition of addition and multiplication by a scalar.

With

the definition

n

°°

I

1lxii

j=-co k=l

xk(j)xk(j))

1/2

this vector space becomes a Banach space, which we denote by (cf. Yosida 1968, Ch. 1.9). Example 4.4: C°(K,ln)

Let

o

be a compact set.

K C IRm

The vector space

is complete with respect to the norm

Iif1l,,= max max If.(x) I j

j

xeK

Here the completeness of the

and is therefore a Banach space.

space results from the fact that, with this definition of the norm, every Cauchy sequence represents a uniformly convergent sequence of continuous functions.

Such a sequence is well

known to converge to a continuous limit function, and thus to an element of the space.

The space

is not complete,

however, with respect to the norm

2= (J 11f11

n 1

K j=1

dx)1/2 f(x)f.(x j j

We can see this from the following counterexample: C°([0,2],4)

the sequence

fu (x) is a Cauchy sequence. limit function.

{fu}u

EIN

in

where

1xu

for

x e [0,1)

l

for

x e [1,2]

It converges, but not to a continuous

4.

Banach spaces

43

In the following, whenever we speak of the Banach space

C0(Ktn), we always mean the vector space of continuous functions

f:K - 4n

Example 4.5:

A = {f:G +

together with the norm

Let

n I

f

be a region in

G

called square-integrable in

n

G

and

]Rm

square-intebrable in

G}, where

f

is

if the integral

[f.(x)f.(x)] dx

E

fG j=1

>

>

exists as a Lebesgue integral and is finite. vector space over

o

11.11a

A

becomes a

with the usual definition of addition

4

and multiplication by a scalar.

The map

111.111:

A - [0,o)

defined by III f III

=

fj (x)x) dx) 1/2

(IG

J

has all the properties of a norm with the exception of

since

III fIII

for all f c N

=0

N = {f c A

I

{x E G

I

1(a),

where

f(x) # 01

has measure zero}.

One eliminates this deficiency by passing to the quotient space A/N.

The elements of

A/N

are equivalence classes of maps in

A, where the elements of a class differ only on sets of measure zero. way.

A/N

becomes a vector space over

4

in a canonical

With the definition

IIfII =IIIfIII,

f e A/N,

fef

this vector space becomes a Banach space, which we denote by L2(G,4n).

Although the vector space and norm properties are

easily checked, the proof of completeness turns out to be substantially more difficult (cf. Yosida 1968, Ch. 1.9).

In


I.

44

order to simplify notation and language, we will not distinguish between the equivalence classes presentatives

f e f

f e L2

and their re-

in the sequel, since the appropriate

meaning will be clear from context.

o

The following definition introduces the important concept of a dense set. Definition 4.6:

subsets of D2

a

if

Let

with

B

for every

B

be a Banach space and let

Dl C D2.

a s D2

Then

be

is called dense in

D1

and for every

such that Ila - b it < e.

b c D1

Dl. D2

c > 0

there exists

o

In our future considerations those vector subspaces of a Banach space which are dense in the Banach space play a significant role.

We first consider several Banach spaces of

continuous functions with norm

Because of Weierstrass's

fundamental theorem, it is possible to display simple dense subspaces.

Theorem 4.7: Weierstrass Approximation Theorem. be a compact set.

K C ]Rm

Then the vector space of polynomials with

complex coefficients defined on space

Let

is dense in the Banach

K

C°(K,4).

A proof may be found in Dieudonne (1960), Ch. VII.4. It follows immediately from this theorem that the spaces k = 1(1)", and

are dense in

they are supersets of the space of polynomials. we even have: Theorem 4.8:

(1)

The vector space

C°(K,1), since In addition,

4.

Banach spaces

45

V = If E

I

f(v)(a)

=

f(v)(b) = 0, v = 1(1)oo}

is dense in the Banach space (2)

The vector space of bounded functions in

is dense in the Banach space of bounded functions in

C`°(IR,c)

C°(IR,4).

The proof requires the following lemma. Lemma 4.9:

Let

with

c1,c2,dl,d2 c 1R

Then there exists a function

h e C °°(JR,4)

(1)

h(x) = 1

for

(2)

h(x) = 0

for x e IR

(3)

h(x) a

for

(0,1)

dl < cl < c2 < d2.

x e

x e

with

[cl,c2]

-

(d1,d2)

(d1,d2)

-

[c1,c2].

A proof of this lemma may be found in Friedman (1969), part 1, Lemma 5.1.

Proof of 4.8(1);

We first show that the space

V = If e V

I

f(a) = f(b) = 0}

is dense in the Banach space f(b) = 0}.

exists a

Now let

6> 0

with

f e W

W = If a C°([a,b],4) and

e > 0

If(x)I < 3 for

Choose

h e C°°(IR,¢)

where

d2 = b

as in Lemma 4.9.

f(a) _

Then there

x e [a,a+d) U (b-6,b].

dl = a, c1 = a+d, c2 = b-d and Since

CW([a,b],4)

there exists a function

IIf-giI,° < 3 Now let g =

be given.

I

Then:

g e

is dense in

with

46


I.

g e C([a,b],4); h(u)(a) = h(u)(b) = 0, u = O(l)eo V

g(v) (x)

I'

E

u=ot

by 4.9(2);

g(v u) (x)h(u) (x),

I

V = 0 (1)";

1)

v = 0(1)";

g(v) (a) = g(v) (b) = 0,

If(x)-g(x)I = If(x)-g(x)I for

x c [a+d,b-6]

Ig(x)I < If(x)-g(x)I + If(x)I < 3 e for

by 4.9(1);

x c (a,a+d) U (b-6,b];

If(x)-g(x)I < If(x)I + Ih(x)IIg(x)I < c x e [a,a+S) U (b-a,b];

for

IIf-gIl,< e And this shows that (1), let an arbitrary function

V

is dense in f e

To prove contention

W.

subject to Lemma 4.9 with

h(x)

Find a

be chosen.

d1 < cl < a < c2
0.

[k,k+l]

(k a 7L)

Let

f

be a bounded function in

By (1) there exist on each of the intervals functions

gk e C"([k,k+l],I)

9(v)(k) = gkv)(k+l) = 0, lf(x)

C°(1R,4)

- gk(x)I < e,

with

v = 1(1)" x e [k,k+l].

It follows from the proof of (1) that the functions be chosen so that additionally

gk

can

Banach spaces

4.

47

gk(k) = f(k),

gk(k+l) = f(k+l).

Thus

g (x) = gk (x) ,

x e [k,k+l)

is a bounded function in C(1R,4) with II f - g Ii, < e.

o

Next we consider two examples of dense subspaces of the Banach space

L2(G,4).

Theorem 4.10: (1)

Let

G

be a region of

The vector space

Co(G,4)

IRm.

Then:

of infinitely differ-

entiable functions with compact support defined on dense in the space (2)

is

L2(G,¢).

The vector space of polynomials with complex co-

efficients defined on G

G

G

is dense in the space

L2(G,4), if

is bounded.

Conclusion (1) is well known.

Proof:

and Theorem 4.7.

o

Definition 4.11:

Let

a subspace of

B1.

B1, B2

(2) follows from (1)

be Banach spaces and let

D

be

A mapping

A: D -. B2 is called a Zinear operator if it is true for all

and all

X,p a 1K

a,b e D

that

A(Aa + ub) = AA(a) + jA(b).

A linear operator

A

is called bounded if there exists an

a > 0 with IIA(a) II . aIIaII for all a e IIAII = inf{a e ]R+

I

IIA(a) II

all all

D.

The quantity

for all a e D)

is then called the norm of the linear bounded operator

A.

We

48


I.

define

L(D,B2) _ {A: D - B2

I

linear and bounded}.

A

In order to define bounded linear operators from

n

B1

B2

to

B1, because

it suffices to define them on a dense subspace of of the following theorem. Theorem 4.12:

Let

and

B1

be Banach spaces, let

B2

B1, and let

a dense vector subspace of

there exists exactly one operator

with

on

A

D.

extension of Proof:

A

on

a C B1.

Let

{a.)

a sequence

be

Then

A E L(D,B2).

which agrees

A E L(B1,B2)

Furthermore, IIAII = IIAII.

D

is called the

A

B1.

Since

B1, there exists

is dense in

D

of elements in

D

converging to

It

a.

follows from

IIA(aj )-A(ak) II = IIA(aj-ak) II _

IIAII

_

11 All

(Ila-ai II + Ila-akll

is a Cauchy sequence in

that

Being a Banach

B2.

is complete, and so there exists a (uniquely deter-

space, B2

mined) limit for the sequence note by

Ilaj-akli

c.

The quantity

c

in

{A(ai )}

B2, which we de-

depends only on

{aJ}; for suppose {a

the particular choice of sequence another sequence of elements of

a, and not on

D

is

a.

Then

{A(a5)}

also

converging to

one estimates

lic-A(a

II_ IIc-A(a IIc-A(a

II + IIA(ai ) -A(ai ) II II

+ IIAII (II a-a ll

+ Ila-i.I1)

Passage to the limit shows that the sequence converges to

c.

We define the mapping

A: B1 -. B2

by the

Banach spaces

4.

rule

A(a) = c.

49

is well defined and agrees with

A

To see the linearity of {a

and

ing to

a

{bJ}

A, let

D.

converg-

D

Then the sequence

b, respectively.

and

on

and let

be two sequences of elements of

converges to the element

ubj}

a,b c B1

A

{Xaj +

It follows that

as + ub.

A(Aa + ub) = lim A(Aa. + ub.) J

J

J9.°°

A lim A(a.) + u lim A(bp) = AA(a) + pA(b).

To see the boundedness of

above.

Then

A, let II,

IIA(aj) II < IIAII IIa

a

and

{aj}

be given as

and by the continuity of the

norm

II = lim IIA(a.) II IIA(a) II = II lim A(a.) 7 3

ja

j4-

_ IIAII dim Ilaj II = IIAIII11im a) II j 4-

= IIAII IIaII

It follows from this that IIAII = IIAII To see that directly. B2

D

is uniquely determined, we proceed in-

be a bounded linear operator from

which agrees with

for which of

A

Let

A

A

on

Suppose there is an

D.

{a.}

A(a), and let

A(a)

converging to

a.

+

to

a e B1-D

be a sequence of elements

Then

IIA(a) - A(a) II_ IIA(a) - A(a

IIA(a)-A(a.) II

B1

IIA(a)-A(a

Ti + IIA(a)

- A(aII

II < IIAII IIa-aj II + IIA1111 a-aj II

Passage to the limit leads to a contradiction.

0

By the theorems of functional analysis (in particular the Hahn-Banach Theorem), a bounded linear operator can be extended to all of

B1, preserving the norm, even when the domain

of definition

is not dense in

D

B1.

However, the extension

I.

50


is not unique in that case. Definition 4.13:

Let

B1

and

a set of bounded linear operators mapping called uniformly bounded if the set

{IIAII

B1 I

to

I

Theorem 4.14: B2

for all

I A (a) II < a IIa II

B1

to

M

a > 0

Let

such

a and

BI

a set of bounded linear operators

Suppose there exists a function

B2.

with IIA(a) II < B(a) for all a e B1 Then the set M is uniformly bounded.

B: B1 + IR+

A e M.

is

is bounded.

and all a E B1.

Principle of uniform boundedness.

be Banach spaces and

mapping

ACM

M

B2.

A 6 M}

This is equivalent to the existence of a constant

that

M

be Banach spaces and

B2

and all

For a proof of Theorem 4.14,

see Yosida (1968), Ch. II.1.

Observe that the function

need not be continuous or

B

linear.

Definition 4.15: real interval.

Let

B

be a Banach space and

[T1,T2] i B

is called differentiable at the point element

a e B

lim to+h a [T1,T21 The element vative of du t ).

u

a

a

A mapping u:

exists an

[T1,T2]

to c [T1,T21, if there

such that

= 0. IhI

is uniquely determined and is called the deriat the point

The main

u

to.

It is denoted by

or

u'(t0)

is called differentiable if it is

differentiable at every point of

[T1,T21.

The mapping

u

is called uniformly differentiable if it is differentiable

4.

51

Banach spaces

and if

1IIu(t+h)-u(t)-hu'(t)II converges uniformly to zero as mapping

u

h + 0

The

is called eontinuousZy differentiable if it is

differentiable and if the derivative [T1,T2].

t e [T1,T2].

for

is continuous on

u'(t)

o

It follows immediately from the above definition that a mapping which is differentiable at a point ous there.

is also continu-

to

It follows from the generalized Mean Value Theorem

(cf. Dieudonne 1960, Theorem (8.6.2)) that for a continuously differentiable function

u:

-1IIu(t+h)-u(t)-hu' (t) II

0

P(B,T,A).

there exists an element

This

c c DE

such that

IIE(t)(c) - E0(t)(E)II < e, (3)

Every generalized solution

for initial value (4)

c c B

belongs to

The linear operators

E(t)

t e [0,T]. E(-)(c)

of

P(B,T,A)

C°([0,T],B). c C°(B,B)

satisfy the

semigroup property: E(r+s) = E(r)E(s), (5)

For all

r,s,r+s a [0,T].

c c DE

E(t)A(c) = AE(t) (c) is satisfied. (6)

For all

c e DE, u(t) = E0(t)(c)

is continuously

differentiable. Proof of (1):

Follows immediately from Theorem 4.12.


I.

58

Of (2):

be given.

e > 0

Let

the norms of the operators

Let

Since

given by (1).

E(t)

DE

such that

c e DE

B, there exists a

is dense in

denote the bound for

L

IIc-c II < L E(t)(E) = Eo(t)(c), we estimate as follows:

Since

IIE(t) (c)-E0(t) (c) II = IIE(t) (c)-E(t) (c) II < L IIc-c II < e Of (3):

Let

s

and

e [0,T]

choose an element

c e DE

for which

t e [0,T] ,

IIE(t) (c) - E0(t) (c) II < 3 ,

is a differentiable map

Since

holds.

By (2), we

be given.

e > 0

fore continuous, there exists a

IIEo(s+h)(c)-Eo(s)(c)II < 3, Altogether, for all

with

h

6

>

such that

0

s+h e [0,T].

IhI < 6, IhI

< 6

and there-

s+h a [0,T], this

and

yields the estimate

IIE(s+h) (c)-E(s) (c) II < IIE(s+h) (c)-E0(s+h) (c) II

+

IIE0(s+h) (c) -E0 (s) (2) II + IIEu(s) (2)-E(s) (c) II < C. Of (4):

r e (0,T].

Let

consider

P(B,r,A).

5.4 which pertain to ditional index

r.

will be identified by an ad-

P(B,r,A)

For a solution P(B,r,A)

u

P(B,T,A), v

of

with

E,r

= span{_ U t

a[r,T]

E (`-r)(D )}. o

E

ob-

v(t) = u(t+r-r),

We consequently define D

P(B,T,A)

The quantities from Definitions 5.3 and

viously is a solution of r e [r,T].

In addition to problem

we

S.

Let

Stability of difference methods

r,s

59

be arbitrary, but fixed in

with

(0,T]

Without loss of generality, 0 < r < s.

r+s < T.

Then

DE C DE's C DE, r C DA C B Eo,r(t)(c) = E0(t)(c),

t e [O,r],

Er(t)(c) = E(t)(c),

t c [O,r], c e B

E0(s)(c) c DE

c e DE.

One obtains for

r'

c e DE

c e DE

E(r)oE(s)(c) = Er(r)oE0(s)(c) = Eo,r(r)oEo(s)(c) = Eo(r+s)(c) = E(r+s)(c).

This proves the contention that c e DE

For

Of (5):

and

DE

t e [0,T]

is dense in

B.

one has the following

estimate:

IIE(t)-A(c) - AoE(t) (c) II < IIE(t)-A(c) +

-

h(t) [E(h) (c)-c] II

E(t) [E (h) (c) -c] - AoE (t) (c) II

II

h

< +

I E ( t ) II IIE(h) (c)-c-hA(c) II

1 IIE (t+h) (c) -E (t) (c) -hA°E (t) (c) II

Since E(-)(c) = interval

Th-T I

.

is differentiable on the entire

[0,T], the conclusion follows by passing to the

limit h * 0. Of (6):

Since

u' (t) = A(u(t)) = AoE(t) (c) = E(t)oA(c) = E(t) (A(c)) the continuity of

u'

follows from (3).

60

I.


With the next two examples we elucidate the relationship of these considerations to problems of partial differential equations. Example 5.6:

Initial value problem for a parabolic differen-

tial equation. Let T > 0, :]R - c, and a e C-(IR, 1R) with a' c Co(JR, IR) and a(x) > 0 (x c IR). It follows that x cIR.

0 < K1 < a (x) < K2 , We consider the problem ut(x,t) = [a(x)ux(x,t)]x

x e1R,

u(x,0) = 4(x)

The problem is properly posed only if

u

t e (0,T).

and

4

to certain growth conditions (cf. Example 1.10).

(5.7)

are subject The choice

of Banach space in which to consider this problem depends on the nature of these growth conditions and vice versa.

choose

B = L2(1R,4)

DA = {f e B

I

We

and

f e C1(R,4),

af' absolutely continuous, (af')' a B}.

DA

is a superspace of

by Theorem 4.10.

C0_(1R,¢)

We define a linear operator f - (af')'.

the assignment into form (5.2).

and therefore is dense in A: DA - B

B

by

Problem (5.7) is thus transformed

That this is properly posed can be shown

with the use of known properties from the theory of partial differential equations. for example.

For this one may choose

DE = C0_(]R,4),

Generalized solutions exist, however, for arbit-

rary square integrable initial functions, which need not even be continuous.

The operators

E0(t)

and

E(t), which by no

5.


61

means always have a closed representation, can be written as integral operators for Example 5.8:

a(x) s constant

(cf. Example 1.10). a

Initial value problem for a hyperbolic differ-

ential equation. Let T > 0, : ]R + with

Ia(x)l

a e C(]R, ]R)

and

We consider the problem

< K, x e]R.

ut(x,t) = a(x)ux(x,t) t c (0,T).

x e IR,

(5.9)

u(x,0) _ fi(x)

For simplicity's sake we choose the same Banach space

B

as

in Example 5.6 and set DA = If c B

I

absolutely continuous, af'

f

We define the linear operator

A

E B}.

by the assignment

f + af'.

All other quantities are fixed in analogy with Example 5.6.

Once again it can be shown that the problem is properly posed. G We are now ready to define the concept of a difference method for a properly posed problem

P(B,T,A)

as well as the

related properties of consistency, stability, and convergence. Definition 5.10:

value problem and

Let

be a properly posed initial

P(B,T,A)

M = {E(t)

I

t e [0,T]}

the corresponding

set of generalized solution operators, as given in Definition 5.4, and (1)

ho a (0,T].

A family

MD - {C(h): B + B

bounded linear operators defined on method for

P(B,T,A)

h c (0,h0])

(O,h

is bounded in

0].

The difference method

if there exists a dense subspace

of

is called a difference

B

if the function

every closed interval of (2)

I

MD DC

is called consistent in

B

such that, for

I.

62

all


c c DC, the expression

II [C(h) - E(h)] (E(t) (c)) fI

1

converges uniformly to zero for (3)

t c [0,T]


MD

h + 0.

as

is called stable if

the set of operators {C(h)'

I

h E (0,h0], n E IN, nh < T}

is uniformly bounded. (4)


MD

is called convergent

if the expression

JJC(hj) i(c) - E(t) (c) lI c e B, for all

converges to zero for all all sequences 0.

Here

if

{njhj}

{hj}

{n.}

of real numbers in

t c [0,T], and for converging to

(0,h0]

is an admissible sequence of natural numbers

converges to

t

and

njhj < T.

a

The following theorem explains the relationship between the above concepts. Theorem 5.11:

Lax-Riehtmyer.

Let

MD

be a consistent dif-

ference method for the properly posed initial value problem P(B,T,A).

Then the difference method

MD

is convergent if

and only if it is stable.

Proof: (a)

Convergence implies stability: We will proceed indirectly,

and thus assume that

MD

there exists a sequence

is convergent but not stable. {h

sequence of natural numbers

of elements in {n.}

(O,h0]

Then and a

related by the condition n

njhj c [0,T], j

E :1N

so that the sequence QC(hj)

ill)

is not


S.

bounded.

Since

[0,T]

and

63

are compact, we may as-

[O,h0]

sume without loss of generality that the sequences and

{hi}

h > 0.

converge to

t

c

and

[0,T]

From a certain index on, nj

5.10(1), IIC(')II n

is bounded in

h c [O,h0].

is constant.

{njhi}

Assume

By Definition

Consequently

[h/2,h0].

n

IIC(h) 'II 0

such that

I Q (h) II

K1 >

0

and

u e IN, h e (O,ho], uh < T

IIC(h)`'II < K1, I

{C(h) + hQ(h)

< K2 ,

h e (0,h0].

On the other hand, we have a representation la)

u

[C(h) + hQ(h)]u =

E

hA

A=0

with operators

P

E K=1 Pa,K

which are products of

XK

factors.

p

,

C(h)

occurs

(u

-

times as a factor, and

x)

C(h), which are not divisible by

We gather the factors as powers.

Now in

Q(h), A

PAK

there are at most

X+l

times.

Q(h),

powers of

,

C(h)

gathered in this way, so that we obtain the estimate

68

I.


IIPx,KII < Ki+1 K2 Altogether, it follows for uh < T

u e]N

and

h c (O,h0]

with

that

II[C(h)+hQ(h)I"II
vO(c,e)

e > 0

there exists a We can

dvK < eLhv.

implies

Eo(t)(c):

I I Eo (t) (c) I I = lim II rvo EO (t) (c) II(v) V-}M

< lim sup IICvor n (C) 11(v) + lim sup V

V"*0°

V+

V

n v-1 I d VK K=O

< L IIcli + eLT.

Since

c > 0 is arbitrary, we also have IIE0(t) (c) II

L IIchI

This inequality, however, was derived under the assumptions -u

t =

2

and

c e DC.

Since the function

differentiable, it is also continuous for fixed viously admitted

t

values are dense in

inequality holds for all sion that

DE C UC.

t c [0,T].

[0,T].

E

0(-)(c)

c.

is

The pre-

Hence the

Finally, use the inclu-

Then it follows for all

c e DE

and

t c [0,T]

72

I.


II_ L II c ll

I I Eo (t) (c)

This proves conclusion (1) of Theorem 5.15.

Again we assume that a fixed

Proof of 5.15(2):

c c DC and

-u

t = u1K2

have been chosen.

2

Similarly to the above, one

can then estimate: n IICvvorv(c)

-

r,eEo(t)(c)II(v)
vo(c,c).

This inequality obviously implies convergence for all Now let

be arbitrarily chosen and let

c e DE

c c DC.

c c DC.

This

yields

IIC VorV(c) - rVoE0(t) (c) II(v) < lICvvorv(c)-Cvvorv(c) II(v) n

+ IlCvvorv(c) - rvoEo(t) (c) II(v) + II rvoEo (t)

(c) - rvoE0 (t) (c) II(v)

< L II rv (c-c) II(v)

+

II rvoEo (t) (c-c) II(v)

n + II Cvvorv (c) - rvoEo (t) (c) II(v) By passing to the limit

Jim v-,-

Here small.

v - -

.

one obtains

sup IICnvorV(c) - rvoEo (t) (c) II(v) < L IIc-cII + 11 E0(t) (c-c) II . E0(t) 13

is bounded and

IIc

-

cII

can be made arbitrarily

6.

Examples of stable difference methods

6.


73

This chapter is devoted to a presentation of several difference methods, whose stability can be established by elementary methods.

We begin with a preparatory lemma and

definition. Lemma 6.1:

C°(]R, ]R+)

f e C°(R,O

Let

bounded, and

be square integrable, a e

Ax a ]R.

Then

(1)

r T{a(x+Qx/2)[f(x+Ox)-f(x)] - a(x-Ax/2)[f(x)-f(x-ox)]}dx -J+W

a(x)lf(x+ox/2)-f(x-&x/2)l2dx.

_

(2)

r+

la(x+Ax/2)[f(x+Ax)-f(x)]

-

a(x-&x/2)[f(x)-f(x-Ax)]l2dx

r+

E

< 4J-: a(x)2lf(x+Ax/2)-f(x-Ax/2)12dx. Proof of (1):

We have

+

r+00

g(x+Ax)dx = J-

g(x) dx 00

for all functions

g

for which the integrals exist.

Thus

we can rewrite the left hand integral as follows: 7{a(x+Ax/2)[f(x+Ax)-f(x)]-a(x-Ax/2)[f(x)-f(x-Ax)]}dx

E 00

x-Ax 2 f(x+.x/2)-a(x)

x-Ax/

f(x-Ax/2)

74


I.

x+ x

a(x)

x+tx

f(x+1x/2)+a(x)

f(x-tx/2)]dx

J+ a(x)lf(x+Ax/2)-f(x-Ax/2)1 2dx. a,s e t

For

Proof of (2):

1a +

we have

012
0

for

ja(x)

a + (x) _

for

a(x) < 0

a (x) = otherwise.

0

This leads to the difference method C(h) = Xa (x)T-1+{1-A[a+(x)+a-(x)]}I+Xa+(x)TAx, Let the function

Theorem 6.9:

a(x)

satisfy a global Lip-

schitz condition with respect to the norm Ia(x)l < K.

X = h/tax.

11.112

and let

Then the Courant-Isaacson-Rees method, for

0 < A < 1/K

(stability condition)

is a consistent and stable difference method for problem P(L2(IR,4),T,A)

Proof:

of Example 5.8 of consistency order

0(h).

The proof of consistency we leave to the reader.

The

stability of the method follows immediately from the methods of Section 8, since the method is positive definite in the terminology of that chapter.

o

The dependencies of the various methods are shown pictorially in Figure 6.10.

The arc in the naive method in-

dicates that the

derivatives are not related to

each other.

x

and

t

6.

87


n J or L "naive" method

Friedrichs method

Courant-Isaacson-Rees method

Figure 6.10

The instability of discretization (6.7) is relatively typical for naive discretizations of hyperbolic differential equations.

Stability is often achieved by means of additional smoothing terms, which remind one of parabolic equations.

We clarify

this in the case at hand by listing the unstable method and the two stabilizations, one below the other: (1)

u(x,t+h)-u(x,t) = ZAa(x)[u(x+Ax,t)-u(x-Ax,t)].

(2)

u(x,t+h)-u(x,t) = ZAa(x)[u(x+Ax,t)-u(x-ox,t)] +

(3)

2[u(x+Ax,t)-2u(x,t)+u(x-&x,t)].

u(x,t+h)-u(x,t) = ZAa(x)[u(x+Ax,t)-u(x-ox,t)] + ZAIa(x)I[u(x+nx,t)-2u(x,t)+u(x-Ax,t)].

The additional terms in (2) and (3) may be regarded as a discretization of viscosity.

cause

c(x,h)uxx(x,t).

They are called numerical

They do not influence the consistency order be-

c(x,h) = 0(h).

For higher order methods, the determina-

tion of suitable viscosity terms is more difficult.

On the

one hand, they should exert sufficiently strong smoothing,

while on the other, they should vanish with order of higher powers of

h.

Let us again consider the domains of dependence and

88


I.

determinancy for the differential equation and difference For the sake of clarity, let

method at hand.

The domain of determinancy for a segment on the

a(x)

__

constant.

x-axis is

then a parallelogram whose right and left sides are formed by characteristics, since the solution of the differential equaThe domain of depen-

tion is constant on the characteristics.

dence of a point, therefore, consists of only a point on the For the discussion of the domains of

x-axis (cf. Example 1.5).

dependence and determinancy of the difference method, we divide the interval Then

[0,T]

Ox = T/(na).

into

n

pieces of length

For

A < 1/jal

and

A > 1/lal

one needs initial values from the interval pute the value

h = T/n.

w(O,h).

[-Ax,Ax]

The determination of

w(O,T)

to comre-

quires initial values from the interval [-ntx,n1x] = [-T/a, T/X],

which is independent of

n.

Thus the domain of determinancy

is a triangle and the domain of dependence is a nondegenerate interval, which contains the "dependency point" of the differential equation only for For

and not for

A < 1/lal

A >1/lal.

A = 1/lal, the domains of determinancy and dependence

of the differential equation and the difference identical, since only a term in

T_'

or

Tax

method are remains in the

expressions

C(h) =

1-Aa

-1 + l+Aa T Ax TAx

C(h) = as T1 +

(1-AIal)I+Aa+T_

Ax.

In this case, the difference method becomes a characteristic

7.

Inhomogeneous initial value problems

89

method.

This situation is exploited in the Courant-FriedrichsLewy condition (cf. Courant-Friedrichs-Lewy, 1928) for testing the stability of a difference method.

This condition is

one of necessity and reads as follows:

A difference method is stable only if the domain of dependence of the differential equation is contained in the domain of dependence of the difference equation upon passage to the limit

h -+

0.

The two methods discussed previously, for example, are stable for exactly those

A-values for which the Courant-Friedrichs-

Lewy condition is satisfied. called optimally stable.

which the ratio

A = h/Ex

Such methods are frequently

However, there also exist methods in must be restricted much more

strongly than the above condition would require.

7.

Inhomogeneous initial value problems So far we have only explained the meaning of consis-

tency, stability, and convergence for difference methods for homogeneous problems u'(t) = A(u(t)) ,

t E [0,T]

u(0) = C. Naturally we also want to use such methods in the case of an inhomogeneous problem u'(t) = A(u(t)) + q(t),

t c [0,T]

u(0) = c. We will show that consistency and stability, in the sense of Definition 5.10, already suffice to guarantee the convergence


I.

90

of the methods, even in the inhomogeneous case (7.1).

Thus it

will turn out that a special proof of consistency and stability is not required for inhomogeneous problems. For this section, let

P(B,T,A)

be an arbitrary but

fixed, properly posed problem (cf. Definition 5.3). tinuous mappings

q:

IIgII = form a Banach space Theorem 7.2:

Let

[0,T] - B, together with the norm max Iiq(t)II te[0,T]

BT = Co([0,T],B).

c e B, q e BT, 8 e [0,1], and a consistent

and stable difference method given. j

The con-

MD = {C(h)

Further let the sequences

= 1(1)", be such that

lim n.h. = t c [0,T]. j+oo ference equations

hi

I

h e (0,h0))

e (O,h0]

and

be

ni cIN,

n.h. < T, lim h. = 0, and 1 3 j*oo Then the solution u j of the dif-

(n)

u(v) = C(hi)(u(v-l)) + hjq(vhj-8hj),

v = 1(1)nj

u(0) = c converges to

u(t) = E(t) (c) +

t r0

E(t-s) (q(s))ds.

I

u

is called the generalized solution of problem (7.1).

Proof: t

We restrict ourselves to the cases

8 = 0

and

< njhj < t+h, j = l(l)co, and leave the others to the

reader.

Further, we introduce some notational abbreviations

for the purposes of the proof: C = C(hi),

tv = vhj,

We will now show the following:

qv = q(vhj), n = nj.

7.

Inhomogeneous initial vale problems

E(t-s)[q(s)]

(1)

{(t,s)

is continuous and bounded on

t e [0,T], s c [O,t]}.

I

ft

ft

E(t-s)[q(s)]ds.

E(nhj-s)[q(s)]ds =

lim

(2)

0

0

For every

(3)

E > 0

IIE(tn-v) (q v) Let

Proof of (1): s

91

-

there exists a

jo e 1N

> jot v < n.

j

Cnv(qv) II < E

IIE(t)II < L, t e [0,T].

such that

For fixed

t

and

we consider differences of the form D = E(t-s)[q(s)] - E(t-s)[q(s)]

E(t-s)[q(s)]

- E(t-s)(q(s)-q(s)]

- E(t-s)[q(s)]

Either

E(t-s) = E(t-s)oE(t-s-t+s) or

E(t-s) = E(t-s)oE(t-s-t+s). In either case,

IIDII < L IIE(It-s-i+sI)[q(s)] - q(s)II + L IIq(s)-q(s)II By Theorem 5.5(3), E(It-s-t+sl)[q(s)] tion of

s-t.

Since

q(s)

is a continuous func-

also is continuous, the right side

of the inequality converges to zero as E(t-s)[q(s)]

Since the set IIE(t-s)(q(s))II

Proof of (2):

is also continuous in {(t,s)

I

t

('t,s) - (t,s).

and

t e [0,T], s e [O,t]}

s

simultaneously. is compact,

assumes its maximum there. By Theorem 4.19(l) we have

to E(nhj-s)[q(s)]ds = E(nhj-t)[I

E(t-s)[q(s)]ds].

Since every generalized solution of the homogeneous problem is continuous by Theorem 5.5(3), the conclusion follows at once.

92

Proof of (3): j


I.

= l(1)-.

Let

IIE(t)II < L

and

IIC(h)nII < L, t e [O,T],

By the uniform continuity of

q, there exists a

such that

d > 0

t,i a [O,T], It-tj < a.

Iiq(t)-q(t)II < 4L'

Furthermore, there are finitely many

such that

u

0 < ua < T.

The finitely many homogeneous initial value problems with initial values

q(p6)

there exists a

jo e]N

can be solved with such that for

IIE(t-pd) [q(ua)] Here the choice of

v

-

j

Therefore

MD.

> jo

and

ua < t

Cn-v[q(ua)] II < 4'

depends on

u, so that

vhj < ua < (v+l)hj.

The functions

E(s)[q(ua)]

Therefore there exists a it-ti

0

such that for all

t, t

s.

with

a

IIE(t) [q(ua)] - E(t) [q (116)] II < q In particular, by choosing a larger

if necessary, one can

jo

obtain

IIE(t) [q(ua)] - E (t-ua) [q(ua)] II < 4

t e [t-u6-2h., t-u6+2hj],

j

> jo.

Since

to-v = nh.-vhj = t-ua+(nhj-t) + (p6-vhj)

we always have

IIE(tn-v) [q(ua)] - E(t-ua) [q(ua)] II < 4 Combining all these inequalities, we get


7.

IIE(tn-v)(gv)-Cn-v(qv)

93

II < IIE(tn-v)(gv)-E(tn-v) [q(ua)] II

(t ua) [q(ua)] II

+ IIE(tn-v) [ Q(ua)] + IIE(t-p6) [q(ua)]

+IICn-v[q(ua)]

Cn-v[q(ua)]II -

Cn-v(gv)II -

E

E

E

E

L 4L + 4 + 4 + L 4L = E. This completes the proof of (1),

(2), and (3).

The solution of the difference equation is n

u(n) = Cn(c) + h.

Cn-v(qv)

E

V=1

It follows from Theorem 5.11 that lim Cn(c) = E(t)(c). j-wu

Because of (2), it suffices to show

n E Cn v(qv) = lim

lim h. j_)._

J vml

t

0

jam-

E(nh.-s)[q(s)]ds,

and for that, we use the estimate

llh

n

i v1

Cn-v (q ) v

-

I

E(nh.-s)[q(s)]dsll

J0

n

vI1Cn-v(qv)

n hj

vIlE(tn-v)(gv)II

IIhJ

n + Ilhj

nh.

V 1 nhj

+ I1J0

E(tn-v)(qv)

-

E(nhj-s)[q(s)1dsll

J

t

E(nhj-s)[q(s)]ds -

I

E(nhj-s)[q(s)]dsll

The three differences on the right side of the inequality converge separately to zero as

j - -.

For the first differ-

ence, this follows from (3); for the second, because it is the

94


I.

difference between a Riemann sum and the corresponding integral (cf. Theorem 4.19(2)).

0

The generalized solutions of (7.1) are not necessarily differentiable, and thus are not solutions of (7.1) in each and every case.

The solutions obtained are differentiable

only if

and

c e DE

are sufficiently "smooth".

q

Exactly

what that means will now be made precise. We define

Definition 7.3:

DA = {c e B A

:

DA } B For

Remark:

For if

I

u(t) = E(t)(c)

A(c) = u'(0).

given by

c c DA, u

is differentiable for t = 0},

is differentiable on all of

tl > t2, then

h [u(t1)-u(t2)] = E(t2){h[E(tl-t2)(c)-c]}. There is a simple relationship between tion

[0,T].

of

u

A

and

is also a solution of

P(B,T,A)

A.

a

Every solu-

P(B,T,A), i.e.

u'(t) = A(u(t)) = A(u(t)). In passing from

P(B,T,A)

to

P(B,T,A), we cannot lose any

solutions, though we may potentially gain some.

are defined may be enlarged under

which the operators

E0(t)

some circumstances.

The operators

changed.

The space on

E(t), however, remain un-

Also nothing is changed insofar as the stability,

consistency, and convergence properties of the difference methods are concerned.

It can be shown that

mapping, i.e., that the graph of This implies that with.

A = A

whenever

A

in

A

B x B

A

is a closed is closed.

is closed to begin

Since we shall not use this fact, we won't comment on

7.


95

the proof [but see Richtmyer-Morton (1967), 3.6 and Yosida (1968), Ch. IX].

In our examples in Section 5, A

is always

closed.

Theorem 7.4:

q E BT

Let

and

q(t) = Jo 0(r)E(r) [q(t)]dr 0

0 e C-(R, IR)

where

Support(4) C (0,T).

and

Then

{ft E(t-s)[q(s)]ds}' = q(t) + A{Jt E(t-s)[q(s)]ds}. 0 0

Remark:

It can be shown

is called a regularization of

q

that with the proper choice of rarily little.

For

and

0, q

q

differ arbit-

c c DE,

u(t) = E(t) (c) +

r0t

E(t-s) [q(s)]ds

J

is obviously a solution of u'(t) = A(u(t)) + q(t),

t

c [0,T]

= c.

u(0)

o

For

Proof of Theorem 7.4:

t c [T,2T]

define

E(t) = F.(t/2)oE(t/2).

There exists an

c >

0

Support(4) C [2E,T-2E].

such that

Let

ft

f(t) =

J

For

IhI

0

0

T-c 4(r)E(t+r-s)[q(s)]drds.

J

E

we obtain

96

I.


f(t+h) = I1(h) + I2(h) t T-e 0(r)E(t+h+r-s)[q(s)]drds 11(h) = j J e

0

t+h T-e I2(h) =

j fe

t

We make the substitution

r = r+h

and exchange the order of

integration (cf. Theorem 4.19(5)), so that reT++h h- e

r0 t

I1(h) =

4(r-h)J

E(t+r-s)[q(s)]dsdi

J

t

r0 T

I,(h) =

E(t+r-s)[q(s))dsdi.

1$(i-h)J 0

has the derivative

I1

T

t

f 0 o

0

E(t+i-s)[q(s)]dsdr.

Ii(0) _

The second integral we split one more time, to get rt+h rT+h-e

0(f)E(t+i-s)[q(s)]drds

I2(h) = 1

+

J

t

e+h

rt+hrT+h-e 1

[(i-h)-q(r)]F.(t+r-s)[q(s)]drds.

1

t

e+h

In the first summand we can again change the limits of the innermost integral to

0

longer depends on

The summand obviously can be differ-

h.

entiated with respect to

and

h.

second summand is of order for

h = 0.

It follows that

T.

Since

The integrand then no

is bounded, the

0(Ih12), and hence differentiable

8.

Difference methods with positivity properties

I2'(0)

rT

4'(r)E(i)[(t)]dr = q(t)

= 1

97

0 t

r0 T

'(r)1 E(t+r-s)[q(s)]dsdi.

+ IZ(0) = q(t)-1

Il'(0)

0

Now let

g(h) = E(h)(

r0t

E(t-s) [q(s)]ds)

1

rt T-C f0

C

0(r)E(t+h+r-s)[q(s)]drds = I1(h)

g'(0) = 11(0). Therefore ft

E(t-s)[q(s)]ds C DA

11

0

and rt

rT

A(I E(t-s)[q(s)]ds) _ -1 1110

8.

t

$'(i)f E(t+i-s)[q(s)]dsdr. 0

o

0

Difference methods with positivity properties The literature contains various (inequivalent!) defini-

tions of difference methods of positive type (cf., e.g.,

Friedrichs 1954, Lax 1961, Collatz 1966, Tornig-Ziegler 1966). The differences arise because some consider methods in function spaces with the maximum norm, and others in function spaces with the

L2-norm.

both categories.

A number of classical methods fit into We will distinguish the two by referring to

positive difference methods in the first case, and to positive definite difference methods in the second. In the hyberbolic case, with a few unimportant excep-

tions, even if the initial value problem under consideration has a

Cm-solution, these methods all converge only to first

98


I.

order (cf. Lax 1961).

However, they allow for very simple

error estimates and they can be carried over to nonlinear problems with relative ease.

We consider positive difference methods primarily on the following vector spaces:

Bln = {f a C°(R,4n) I lim II f(x)

0}

x-+m

B2n = { f a co R

2n-periodic}

B3n = {f E B2nIf(x) = f(-x),

x c 1R)

B4n = {f c B2nIf(x) = -f(-x),

x E]R)

BSn = {f E C°([-n/2,3n/2],'n)If

Definition 8.1:

satisfies the equations in Def. 8.1).

Functional equations.

(Bo + aox)f(-x) = (8° - aox)f(x) x e (0,n/2)

(Sn + anx)f(n+ x) _ (an Here we let with

a

+B 0

o

B5n

ao, Bo, an, Bn >

0

and

an +S

n

- anx)f(n- x).

be real, nonnegative constants > 0.

o

naturally depends on the given constants.

Since

we shall think of these as fixed for the duration, we suppress them from the notation and use the abbreviation

B5n.

of the various functional equations, the functions in through

Because B2n

B5n are determined by their values on a part of the

interval of definition.

Therefore we define the primary in-

tervals to be G1 =IR,

G2

G3 = G4 = G5 = [0,n]

and the intervals of definition to be

Difference methods with positivity properties

8.

Dl = D2 = D3 = D4 For

sup IIf(x)II_= max

IIf(x)II.,

fe

un

B

xeG u

xcDU

Bun, which we write

These suprema are norms in

The

IIfII,;

become Banach spaces by virtue of these norms

Bun

and are subspaces of

C°(DU,4n).

study of functions on ditions.

D5 = [-r/2, 31T/2].

IR, =

we obviously have

p = 1(1)5

spaces

99

Thus if

GU

They are useful for the

which satisfy certain boundary con-

we have the following

f c Bun (1 C1(DU,4n)

relations, by cases:

and

p = 2:

f (-7T) = f (7r)

p = 3: p = 4:

f'(0) = f'(r) = 0

p = 5:

aof(0)

f'(-7r) = f' (r)

f (0) = f (r) = 0 -

sof'(0) = arf(7r)+$rf'(r) = 0.

Obviously the boundary conditions for

p =

special cases of the boundary conditions for spaces

and

B3n

B

4n

Let

g(x) = f(x)

f"(0) = 0

sup

x e G5.

c

Further let

and simultaneously, Sr # 0

xe[-tx,r+Ax] Proof:

g e B5n n C1(DS,cn), f

for all

g e C2(DS,tn)

DU

or

to

B5n

and

# 0

or

f"(r) = 0.

Then

and

IIf(x) - g(x)IIm= 0((Ax)3),

g 6 BSn fl C1(DS,4n)

implies

if

[-r/2,3r/2].

C3(DS,1n) 0°

are

The

p = S.

also become special cases of

we restrict the intervals of definition Lemma 8.2:

p = 4

and

3

Ax a (O,r/2).

100


I.

aog(0)

-

a,g(n) +

Sog'(0) = aof(0) B,ffg' (n)

-

B0f'(0) = 0

= aof(n) + anf' (n) = 0.

Furthermore, by 8.1

g(x)

I

f(x)

for xe[0,7r]

f(-x)(So+aox)/(eo-a0x)

for xe[-7/2,0)

I f(2r-x)[6 n-a7r(x-1r)]/[Bn+a7r (x-i)] for xe[a,3n/2].

First let

B0 # 0.

For

x < 0

we obtain

g'(x) = 2a0B0f(-x)/(80-a0x)2 - f'(-x)(B0+aox)/(a0-aox) g"(x) = 4a2B0f(-x)/(B0-a0x)3 - 4a0a0f'(-x)/(Bo-a0x)2 + f"(-x)(a0+aox)/(Bo-a 0x)

g"(-O) = 4aof(0)/ao - 4a0f'(0)/B0 + f"(0) = 4a0[aOf(0) =

-

a0f'(0)]/Bo + f"(0)

f"(0) = g"(+0). B0 = 0, we have for

In the exceptional case

x
a

f - b

L2(IR,Cn); i.e., the mappings

are linear, injective, surjective, and satisfy the condition II a II = II f II

= II b II.

The second mapping is the inverse of the

We denote the first by J'

first.

and the second by

_Znl.

In order to simplify our notation, we will be somewhat imprecise in the sequel and write _F n(f)(Y) =

(2n)-1/2 ff(x)exp( iyx)dx = a(Y)

,

Y 1(a)(x) = (2r)

(9.5)

f(x)

(9.6)

J

This ignores the fact that the integrals converge only in the mean, in general.

Pointwise convergence of the integrals only

occurs in special cases, e.g., when support. of

f.

f

or

a

has compact

Representation (9.6) is called the Fourier integral From a physical point of view, it says that

f(x)

cannot be built out of harmonic oscillations alone, but that (complex) oscillations of all frequencies

y

arise.

There-

fore, the infinite series (9.3) has to be replaced by an integral, where the "infinitesimal" factor to the previous coefficient

a(v).

a(y)dy

The following

corresponds lemmas

describe important computational rules for Fourier series and Fourier integrals.

Fourier transforms of difference methods

9.

Ax CIR+, define

Lemma 9.7:

For

exp(ixAx).

Then

6Ax:

123

1R - I by eAx(x) =

sAX(v) .f2 w,n(f)(v)

(1)

f e L2((0,27r),n), v e f c L2(R,4n).

n[TAX (f)] =

(2)

Proof:

Tl

Conclusion (1) follows from the relation

(2Tr)-1/2 j

2x

Tox(f) (x)exp(-ivx)dx

0

2ir (2fr)-1/2

f(x)exp(-ivx)exp(ivex)dx.

j

0

To prove (2), let

support, i.e., fi(x) = 0 u c]N.

for

u-tx

for suitable

x e]R

Then we can rewrite

u+Ax J

be a function with compact

0 e C°(]R,In)

ru O(x+Ax)exp(-iyx)dx =

D(x)expf-iy(x-Ax)]dx

1-

u fru

eAx(Y)- 4,(x)exp(-iyx)dx. u

The conclusion follows from this via Theorem 4.10(1), since the

mappings n , TAX, and Lemma 9.8:

Let

f -

n(x) = ix.

e

If the function

f e Cm(II2,¢n)

satisfies the growth condition

sup x e 1R

for all

j

c 1N

IIP(x)fO)(x)II

and all polynomials

_W n(f(q))

K2f(w)Hf(w). Since

is continuous, there exists a nondegenerate interval

f

such that

[yl,y2]

Y E [Y1,y2].

f(Y)HS(Y)HS(Y)f(Y) > K2f(Y)Hf(Y),

(9.14)

We define

for y C [Y1.Y2] for y c ]R - [yl'y2]'

( f(y)

f(Y)

I

0

=-9 1(f).

g

By (9.14) we have

K2IIfil, so upon applying Theorem

9.4 and Lemma 9.7(2), we obtain

IIS(')fII = II G(h,')'. =

(g) II = II .sTn[C(h)'(g)] II

IIC(h)R(g)II > K IIf1I = K IIgh

.

Therefore, the difference method cannot be stable.

Case 2, IF = ZZ

or IF =IN:

The proof is analogous to Case 1.

Instead of the Fourier integral

series $(f). n(f). 9.1.

we have the Fourier

Instead of Theorem 9.4, we apply Theorem

Lemma 9.7(2) is replaced by Lemma 9.7(1).

o

It follows from the preceding theorem that a difference method which is stable in the space in the spaces

L2((0,27r),4n)

and

L2c1R,4n)

is also stable

L2((O,rr),4n).

However, the

converse of this statement need not be true, although the difference methods where it fails are all pathological ones.

As

a result, in practice one tests the boundedness of the norm of the powers of the amplification matrix for for

y cZZ

or

y EIR, and not

y EIN, even when the methods belong to the

9.


spaces

or

L2((0,2n),In)

131

L2((O,n),tn)

The necessary and sufficient condition for the stability of a difference method with coordinate-free coefficients,

given in Theorem 9.13, is one which is not easily checked, in The following theorem provides a simple necessary

general.

condition.

Von Neumann condition.

Theorem 9.15:

Let

MD

ence method with coordinate-free coefficients.

values of the amplification matrix

G(h,y)

be a differThe eigenare de-

MD

for

noted by

j

X. (h,y), If

MD

h e (O,h0], y e IF.

is stable, there exists a constant

Ix.(h,y)I < 1+Kh, Proof:

= 1(1)n,

Let

MD

j

= l(l)n,

be stable.

k > 0

h e (O,ho],

such that

y E IF.

By Theorem 9.13 it follows that

y e IF, h e (0,ho], u e IN, uh < T.

IIG(h,y)"II2 < K, Since

p(G(h,y)u) < IIG(h,y)''II2 it follows that Iaj(h,y)Iu < K

and hence, for

ph > T/2, that

Iaj(h,y)I < K1/V < K2h/T = exp[2h(log K)/T] < 1+Kh.

Theorem 9.16:

o

The von Neumann condition

Iaj(h,y)I < 1+Kh,

j

= 1(1)n,

h c (0,h0],

y LIF

of Theorem 9.1S is sufficient for the stability of difference

132


I.

method

MD

if one of the following conditions is satisfied:

(1)

The amplification matrix

(2)

There exists a similarity transformation, inde-

pendent of

h, which simultaneously transforms all the mat-

rices

AV(h) (3)

and

to diagonal form.

BV(h)

where

G(h,y) = G(w)

Further, for each

Ax =

is always normal.

G(h,y)

w = y6x

and

Ax = h/A

or

one of the following

w c]R

three cases holds: has

different eigenvalues.

(a)

G(w)

(b)

G(u)(w) = yuI has

(c)

n

n

for

u = 0(1)k-1, G(k)(w)


P(G(w)) < 1.

Three lemmas precede the proof of the theorem.

Lemma 9.17:

be a matrix with p(A) < 1.

Let A c MAT(n,n,4)

Then there exists a matrix norm

for all

B,C

11-11s

with 1IAIIs < 1

and

E:

For a proof, see Householder 1964. Lemma 9.18:

Let

G(w0)

have

e >

and maps

0

for all

n

G e C°(1R,MAT(n,n,c)) and


w0 e1R.

Also let

Then there exists an

S, D c C°((wo-e,wo+e), MAT(n,n,4)), such that

w e (wo-e,wo+e): (1)

D(w)

is a diagonal matrix.

(2)

S(w)

is regular.

(3)

D(w) = S(w)-1G(w)S(w).

The proof is left to the reader.

It is possible to weaken the

hypotheses of the lemma somewhat. Lemma 9.19:

Let

u = 0(1)k-1, let

G c Ck(IR,MAT(n,n,t)) G(u)(wo) = yuI

and

and let

w° e1R. G(k)(wo)

For have

n

9.


different eigenvalues. pings

Then there exists an

133

c >

and map-

0

S, D E C°((w°-e,wo+e), MAT(n,n,¢)), so that for all

w e (wo-e,wo+e):

Proof:

(1)

D(w)

is a diagonal matrix.

(2)

S(w)

is regular

(3)

D(w) = S(w)-1G(w)S(w)

By Taylor's theorem, k-l

G(w) = G

I

is continuous.

1jI0 =

yu(w-W0+

u,

'ET

has

G(wo)

n

(w-wo)kG(w).


conclusion follows by applying Lemma 9.18 to Proof of Theorem 9.16(1):

G.

The

o

Since the spectral radius and the

spectral norm are the same for normal matrices, the bound for the eigenvalues implies a bound IIG(h,y)u112 < (1+Rh)u < exp(Ruh) < exp(RT)

for the norm of the powers of the amplification matrix. Proof of 9.26(2):

Let

S

be the matrix which simultaneously

transforms all the matrices

Av(h)

and

Bv(h)

to diagonal

form:

S 1Av(h)S = Dv(h) v = -k(l)k S-1Bv(h)S = Dv(h). Then,

1-1

k

S-1G(h,y)S

=

I

v=-k

v=-k

The transformed matrix same eigenvalues as

k

exp(ivy0x)Dv(h)E exp(ivyex)Dv(h) S-1G(h,y)S

G(h,y).

is normal and has the

It follows that

134


I.

II[S 1G(h,y)S]"II2 < exp(KT) 115-1112

1 1 G (h,y)" 1 1 2 _ 11 S 112

Proof of 9.16(3):

exp (KT) .

is arbitrarily often differentiable and first prove the following assertion. there exists an

v e IN

exp(iw),

G(w), as a rational function of

e >

and a

0

K > 0

2n-periodic.

We

wo EIR

For every

such that for all

and all w e (wo-E,wo+E) , 11G(')"112 < K.

The constants

and

c

K

depend on

Since finitely many open intervals the interval

wo

to begin with.

[0,2n], we can find a different

inequality holds for all

w EIR.

will cover

(u0-E,wo+e)

so that the

K

To establish conclusion (3),

we have to distinguish three cases, depending on whether hypothesis (a), (b), or (c) applies to Case (a):

A special case of (b) for

Case (b):

G

The quantity

and E

wo

wo.

k = 0.

satisfy the conditions of Lemma 9.19.

of the lemma we denote by

w e (wo-2c,wo+2E), G"

here.

2c

For

then has the representation

(w)" = S(w)D(w)"S(w) 1. Let

K= The diagonal of

max IIS(w)112 IIS(w)-111 2. wE[W0-E,W0+c] D(w)

They depend only on

contains the eigenvalues of w, and not explicitly on

follows from the von Neumann condition that IID(W) 112

0, since the conditions are contradictory.

Letting

66-6Oaa > 0

142

1.

n = (33+30aa) +


(26-2Oaa)cos w + (1-lOaa)cos 2w > 0

we have on the one hand that G(h,y) = 1-20aX(3-Zcos w - cos 2w)/n
-1.

The method is stable, therefore, for all Example 9.23:

A > 0.

0

(cf. Equation 6.7).

Differential equation: ut(x,t) = aux(x,t),

a eIR.

Method: C(h) =

I

aa(TOx - TAX

+

2 Amplification matrix: G(h,y) = 1 + iaa sin w.

The method is unstable, as already asserted in Section 6, for the norm of the amplification matrix is Example 9.24:

1 + a2A2 sin2w.

Friedriche method.

Differential equation: ut(x,t) = Aux(x,t) where

is real diagonalizable.

A e MAT(n,n,IR)

Method:

C(h) = 1[(I+AA)TQx + (I-AA)T-l AX 2

Amplification matrix: G(h,y) = I cos w + iXA sin w.

o

9.


Since the coefficients of

143

are simultaneously diagon-

C(h)

alizable, the von Neumann condition is also sufficient for stability.

For every eigenvalue

ponding eigenvalue

u(A,w)

u

of

there is a corres-

A

G(h,y), and vice-versa.

of

We

have

u(A,w) = cos w + iAU sin w, I u(a,w)12 = 1 + (A2 11

2-1)sin2W.

Thus the method is stable for

AP(A) < 1 This condition corresponds to the Courant-Friedrichs-Lewy condition in Section 6.

When the method is stable, it is

also positive, and is positive definite if Example 9.25:

A

is symmetric.

Courant-Isaacson-Rees method.


A E MAT(n,nfit)


Method:

C(h) =

[I-AA+-AA ]+AA TAX.

Amplification matrix: G(h,y) = AA+exp(iw) + I-AA+-AA i`.A exp(-iw).

In analogy with the previous example, we compute For

u > 0,

value of

A

ti,

is an eigenvalue of

A+

for the same eigenvector.

and

0

ji()',w).

is the eigen-

We obtain

u(A,w) = Au exp(iw) + 1-AU, ,11(A,w)l2

= 1 + 2AU(Ap 1)(1- cos w).

o

144


I.

Similarly, for

u < 0

we get

u(X,w) = 1-ajuj+ajujexp(-iu), Iu(a,w)I2 = 1+2AIuI(XIiI-1)(1-cos w).

Stability holds exactly for Ap(A) < 1.

This again is the Courant-Friedrichs-Lewy condition from Section 6.

Here again stability implies positive, and potenti-

ally, positive definite. Example 9.26:

o

Lax-Wendroff method.


A e MAT(n,n,1)


Method: C(h) =

I

+ 2LA(TLx-T_x) +

Amplification matrix: G(h,y) =

I

+ iXA sin w - A2A2(1

Because the method converges like in practice.

O(h2)

-

cos u).

it is very popular

But it is positive or positive definite only in

unimportant exceptional cases. is symmetric and that

Assume, for example, that is an eigenvalue of

u # 0

A.

A

C(h)

is positive or positive definite only if the three matrices I-a2A2,

are positive

AA+a2A2

semidefinite.

l - A 2 1 u 1 2 > 0

and

and

-aA+a2A2

This means

- I u I

+

X 1 02 > 0.

9.


All eigenvalues value, and

p # 0

of

A

145

must have the same absolute

is the only possible choice for the

A = 1/ILK

In this special case, the method can be

step size ratio.

In Section 11 we will

regarded as a characteristic method.

show that the Lax-Wendroff method can be derived from the Friedrichs method by an extrapolation.

The von Neumann condition leads to some necessary and We have

sufficient conditions for stability. u(X,w) = l+iap sin w-X2p2(1-cos w), ,u(A,w)12

l+A4p4(1_ cos w)2-2A2p2(1-cos w) + X2p2sin2w.

=

and obtain

w = w/2

We substitute

1-cos w = 2 sin sin2w = 4 sin2w jp(A,w)I2

=

1

2w, -

4 sin

4w,

4A2u2(1-A2)12)sin4w,

Stability is now decided by the sign of

1-A211 2.

In agree-

ment with the Courant-Friedrichs-Lewy condition, we obtain the stability condition Ap(A) < 1.

Example 9.27:

a

(cf. Example 8.10).


A E MAT(n,n,R)


Method:

C(h) _ {-[r2I+(1-a)AA]T6x+(2+2r2)I-[r [(rlI+a),A)TAX+(2-2r1)I+(r II-aAA)T-lI

where

a e [0,1], r1 = aap(A), and

r2 = (1-a)ap(A).

2I-(1-a)XA]T4z}-1

146


I.

Amplification matrix:

G(h,y) = [(1+r2-r2 cos w)I-i(1-a)AA sin w11 [(1-r1+r1cos w)I+iaAA sin w]. In Example 8.10 we only considered the special case 0

al

a

0j

A

,

albeit for nonconstant coefficients.

Implicit methods are of

practical significance for initial boundary value problems, at best.

Nevertheless, in theory they can also be applied to

pure initial value problems. positive.

arbitrary.

For

aAp(A) < 1, the method is

In particular, this is so for

a = 0

We now compute the eigenvalues

amplification matrix

-cos w)

+r2

u(A,w)

of the

G(h,y):

1-rl(1-cos w) u(a'w)

A > 0

and

+ iaau sin w -

u sin w

-a

i

[1-rl(1-cos w)]2+a2A2u2sin2w lu(a,w )l2 =

[l+r2(1-cos w)] +(1-a) A u sin w

We have stability so long as the numerator is not greater than the denominator, for all

lul

< p(A).

The difference

of the numerator and denominator is D = -2(rl+r2)(1-cos w)

+

(r12-r2)(1-cos w) 2

-a2u2sin2w + 2a a2u2sin2w. Since

rl+r2 = Ap(A)

and

ri-r2

=

(2a-1)A2p(A)2, we get

D = -2Ap(A)(1-cos w) + (2a-l)A2p(A)2(1-cos w)2 + (2a-l)X2u2sin2w. For

a < 1/2, D
0.

We have

a > 1/2.

D < Ap(A)[-2(1-cos w)+(2a-l)Ap(A)(1-cos w) 2 +(2a-l)Ap(A)sin2w].

To simplify matters, we

again substitute

w = w/2.

We get

the inequality

D < 4Ap(A)[-sin2w + (2a-l)Ap(A)sin4w + (2a-1)Ap(A)sin2w-(2a-1)Ap(A)sin4W.

method

(D < 0).

tute the values D.

is sufficient for stability of the

(2a-1)Ap(A) < 1

Thus

To obtain a necessary condition, we substiw =

it

and

P2 = P(A)

2

in the equation for

We obtain

D = 4Ap(A)[-l + (2a-1)Ap(A)] Thus, the given condition is also necessary.

The stability

condition (2a-1)Ap(A) < 1 a e (0,1)

in part for

may deviate substantially from the

positivity condition aAp(A) < 1.

Example 9.28:

o

(cf. Example 1.9).

Differential equation: ut(x,t) = Aux(x,t) where A = a Method: C(h)

rLI-

1

ZaAB1(TAx-TAX )J o[I+

.aAB2(TAx-TAx )]

148


I.

where

B1 = (0

B2 = r0

,

00 1

11

l

Amplification matrix:

J

G(h,y) _ [I-iaAB1 sin w]-1[I+iaAB2 sin w]. In this case it is not very suggestive to represent the dif-

ference method with the translation operator, so we shall switch to componentwise notation.

v(x,t)

u(x,t) _ The method

Let

w(x,t)

.

now reads

v(x,t+h) = v(x,t) + Zaa[w(x+Ax,t)-w(x-Ax,t)] w(x,t+h) - ZaA[v(x+Ax,t+h)-v(x-Ax,t+h)] = w(x,t).

In the first equation, the space derivative is formed at time

t, and in the second, at time

would first compute

t+h.

In practice, one

on the new layer, and then

w.

Then

one can use the new v-values in the computation of

w.

The

v

method thus is practically explicit. any

A > 0.

Since

It is not positive for

B1 = 0,

G(h,y) = [I+iaAB1 sin w][I+iaAB2 sin w] = I+iaa(B1+B2)sin w - a2a2B1B2 sin2w

B1

and

B2

1

iaA sin w

iaa sin w

1-a2X2 sin2w

I

.

obviously are not exchangable.

Thus the coeffici-

ents of the method cannot be diagonalized simultaneously. addition, we will show that

C(h,y)

is not normal for all

In w

9.


and that double eigenvalues occur.

149

The von Neumann condition

therefore is only a necessary condition. Let

The eigenvalues of

n = a2A2sin2w.

G(h,y)

sat-

isfy the equation (2-n)11+l = 0.

-

112

The solutions are u1,2 = 1 Case 1:

as > 2.

are real.

11

-

If

Zn

-

(4 n2-n)1/2

w = n/2, then (4n2-n)1/2I

-

?n

+-

Both eigenvalues

n > 4.

is greater than

1.

The

method is unstable. as < 2.

Case 2:

If

w # vn

where

are different and of absolute value

The derivative of

G(h,y) = I.

v c22, the eigenvalues For

1.

w = vn,

with respect to

G(h,y)

w

has distinct eigenvalues at these points, namely

u1,2

_ Iiaa.

By Theorem 9.16(3), the method is stable. Case 3:

value

1.

stable.

All eigenvalues of

as = 2.

G(h,y)

The method is weakly stable.

Suppose it were also

Then every perturbation of the method in the sense

of Theorem 5.13 would also be stable. with

have absolute

We replace matrix

B2

and obtain a method with amplification matrix

B2(l+h)

1

iaa(l+h)sin w

iaX sin w

1-a2A2(1+h)sin2w,.

In the special case

w = n/2, we get, for

1

2i+2ihl

2i

1-4-4h

as = 2,

150

I.


The eigenvalues of this matrix include -1

2h -

-

2(h+h2)1/2.

Obviously there is no positive constant 1+Kh.

shown:

K

The perturbed method is not stable.

such that

Thus we have

as = 2, one obtains a weakly stable method

(1) For

which is not stable, and (2) there is no theorem analogous to 5.13 for weakly stable methods.

Thus, the stability condi-

tion for our method is

The Courant-Friedrichs-Lewy

condition yields

as
0.

The

Richtmyer-Morton pre-

fer the implicit method of the previous example, since the present stability condition,

A = h/(Ax) 2 < 1/(21al), is very

strong.

When the coefficients of a difference method do depend on the coordinates, one cannot automatically apply a Fourier transform to the method.

Although the amplification matrix

appears formally the same as for coordinate-free coefficients, it is not the Fourier transform of the method.

In these cases,

the amplification matrix can only be used to investigate local stability of the method.

Then the variable coefficients of

the difference method are "frozen" with respect to

x, and

the stability properties of the resulting method with coordinate-free coefficients become the subject of investigation. The following theorem 9.31 shows that under certain additional conditions, local stability is necessary for stability.

For simplicity, we restrict ourselves to explicit

methods and

B = L2c1R,¢n).

There also exist a number of sufficient stability criteria which depend on properties of the amplification matrix.

We refer to the work of Lax-Wendroff (1962), Kreiss (1964), Widlund (1965), and Lax-Nirenberg (1966).

The proofs are all

9.


155

We therefore will restrict ourselves to a

very complicated.

result on hyperbolic systems (Theorem 9.34) due to Lax and Nirenberg.

Let

Theorem 9.31: -k(1)k

AV E C°(IR X (O,ho] , MAT(n,n jR)) , v = be an explicit difference

MD = {C(h)lh c (O,ho])

and

P(B,T,A), where

method for a properly posed problem k

C(h) = v=-k

Further assume that as

AV(x,h)Tex'

h - 0, each

formly on every compact subset of bounded norm.

converges uni-

Av(x,h)

IR

to a mapping MD

Then the stability of method

Av(x,0) of

implies the

MD = {C(h)ih a (O,h0]}, where

stability of the method k

Av(x,0)Tnx

C(h) = v=-k

for every (fixed) point

of

IR.

The proof will be indirect and so we assume that

Proof:

there exists an

x £IR

By Theorem 9.13, for each constant

stable.

y eIR, an

exists a

MD

for which the method

h c (0,h0], an

is not

K c]R+

N c IN with

there

Nh < T,

V c 4n, such that

and a vector

(9.32)

I1r(x)NV112 > K IIVII 2' where k

r(x) =

E

exp(iv&)Av(x,0),

& = ytx.

v=-k Since

Av(x,O)

is continuous, there is a

inequality (9.32) also holds for all now fix

t

and pass to the limit

d eIR+

such that We

x e S6 =

h -+ 0

(and hence

Inequality (9.32) remains valid throughout for all

Ax -

x e Sd.

0).

156


I.

Now let

p:]R + ]R

be an infinitely often differenti-

able function with

p(x) = 0 p(x) $ 0

for

x c 1R

in

Sd.

-

S6

Set

v(x) = Vp(x)exp(iyx). Then k

C(h)(v)(x) =

Av(x,h)VP(x+vAx)exp[iy(x+vAx)] v=-k

= C(x)v(x) + c1(x,h). Here

such that: h

is a function for which there is a

c1(x,h)

El(x,h) = 0

(1)

for

sufficiently small, and (2)

to zero as

h -

0

for

'd

cIR+

x e]R (x-d S,x+d+d)

and

converges uniformly

c1(x,h)

x e (x d d,x+d+S),

Applying

C(h)

repeatedly, we obtain C(h)N(v)(x)

Here

eN(x,h)

=

P(x)Nv(x) + eN(x,h).


sufficiently small

eI(x,h).

Choose a

h, and then it follows from (9.32) that

IIC(h)NII > K. This contradicts the stability of Example 9.33:

0

MD.

Application of Theorem 9.31.

Differential equation (cf. Example 5.6): ut(x,t) = [a(x)ux(x,t)]x where

ae

IR),

a'

a Co(]R, IR)

and

a(x) > 0,

x c 1R.

9.


Method (cf.

(6.3) for

157

a = 1):

C(h) = Aa(x-Ax/2)T-1 + [1-Aa(x+tx/2)-Aa(x-tx/2)]I A = h/(0x)2.

+ Aa(x+Ax/2)TAx,

By Theorem 6.5, the condition

a(x) < 1/2

0 < Amax x e]R

is sufficient for the stability of the above method.

It

follows from Theorem 9.31 that this condition is also necesThus, for fixed

sary.

x cIR, consider the method

C(h) = ;,a(x)T-1 + [1-2Aa(x)]I + aa(X)TOx.

The corresponding amplification matrix is G(h,y) = 1 + 2Aa(x)[cos w - 1]. Since the above stability condition is necessary and sufficient for h c (0,h0],

y eiR,

IG(h,y)I < 1,

the conclusion follows from Theorem 9.13. Theorem 9.34:

Lax-Nirenberg.

Let

a

MD = (C(h)Ih > 01

difference method for a properly posed problem

be a

P(L2(R,]Rn),

T,A), where

k C (h) and

Ax = h/a, with

=

E

A > 0

u T11 B(x) 1x fixed.

Let the following condi-

tions be satisfied: (1)

Bu c C2(1R,MAT(n,n,IR)), u = -k(1)k.

(2)

All elements of the matrices

-k(l)k, x cIR

are uniformly bounded.

B(v)(x), v = 0(1)2,

158

I.

(3)


for h > 0, y c ]R, x e 1R where

II G(h,y,x) 112 < 1

G(h,y,x)

is the amplification matrix for

Then

is stable.

MD

C(h).

Although we have a real Banach space, we form the

Remark:

amplification matrix exactly as in the case of

L2(JR,4n),

namely k

B (x)exp(iuyAx). u=-k u

G(h,y,x) _

For fixed

x

0

c]R, it follows from condition (3) that k

u=-k

Bu (xo)TAx II2

For the proof, we embed

< 1.

L2(1R,IRn)

in

L2(]R,4n)

in the

The conclusion then follows from

canonical way. k k

.G(h,y,xO)o Y .

-9

Bu(xo)TAx

o

Before we prove Theorem 9.34, we need to establish several

We begin by introducing a

further observations and lemmas.

scalar product for u,v a

L2 (R, IRn)

:

= f u(x)Tv(x)dx = . IR

With respect to this scalar product, there exists an operator C(h)T

which is adjoint to

C(h), namely

k

C(h)'

T-x°BU(x)T.

=

k

Using the symmetry of the matrices

Bu(x), we obtain

k

_

J [B(x)u(x+pAx)]Tv(x)dx

u=-k k

1R

f

E

u=-k

J

u(x)T[BV(x-UAX)Tv(x-1Ax)]dx

]R

_ .

9.


159

In particular,

_ . In addition to the difference operators

C(h), we also have

to consider the difference operators with "frozen" coefficiFor fixed

ents.

a cIR

let

k

I

Ca (h)

u= k

Ca(h)T =

Bu (a) T,x

k u=_k

B (a)TT. x

As remarked above, it follows from (3) that

II Ca(h) (u) II2 1 II u11 2, which is to say < or

> 0.

Our goal is to establish a similar inequality for the operaC(h), instead of

tors

Ca(h).

Lemma 9.35: I-C(h)ToC(h) = Q(h) + Ox R(h)

where

Q(h)

into itself. ex.

and

are bounded linear mappings of

R(h)

is bounded independently of

IIR(h)II2

The coefficients of

Q(h)

are independent of

h h.

L2(IR,IRn)

and

We

then have 2k

Q(h) =

E

u=-2k Proof:

form

The product

D (x) T11 Ax u

C(h)TOC(h)

consists of summands of the

160

I.

TAxoB-r(x)TBs(x)-TAx

+

Q(h)


B-r()TBs(x),,TAxs

=

[B_r(x+rAx)TBs(x+rAx)-B_r(x)TBs(x)]TAxs.

contains the first summand on the right side.

in square brackets is divisible by bounded independently of

The quotient is

Ax.

because

Ax

The term

JJB,1(x)JJ

and

JIB 1j(X)II

0

are bounded.

Analogously to

Ca(h), we can define 2k

Qa(h) =

E

u=-2k

Du(a)T"x

Obviously,

Qa(h) = I-Ca(h)ToCa(h) and therefore, > 0. Lemma 9.36:

Let

Eu =

Z[DU(a) + D_u(a)]

Fu = Z[DU(a)

- D u(a)]

Then: c2k

(1)

Qa(h) =

[Eu(TAx+T Ax) + Fu(TAX TAX))

uL- 2k (2)

ET = Eu, 11

Proof:

FT =

Fu,

P

(1) is trivial.

u = -2k(1)2k.

As for (2), note that

Qa(h)

con-

sists of summands of the form B_r(a)TBs(a)Tr+s and

Bs (a) TBs (a) TO

s(a)TB-r(a)T-r-s.

u = r+s >

0

9.


The corresponding summands in B_r(a)TBs(a)

161

are

ZEU

+ Bs(a)TB-r(a),

u > 0

or

Bs (a)TB5 (a) , These matrices are symmetric.

contains the antisymmetric

2F 11

terms

B_r(a)TBs(a)

-

Bs(a)TB_r(a).

a

One important tool for the proof of Theorem 9.34 is a special partition of unity. Lemma 9.37:

There exists a

0 e C`°(1R,]R)

with the following

three properties. (1)

Support(4) _ [-2/3,2/3].

(2)

0 < $(x) < 1,

x c 1R. x e1R.

(x-u)2 = 1,

(3)

u=-m Proof:

Choose

4 c C"OR,]R)

(cf. Lemma 4.9) such that

fi(x)

= 1

for

jxj

< 1/3

fi(x)

= 0

for

lxi

> 2/3

0 < ¢(x) < 1

for

x e (1/3,2/3)

for

x c (-2/3,-1/3).

fi(x)

= 1-;(x+l)

It follows that +m r

(x-u) = 1,

x c]R.

u=-w

All the derivatives of

fore, 4(x) = (x)1/2 For if

1xo1

vanish at the zeroes of

xo,

There-

is arbitrarily often differentiable.

> 2/3, i.e., xo

neighborhood of

;.

a zero of , then in some

162

I.


(x) = (x-x o) 2s ps (x) Ix-xols

4(x) = Here

s

y 5(x.

is arbitrary and

IN

easily shown that

is continuous and

$s x

times differentiable at

(s-1)

is continuous.

Vs

x0.

It is

is at least

0

a

Like , the function ' has compact support. fore,

attains its maximum, which we denote by

Ic'(x)I

10 (X)

-

0(x)I < Lax-xI.

x,x c 1R.

Lemma 9.38:

nv(x) = 0(yx-v), v = --(1)-

Let

as in the preceding lemma. all

L

By the Mean Value Theorem,

in the following.

for all

There-

where

y eIR+

Then for all

where

h = Atx > 0

0

is

and

6ky6x < 1,

I -

u e L2@i, IRn).

M1y2L2(6x)21IuI12, The constant on

u, y, h, or For

Proof:

depends only on the method

M1 > 0

MD, and not

6x.

u = -2k(1)2k

we have

M

D (x)u(x+utx) u

=

[1

-

E

-

nv (x)D

E

u

(x)nv (x+uAx)u(x+uox)

nv(x)nv(x+u6x)IDU(x)u(x+uox).

v=

Using 9.37(3), we replace the

2

nv(x)2

v=-m

+

1

in the square brackets by

nv(x+u6x)2


9.

163

The term in the brackets is then a sum of squares, 1

1

nv(x)nv(x+uox) =

-

2 [nv(x)-nv(x+udx)].

v=

nv(x) = 0

v=

Iyx-vj > 2/3.

for

(6yk)-l, nv(x)-nv(x+utx) = 0

Since for

Jul

< 2k

Iyx-vI > 1.

and

Ax
0

Let

LIx-xI

u e L2(R,IRn)

with

for all

establishes

o

iy

c C° (1R, IR)

be such that

x,x a (-28,28).

Support(u) C [-8,8]

Then for all

and for all

h

2kex < 8,

with

l -I

I

I-'I

< [(4k+1)M282 + 2K8-2(0x)2] IIuoIIZ _ (2kM2 + 2K/A2)h IIuoIIZ = M3h IIuoIIZ. The scalar products

<W2uo,Q38(h) (p2uo)> are both nonnegative, which proves that

> -M3h IIuoIIZ.

9.


Analogously, we have for all

v = --(1)-

167

that

> -M3h Lemma 9.38 then implies that

Since

-M1y2L2 (Ax) 2 IIuII2 - M3h

>

y2(Ax)2

=

h/A2

v=-.

H uV II2.

and since by Lemma 9.37(3),

IIuv112 =IIuII2 we have that

> -M4h IIuII2 where

M4 = M1L2/a2 + M3. Applying Lemma 9.35 yields

> -M4h

IIuII2 - M5h IIuII2

- > -M6h

< (l+M6h) IIC(h)112

x1,

xm.

t

explain the situation for

m

We have avoided the case

until now for didactic and notational reasons.

1

and

We will

in this section with the

m > 1

aid of typical examples.

Initial boundary value problems, in contrast to pure initial value problems, are substantially more complicated when

m >

1.

The additional difficulties, which we cannot

discuss here, arise because of the varying types of boundaries.

The problems resemble those which arise in the study of

boundary value problems (undertaken in Part II).

Throughout this chapter we will use the notation m

x = (xl,...,xm) eIR

y = (YI,...ym) eIRm m

<x,y> =

dx = dxl...dxm,

I

xuYU'

u=1

In addition, we introduce the multi-indices e a

s = (sl) .... s

The translation operator replaced by e{u)

m

TAx of

IR1

different operators

_ (eemu)),

e(u) = 6 uv$

(cf. Definition 6.2) is Tku

in

IRm:

u = l(1)m u,v = 1(l)m

V

Tku(x) = x + ke(u),

m

m x e IR,

k eIR,

u = l(1)m.

10.

Problems in several space variables

For all

169

let

fE

x E Rm.

Tku(f)(x) = f(Tku(x)),

With this definition, the translation operators become bijective continuous linear mappings of

L2(IR m,tn)

For all

They commute with each other.

v e 7l

into itself.

we have

Tku = Tvk,u Let

have bounded spectral norm

B E

II B (x) 112 .

The map

f(') - B(.)f(.) is a bounded linear operator in relations for

Tku

and

B

L2ORm,¢n).

The commutativity

are

Tku°B(x) = B(Tku(x))Tku B(x)'Tku

In many cases, B

Tku0B(T-ku(x)). =

will satisfy a Lipschitz condition

IIB(x)-B(Y) 112 < L I1x-Y112. Then, IIB(x)°Tk1i -Tku°B(x)112 < L1kj.

For

k cIR

and arbitrary multi-indices m s

Tk =

su

fl Tku

.

Vj=l

The difference method MD = {C(h)Ih e(0,ho]}

can now be written in the form

s, we define

170


I.

C(h) _ (I B5(x,h)Tk)

(E A5(x,h)Tk).

All sums, here and henceforth, extend only over finitely many Also we assume that for all

multi-indices.

s, x, and

h,

As(x,h),Bs(x,h) E MAT(n,n,IR) k = h/A

A IR+.

where

k =

or

Analogously to Definition 9.12, we can assign to each difference method an amplification matrix exp(ik<s,y>)Bs(x,h))-1(1

G(h,y,x) =

exp(ik<x,y>)As(x,h)).

(X

s If the matrices of

s As(x,h)

and

B5(x,h)

are all independent

x, we speak of a method with space-free coefficients.

Then we abbreviate As(x,h), Bs(x,h), G(h,y,x) to

As(h), Bs(h), G(h,y). The stability of a method with space-free coeff-

icients can again be determined solely on the basis of the amplification matrix

G(h,y).

Theorems 9.13 and 9.15 extend

word for word to the Banach spaces placed by

case the

m = 1.

if

IF

is re-

Theorems 9.16, 9.31, and 9.34 also carry over

IItm.

in essence.

L2(Rm,cn)

All the proofs are almost the same as for the Basically, the only additional item we need is

m-dimensional Fourier transform, which is defined for

all f e L2

by (2")-m/2

a,(y) = 7n_(f)

f(x)exp(-i<x,y>)dx

rIl

xll2

= lim a Viw


10.

171

The limit is taken with respect to the topology of As in the case

L2(Iltm,n).

m = 1, we have: is bijective.

(1)

jn

(2)

11-9n11 = II_V -11I = I.

(3)

Fn(Tk(f))(')

=

For differential equations with constant coefficients, the best stability criteria are obtained from the amplification matrix.

Even when the coefficients are not constant,

this route is still available in certain cases, for example, with hyperbolic systems.

Also, one can define

positive definite methods.

positive and

They are always stable.

For positive definite methods, we need (1)

C(h)

I As(x)TS

with

k = h/a.

S

(2)

1

=

E A5(x) s

(3)

All matrices

As(x)

are real, symmetric, and

positive semidefinite. (4)

For all multi-indices

s

and all

x,y EIRm

we

have

IIAs(x) -As (Y) 112 0.

We consider positive methods only in the scalar case If

m = 1.

m > 1, they are of little significance for systems of

differential equations.

This is due to Condition (3) of De-

finition 8.4, which implies that the coefficients of the difference operators commute.

For

m > 1, the coefficients of

most systems of differential equations do not commute, and

172


I.

hence neither do the coefficients of the difference operators. The positive methods occur in the Banach space B = If e

lim

If(x)j

= 01

11xL -Here the norm is the maximum norm. also defined in this space. (1)

e(x,h)C(h) _

k= h/h (2)

The operators

For positive

are

methods, we need

as(x,h)Tk + I bs(x,h)Tk0C(h)

k= h X

or

Tk

where

e(x,h) = E [as(x,h) + bs(x,h)],

A e1R+.

x e

he(0,ho]

s

(3)

as ,bs a CO(JRm, 1R) as(x,h) > 0,

bs(x,h) > 0

E as(x,h) > 1. S

For

m > 1, the so-called product methods occupy a

special place.

methods for

They arise from the "multiplication" of Their stability follows directly from

m = 1.

the stability of the factors.

More precisely, we have the

following.

Theorem 10.1:

MD,u

Let

B

be a Banach space and

{Cu(h)lh c (O,ho]},

a family of difference methods for properly posed problems.

u = 1(1)m m

(possibly different)


MD = {C(h)lh a (O,ho]} is defined by

C(h) = C1(h)C2(h) ... Cm(h). MD

is stable if one of the following two conditions is


10.

173

satisfied. (1)

For fixed

h e (0,h01, the operators

C}.(h), u =

1(l)m, commute. (2)

There exists a

such that

K > 0

p = 1(1)m,

IICp(h)II < 1+Kh,

h e (O,h0J.

If (1) holds, we can write

Proof:

m

IIC(h)nil < Since each of the

m

p n IIC(h)nil

p=1

factors is bounded, so is the product.

If (2) holds, we have the inequalities IIC(h)nil

< (1+Kh) mn < exp(mKT).

We now present a number of methods for the case In all the examples, A = h/k

or

a = h/k2, depending on the

order of the differential equation. Example 10.2:

Differential equation: m

ut(x,t) =

E

ap[ap(x)apu(x,t)],

ap = a/ax

p=1

where and

ap a C1(IRm, IR) , 0 < S < ap (x) v = 1(1)m. Iavap(x)I < K,

1.

174

I.


Amplification matrix: G(h,y,x) = [l+aaH]/[1-(1-a)Xi} where m

H =

I (au(x+ zkeu)[exp(ikyu)-1] u=1

+ au (x- Zkeu)[exp(-ikyu)-1]}. For

2mKaa
-4mK.

au

u =1

-

u

u

Precisely when 2mK(2a-1)d

0

at times.

The latter depends on the amount

of effort required to solve the system of equations. case, m+l+c >

2.

In any

To improve the precision by a factor of

thus is to multiply the computational effort by a factor of q(m+l+e)/k

q

11.

Extrapolation methods

193

In solving a parabolic _'ifferential equation we have

as a rule that 0(h

-m/2 -

1

-

h/(Ax) 2 = c)

A

The growth law

- -or.stant.

for the computational effort appears more

However, a remainder of O(hk) + O((tx)k) _ q(m+2+2e)/k. q = q(m+2+2e)/2k implies q = is only

favorable.

O(hk/2)

achieved with a remainder

O(hk)

0((Ax)2k)

+

=

O(hk).

How then is one to explain the preference for simpler methods in practice?

There are in fact a number of import-

ant reasons for this, which we will briefly discuss. (1)

involved.

In many applications, a complicated geometry is The boundary conditions (and sometimes, insuffici-

ently smooth coefficients for the differential equations) lead to solutions which are only once or twice differentiable. Then methods of higher order carry no advantage.

For ordin-

ary differential equations, there is no influence of geometry or of boundary conditions in this sense; with several space variables, however, difficulties of this sort become dominant. (2)

The stability question is grounds enough to re-

strict oneself to those few types of methods for which there A method which is stable

is sufficient experience in hand.

for a pure initial value problem with equations with arbitrarily often differentiable coefficients, may well lose this stability in the face of boundary conditions, less smooth coefficients, or nonlinearities.

In addition, stability is a

conclusion based on incrementations quite unclear how

h

0

h < h

-

.

o

It is often

depends on the above named influences.

In this complicated theoretical situation, practical experience becomes a decisive factor. (3)

The precision demanded by engineers and physicists

194

I.

is often quite modest.


This fact is usually unnoticed in the

context of ordinary differential equations, since the computing times involved are quite insignificant.

As a result, the

question of precision demanded is barely discussed.

As with

the evaluation of simple transcendental functions, one simply uses the mantissa length of the machine numbers as a basis for precision.

The numerical solution of partial differential

equations, however, quickly can become so expensive, that the engineer or physicist would rather reduce the demands for This cost constraint may well be relaxed with

precision.

future technological progress in hardware.

These arguments should not be taken to mean that higher order convergence methods have no future.

Indeed one

would hope that their significance would gradually increase. The derivation of such methods is given a powerful assist by extrapolation methods.

We begin with an explanation of the

basic procedure of these methods.

In order to keep the for-

mulas from getting too long, we will restrict ourselves to problems in and

1R2, with one space and one time variable, x

t.

The starting point is a properly posed problem and a corresponding consistent and stable difference method. solutions for considered.

noted by

h.

s-times differentiable initial functions are The step size of the difference method is deThe foundation of all extrapolation methods is

the following assumption: Assumption:

Only

The solutions

w(x,t,h)

method have an asymptotic expansion

of the difference


11.

r-1

w(x,t,h)

195

y

T,(x,t)h j + p(x,t,h),

=

(x,t)

E G,

v=0

h e (O,h01

where

r >

and

2

11p(x,t,h) JI Tv

= 0(hy r),

G + ¢n,

:

v = 0(l)r-1

0 = Yo < Y1 To

(x,t) a G, h e (O,ho]

.

< Yr.

is the desired exact solution of the problem.

a

We begin with a discussion of what is called global extrapolation.

method for

r

For this, one carries out the difference different incrementations

for the entire time interval. dependent of each other. tk/hj c2Z

for all

j

= 1(1)r, each

computations are in-

r

For each level

t = tk, where

= 1(1)r, one can now form a linear com-

w(x,tk,hl,...,hr)

bination

The

hj, j

of the quantities

w(x,tk,hj)

so that w(x,tk,hip.... hr) = T0(x,y) + R.

Letting

by = qvh, v = 1(1)r, and letting

h

converge to

zero, we get

R = 0(hlr) w

is computed recursively:

Tj,o = w(x,tk,hj+l), T.

j

T J.,v-1 B Jv[T J.

= 0(1)r-l l,v-1-T J.,v-1 ]'

J ,v=

1(1)r-1,

v

j

= v(1)r-1

w(x,tk,hl,...,hr) = Tr-l,r-1' In general the coefficients ways on the step sizes

hj

8jv cIR

depend in complicated

and the exponents

yv.

In the

196

I.


following two important special cases, however, the computation is relatively simple. Case 1:

hi = lhj 1, _

= 2(1)r,

Yv

Y 2

v-1

Yv = vy, y > 0, v = 1(1)r, hj

6jv

arbitrary

1

Sjv Case 2:

j

=

arbitrary

1

(h.

Y

-Jh

-1

J

l

The background can be found in Stoer-Bulirsch, 1980, Chapter 2, and Grigorieff (1972), Chapter 5.

This procedure, by the way,

is well-known for Romberg and Bulirsch quadrature and mid-

point rule extrapolation for ordinary differential equations (cf. Stoer-Bulirsch 1980).

In practice, the difference method is only carried out for finitely many values of sible for those

x

Extrapolation is then pos-

x.

which occur for all increments

The

h j*

case

hj/(tx)2 = constant

ratios of the

hj's

presents extra difficulties.

The

are very important, both for the size of

the remainder and the computational effort.

For solving hy-

perbolic differential equations one can also use the Romberg or the Bulirsch sequence. Romberg sequence: hj = h/2j-l,

j

= 1(1)r.

Bulirsch sequence: hl = h, h2j = h/2J,

h2j±1 = h/(3.2J 1),

j > 1.

Because of the difficulties associated with the case hj/(Ax)e - constant, it is wise to use a spacing of the

(Ax)j

11.


197

based on these sequences for solving parabolic differential equations.

In principle, one could use other sequences for

global extrapolation, however.

Before applying an extrapolation method, we ask ourselves two decisive questions: expansion?

Does there exist an asymptotic

What are the exponents

would be optimal.

yv?

Naturally

yv= 2v

Usually one must be satisfied with yv = v.

In certain problems, nonintegral exponents can occur.

In

general the derivation of an asymptotic expansion is a very difficult theoretical problem.

This is true even for those

cases where practical experience speaks for the existence of such expansions.

However, the proofs are relatively simple

for linear initial value problems without boundary conditions.

As an example we use the problem ut(x,t) = A(x)ux(x,t) + q(x,t),

u(x,O)

x SIR, t c (0,T)

x SIR.

4 (X)'

The conditions on the coefficient matrix have to be quite strict.

We demand

A e C (IR, MAT(n,n,IR)) A(x)

real and symmetric,

IIA(x)-A(R) Ij < L2Ix-XI , Let the

w(x,t,h)

IjA(x)II < L1

x,R a IR.

be the approximate values obtained with

the Friedrichs method.

Let a fixed

A = h/Ax > 0

be chosen

and let A sup

p(A(x)) < 1.

x SIR The method is consistent and stable in the Banach space L2OR,4n)

(cf. Example 8.9).

In the case of an inhomogeneous

198


I.

equation, we use the formula w(x,t+h,h) = 2[I+XA(x)]w(x+Ax,t,h) +

Theorem 11.1:

Let

Z[I-XA(x)]w(x-Ax,t,h) + hq(x,t).

r e 1N, 0 e Co (R, IRn)

h c (O,h0]

Then it is true for all

[O,T],IRn).

q c Co (IR x

and

that

r-l

w(x,t,h) _

TV(x,t)hV + p(x,t,h),

I

v=0

x cIR, t e [O,T],

t/h c ZZ

TV e co(R x (0,T1, ]Rn) O(hr)

uniformly in

t.

Since there is nothing to prove for

Proof:

pose that

r = 1, we sup-

We use the notation

r > 1.

V = Co(dt, IRT),

W = Co(JR x

[0,T], IRn).

The most important tool for the proof is the fact that for $ c V

q c W, the solution

and

longs to

W.

u

of the above problem be-

This is a special case of the existence and

uniqueness theorems for linear hyperbolic systems (cf., e.g., Mizohata 1973).

For arbitrary

v e W, we examine the differ-

ence quotients

Q1(v)(x,t,h) = h-1{v(x,t+h)

-

Z[v(x+Ox,t)+v(x-Ax,t)]}

Q2(v)(x,t,h) = (2ox)-1{v(x+Ax,t)-v(x-Ax,t)} Q(v) = Q1(v) - A(x)Q2(v)

Although

w(x,t,h)

apply

to

Q

is only defined for

t/h c2Z, one can

w:

q(x,t), x cIR, tc[0,T], t/h cZZ, hc(O,h01.

11.


For

v e W, Q1(v)

and

199

can be expanded separately

Q2(v)

with Taylor's series

Q(v) (x,t,h) = vt(x,t) - A(x)vx(x,t) s

+

hv-1DV(v)(x,t)

+ hsZ(x,t,h).

s

v2 Here s

s

is arbitrary.

c IN

vanishes.

The operators

operators containing order

We have

v.

For fixed

h,

x, t, and

h.

A(x)

For

The quantities

2

to

DV, v = 2(1)00, are differential

as well as partial derivatives of

DV(v) c W. e W.

s = 1, the sum from

The support of

Z(x,t,h)

Z

is bounded.

is bounded for all

tv e W, v = 0(1)r-1

are defined re-

cursively:

v=0: a to(x,t) = A(x)Bz T0(x,t)+q(x,t)

te

x e IR,

To(x,0) = fi(x)

[0,T]

V-1 v>0:

8t TV(x,t) = A(x)8z TV(x,t)-uIODV+1-u(tu)(x,t)

t e [0,T]

x e IR, TV(x,0) =

It follows that tients

Q(TV)

0

TV E W, v = 0(1)r-1.

The difference quo-

yield 2r-1

U(T0)(x,t)+h2r-1z0(x,t,h)

hµ-'D

Q(T0)(x,t,h) = q(x,t)+ u=2

2r-2v-1

V-1

E Dv+1

Q(t)(x,t,h)

u(tu)(x,t)+

+

h2r-2v-lz(x,t,h),

In the last equation, the sum from when

v = r-l.

I

h11- D

u=2

P=O

2

v = 1(1)r-1. to

2r-2v-1

vanishes

Next the v-th equation is multiplied by

hV


206

I.

and all the equations are added.

Letting

r-2

u=0

v r-v u-1

h

h

h u-1 D u (t

h

v=0

u=r-v+l

r-1 h

+

u

v 2r-2v-1

r-2 F

V D (T) (x,t)

u=2

v=O

+

Dv+1-u(ru) (x,t)

I

11v

v=l

+

we get

v-1

r-l

Q('1) (x,L,1i) = q(x,t)- E

T = Ervhv

2r v 1

Zv

) (x,t)

(x,t,h).

v=0

The first two double sums are actually the same, except for sign.

To see this, substitute

in the second,

v+u-l

obtaining r-1

r-2

I

v=0 1=v+1

hVD=_v+1 (TV)(x,t)

Then change the order of summation: r-1

u

h

u=1

u-1 1

Du+l-v(TV)(x,t)

v=0

Now the substitution

(u,v)

-

(v,u)

yields the first double

sum.

While the first two terms in this representation of Q(T)

cancel, the last two contain a common factor of

hr.

Thus we get

Q(T)(x,t,h) = q(x,t) + hrZ(x,t,h), x E IR,

Z

t

c

[0,T], t+h E [0,T],


ous for fixed h c (0,h0]. tion

Z

v

h, bounded for all The quanity

T-W

:

h e (0,h01.

bounded support, continux E IR,

t

e [0,T], and

satisfies the difference equa-

11.


Q(T)(x,t,h) r(x,0,h)

Thus, r-w

-

201

Q(w)(x,t,h) = hrZ(x,t,h)

- w(x,0,h) = 0.

is a solution of the Friedrichs method with initial

function

and inhomogeneity

0

hrZ(x,t,h).

It follows from

the stability of the method and from t/h e2Z

and

h e (0,ho], that for these

IIT(',t,h)

for

L t

and

h,

o

-

From the practical point of view, the restriction to functions

and

q

with compact support is inconsequential

because of the finite domain of dependence of the differential equation and the difference method.

Only the differen-

tiability conditions are of significance. do not have a finite dependency domain. V

W

and

Parabolic equations The vector spaces

are therefore not suitable for these differential

equations.

However, they can be replaced by vector spaces of

those functions for which sup

1 0 )(x)xkI
1.

On the other hand, for H2(2h,y,x) =

In3I < 1, that is,

w = n/2 we have

I

H2(h,y,x) = I-2A2A(x)2

n3 = 1 + 3 (u4-u2) and hence the condition

IuI

< 1.

Thus

if by chance all of the eigenvalues of 0

or

-1, for all

x eIR.

E3 XA(x)

is stable only are

+1

or

In this exceptional case, the

Friedrichs method turns into a characteristic method, and thus need not concern us here.

For characteristic methods, local extrapolation is almost always possible as with ordinary differential tions.

present.

This is mostly true even if boundary conditions are The theoretical background can be found in Hackbusch

(1973), (1977).

PART II. BOUNDARY VALUE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

12.

Properly posed boundary value problems Boundary value problems for elliptic differential equa-

tions are of great significance in physics and engineering.

They arise, among other places, in the areas of fluid dynamics, electrodynamics, stationary heat and mass transport (diffusion), statics, and reactor physics (neutron transport).

In

contrast to boundary value problems, initial value problems for elliptic differential equations are not properly posed as a rule (cf. Example 1.14).

Within mathematics itself the theory of elliptic differential equations appears in numerous other areas.

For a

long time the theory was a by-product of the theory of functions and the calculus of variations.

To this day variational

methods are of great practical significance for the numerical solution of boundary value problems for elliptic differential equations.

Function theoretical methods can frequently be

used to find a closed solution for, or at least greatly simplify, planar problems.

The following examples should clarify the relationship 207

208

BOUNDARY VALUE PROBLEMS

II.

between boundary value problems and certain questions of function theory and the calculus of variations. G

Throughout,

will be a simply connected bounded region in

continuously differentiable boundary

IR2

with a

aG.

EuZer differential equation from the calculus

Example 12.1:

of variations.

Find a mapping

u: G -+]R

which satisfies the

following conditions: (1)

is continuous on

u

entiable on

and continuously differ-

G

G.

(2)

u(x,y) = (x,y)

(3)

u

for all

(x,y) E aG.

minimizes the integral

I[w] = If

[a1(x,Y)wx(x,y)2

+

a2(x,y)wy(x,Y)2

G +

c(x,y)w(x,y)2

2q(x,y)w(x,y)]dxdy

-

in the class of all functions

Here

al,a2 a

C1 (G, ]R)

,

c,q c

w

satisfying (1) and (2).

C1 (G, IR)

al(x,y) > a >

a2 (x,y) > a > c(x,Y) > 0.

,

and ip E C1 (aG, IR)

with

0

0

(x,y)

E

It is known from the calculus of variations that this problem has a uniquely determined solution (cf., e.g., GilbargTrudinger 1977, Ch. 10.5). u

In addition it can be shown that

is twice continuously differentiable on

G

and solves the

following boundary value problem: -[al(x,y)ux]x -

[a2(x,Y)uy]y + c(x,y)u = q(x,y), (x,y) E G

u(x,y) = 'P(x,y),

(x,y)

a

G.

(12.2)

12.

Properly posed boundary value problems

209

The differential equation is called the Euler differential equation for the variational problem.

Its principal part is

-aluxx - a2uyy.

The differential operator 2

a2

__7 ax

ay

2

is called the Laplace operator (Laplacian).

In polar coor-

dinates,

x=rcos0 y = r sin it looks like a2

Dr

1

+

a

r 3r

1

+

a2

r 7 ao2

The equation -°u(x,y) = q(x,y)

is called the Poisson equation and -°u(x,y) + cu(x,y) = q(x,y),

c = constant

is called the Helmholtz equation.

With boundary value problems, as with initial value problems, there arises the question of whether the given problem is uniquely solvable and if this solution depends continuously on the preconditions.

In Equation (12.2) the

preconditions are the functions

and

q

ip.

Strictly speak-

ing, one should also examine the effect of "small deformations" of the boundary curve.

Because of the special prob-

lems this entails, we will avoid this issue.

For many bound-

ary value problems, both the uniqueness of the solution and its continuous dependence on the preconditions follows from

210


II.

the maximum-minimum principle (extremum principle). Maximum-minimum principle.

Theorem 12.3:

q(x,y) > 0 (q(x,y) < 0) for all

and

every nonconstant solution

If

c(x,y) > 0

(x,y) c G, then

of differential equation (12.2)

u

assumes its minimum, if it is negative (its maximum, if it is positive) on

DG

and not in

G.

A proof may be found in Hellwig 1977, Part 3, Ch. 1.1.

Let boundary value problem (12.2) with

Theorem 12.4:

c(x,y) > 0 (1)

for all

(x,y)

e G

be given.

Then

It follows from q(x,y) > 0,

(x,y) E U

i4(x,y) > 0,

(x,y) e DG

u(X,y) > 0,

(x,y) E G.

and

that

(2)

Iu(x,y)I

There exists a constant

0

such that

max Iq(X,Y)l, (X,y)cG (x,y) E

The first assertion of the theorem is a reformulation of the maximum minimum principle which in many instances is more easily applied.

The second assertion shows that the boundary

value problem is properly posed in the maximum norm. Proo

:

(1) follows immediately from Theorem 12.3.

(2), we begin by letting w(x,y) = ' + (exp($ ) - exp(sx))Q where

To prove

12.


'1'

=

max lb(X,Y)1, (x,y)c G const. > 0,

a

211

max jq(x,Y)j (x,y)EG

Q =

a const.

>

max (x,y)EG

Further, let maxc_ {1aX al(x,Y)I, c(x,Y)}.

M

(X,y)

Without loss of generality, we may suppose that the first component, x, is always nonnegative on

Since

G.

a1(x,y) > a,

we have

r(x,y) _ -[al(x,Y)wx(x,Y)]x -

[a 2(x,Y)wy(x,Y)]y

+ c(x,Y)w(x,Y)

= Q exp(Bx)[al(x,Y)s2 +

+ c(x,Y) [Q exp(SC) +

s

ax ai(x,Y) - c(x,Y)J

Y']

> Q exp(Bx)[as2 - M0+1)).

Now choose

a

so large that as2 - M(0+1) > 1.

It follows that r(x,Y) I Q,

(x,y)

E G.

In addition,

w(x,y) >

'l,

(x,y) E 9G.

From this it follows that q(x,y) + r(x,y) > 0 (X,y)

q(x,y)

E G

- r(x,y) < 0

u(x,Y) + w(x,Y) = V'(x,Y)

+ w(x,Y) > 0

- w(x,Y) = i,(x,Y)

- w(x,Y) < 0

(x,y) E U(x,Y)

G.

212


II.

Together with (1) we obtain u(x,y) + w(x,y) > 0 u(x,y)

- W(X,Y) < 0

which is equivalent to (x,y) e G.

Iu(x,y)I < W(x,Y),

u, and its continu-

To check the uniqueness of the solution ous dependence on the preconditions ferent solution

u

and

for preconditions

Theorem 12.4(2), for Iu(x,Y)

1

- u(x,Y)I
0,

0 > 0,

a+B > 0

the problem has a unique solution. tinuously on the preconditions

The solution depends con-

q(x,y)

is a valid monotone principle: q(x,y) > implies

and

*(x,y). and

0

4i(x,y)

There > 0

u(x,y) > 0. (2)

If

a = 0 = 0, then

a solution whenever uniquely solvable.

u(x,y)

is.

u(x,y) + c, c = constant, is Therefore the problem is not

However, in certain important cases, it

can be reduced to a properly posed boundary value problem of the first type.

To this end, we choose

gl(x,y)

and

g2(x,y)

so that

3x gl(x,Y) + ay g2(x,Y) = q(x,y).

The differential equation can then be written as a first order system:

-ux(x,Y) + vy(x,Y) = gl(x,Y), -uy(x,Y) v

- vx(x,Y) = g2(x,Y)

is called the conjugate function for

u .

If

q e C1(G,IR),


12.

v

219

satisfies the differential equation - v(X,Y) = g(x,Y) = ax g2(x,Y)

-

y gl(x,y).

We now compute the tangential derivative of point.

Let

(wl,w2)

the outward normal.

v

at a boundary

be the unit vector in the direction of Then

is the corresponding tan-

(-w2,wl)

gential unit vector, with the positive sense of rotation. -w2vX(X,Y) + wlvy(X,Y)

= -w2[-uy(x,y)-g2(x,y)] + wl[ux(x,Y)+gl(x,Y)]

= (X,Y) + wlgl(x,Y) + w2g2(x,Y) _ 'P(X,Y) thus is computable for all boundary points

'P(x,y)

given

'P(x,y), gl(x,y), and

g2(x,y).

(x,y),

Since the function

v

is unique, we obtain the integral condition ds = arc length along

faG (x,y)ds = 0,

G.

If the integrability condition is not satisfied, the original problem is not solvable. obtain a

E

Cl(aG,]R)

Otherwise, one can integrate

P

to

with

4)

a s

'P

is only determined up to a constant.

Finally we obtain

the following boundary value problem of the first type for v: -AV(X,y) = g(X,Y), v(x,Y) = T(X,Y),

One recomputes tem.

u

from

v

(X,y) E G (x,y) E

G.

through the above first order sys-

However, this is not necessary in most practical in-

stances (e.g., problems in fluid dynamics) since our interest

220

II.

is only in the derivatives of a < 0

For

(3)


u.

a < 0, the problem has unique

or

solutions in some cases and not in others.

a = 0, -a = v eIN, q = 0, and

5

For example, for

0, one obtains the family

of solutions

y eIR

u(x,y) =

x = r cos , y = r sin Q.

r2 = x2+y2,

Thus the problem is not uniquely solvable.

In particular,

there is no valid maximum-minimum principle. Example 12.8: geneous plate.

o

Biharmonie equation; load deflection of a homoThe differential equation

06u(x,y) = u

xxxx

+ 2u

xxyy

+ u = 0 yyyy

is called the biharmonie equation.

As with the harmonic equa-

tion, its solutions are real analytic on every open set.

The

deflection of a homogeneous plate is described by the differential equation MMu(x,y) = q(x,y),

(x,y) c G

with boundary conditions u(x,y) _ *1(x,y) (x,y) c 3G

(1)

(x,y) c DG.

(2)

-Du(x,y) = Yx,y) or

u(x,y) = *3(x,y) auan,y) _ 4(x,y)

Here

q c C°(U,IR), ip 1

c

C2(3G,IR), *2,'P4 a C°(3G,IR), and

12.


3 E C1 (BG,IR).

221

The boundary conditions (1) and (2) depend In the first case,

on the type of stress at the boundary.

the problem can be split into two second-order subproblems: -Av(x,y) = q(x,y),

(x,y) e G

(a)

v(x,y) = Yx,y),

(x,y)

e aG

and -tU(X,y) = V(X,y),

(X,y) C G

(b)

u(x,y) = P1(x,y),

(x,y)

c

G.

As special cases of (12.2), these problems are both properly posed, since the maximum minimum principle applies.

All prop-

erties--especially the monotone principle--carry over immediately to the fourth-order equation with boundary conditions (1).

To solve the split system (a),

t'I E C° (BG,IR) problem (2)

(b), it suffices to have

instead of l e C2 (aG,IR). Boundary value

is also properly posed, but unfortunately it can-

not be split into a problem with two second-order differential equations.

Thus both the theoretical and the numerical treat-

ment are substantially more complicated.

There is no simple

monotone principle comparable to Theorem 12.4(1). The variation integral belonging to the differential equation AAu(x,y) = q(x,y) is

I[w] = ff

[(Aw(x,y))2 - 2q(x,y)w(x,y)]dx dy.

a The boundary value problem is equivalent to the variation problem

I [u] =min {I [w] with

I

w e W}

222


II.

W = (w C C2(G,IR)

I

w

satisfies boundary cond. (1)}

or

W = {w C C1(G, IR) n C2(G, IR)

I

w

satisfies boundary cond. (2)}.

It can be shown that

differentiable in

u

G.

is actually four times continuously o

Error estimates for numerical methods typically use higher derivatives of the solution problem.

u

of the boundary value

Experience shows that the methods may converge ex-

tremely slowly whenever these derivatives do not exist or are This automatically raises the question of the

unbounded.

existence and behavior of the higher derivatives of

u.

Matters are somewhat simplified by the fact that the solution will be sufficiently often differentiable in

G

if the bound-

ary of the region, the coefficients of the differential equation, and the boundary conditions are sufficiently often differentiable.

In practice one often encounters regions with

corners, such as rectangles

G = (a,b) x (c,d) or L-shaped regions G = (-a,a)

x (O,b) U (O,a) x (-b,b).

The boundaries of these regions are not differentiable, and therefore the remark just made is not relevant.

We must first

define continuous differentiability for a function on the boundary of such a region. set

U dIR2

properties: G.

and a function

defined

*

There should be an open

f C C1(U,]R)

with the following

(1) 3G c U, and (2) T = restriction of

f

to

Higher order differentiability is defined analogously.


12.

223

For the two cornered regions mentioned above, this definition is equivalent to the requirement that the restriction of

to each closed side of the region be sufficiently often

*

continuously differentiable. Poisson equation on the square.

Example 12.9:

-Au(x,y) = q(x,y),

(x,y)

c G = (0,1) x (0,1)

u(x,Y) = i(x,Y),

(x,y)

a aG.

v = 1(1)k

u c C2k(G,]R), then for

'Whenever

(-1)v-1(ay)2vu(x,Y)

(DX)2vu(x,Y) +

(-1)v- j-1

x)2j(

(

y)2v-2j

2 ]Au(x,Y)

v=o j Let

(xo,yo)

let

*

be one of the corner points of the square and 2k-times continuously differentiable.

be

left side of the equation at the point mined by and

alone.

*

(xo,yo)

Then the is deter-

We have the following relations between

q:

*xx(xo,Yo) + 4) yy(xo,Yo) = -q(xo,Yo) IPxxxx(xo,Yo)

-

Ip

-gxx(xo,Yo) + gyy(xo,Yo)

YYYY(xo,Yo)

etc.

does not belong to

When these equations are false, u

On the other hand a more careful analysis will

C2k(G,]R).

show that

u

does belong to

tions are satisfied and

q

C2k(G,]R) and

p

if the above equa-

are sufficiently often

differentiable.

The validity of the equations can be enforced through

224

II.


the addition of a function with the "appropriate singularity". v = 1(l)-, let

For

v

Im(z2vlog

vv(x,Y) = 2(-1)

log z = log r+i4 For

x > 0

and

where

y > 0

z)

r = IzI, 4 = argIzI,

-n < 4

< n.

we have

vv(x,0) = 0 y2v vv(O,Y) = Set

cpv = xx(li,v)+'pyy(u,v)+q(p,v), u = 0,1 and v = 0,1

i

u(x,Y) = u(x,Y) + n

V+(x,Y) = V'(x,Y)

+

n

1

1

1

1

2

cpv Im(zpvlog zpv)

p=0 v=o 1

1

E

E

2

cpv Im(zpv log zpv)

p=0 v=0

where z00 = z,

z10 = -i(z-l),

z01 = i(z-i),

zll = -(z-i-1).

The new boundary value problem reads -au(x,y) = q(x,y),

u(x,Y) = kx,Y),

We have

u e C2 (G, IR)

(x,y)

e G

(x,y) c DG.

.

The problem -Eu(x,Y) = 1,

u(x,Y) = 0,

(x,Y) e G (x,y)

c DG

has been solved twice, with the simplest of difference methods (cf. Section 13), once directly, and once by means of u.

Table 12.10 contains the results for increments

the points

(a,a).

h

and at

The upper numbers were computed directly


12.

225

with the difference method, and the lower numbers with the given boundary correction.

a

1/2

h

1/32

1/8

1/128

0.7344577(-l)

0.1808965(-1)

0.7370542(-l)

0.1821285(-l)

0.7365719(-l) 0.7367349(-1)

0.1819750(-1)

0.1993333(-2)

0.1820544(-1)

0.1999667(-2)

1/256 0.7367047(-1)

0.1820448(-1)

0.1999212(-2)

0.1784531(-3)

0.7367149(-1)

0.1820498(-1)

0.1999622(-2)

0.1788425(-3)

1/16

1/64

Table 12.10

h

a

1/64

1/2

1/128

1/32

1/8

0.736713349(-1)

0.182048795(-l)

0.199888417(-2)

0.736713549(-1)

0.182049484(-1)

0.199961973(-2)

1/256 0.736713532(-1)

0.182049475(-1)

0.199961516(-2)

0.178796363(-3)

0.736713533(-1)

0.182049478(-1)

0.199961941(-2)

0.178842316(-3)

Table 12.11

Table 12.11 contains the values extrapolated from the preceding computations. pure

Extrapolation proceded in the sense of a

h2-expansion:

wh(a,a) =

3[4 uh(a,a)

With the exception of the point

- u2h(a,a)]

(1/128,1/128), the last line

is accurate to within one unit in the last decimal place.

the exceptional point, the error is less than 100 units of the last decimal.

The values in the vicinity of the

At

226

II.


corners are particularly difficult to compute. that the detour via

and

'

is worthwhile.

u

It is clear Incidentally,

these numerical results provide a good example of the kind of accuracy which can be achieved on a machine with a mantissa length of 48 bits.

With boundary value problems, round-

ing error hardly plays a role, because the systems of equations are solved with particularly nice algorithms. Example 12.12:

Poisson equation on a nonconvex region with

corners.

(x,y) E G

-ou(x,y) = q(x,y),

u(x,y) _ (x,y), Ga = {(x,y) cIR2

1

(x,y) s DGa and

x2+y2 < 1 y

Figure 12.13

jyj

for

> x tan Z} a e (r,2a).


12.

227

The region (Figure 12.13) has three corners (0,0), (cos a/2, sin a/2), (cos a/2, -sin a/2).

The interior angles are a,

n/2,

n/2.

The remarks at 12.9 apply to the right angles. interior angle of

arise.

u

a > n

But at the

other singularities in the derivatives

Let

t (x,y) = Re(zn/a) = Re exp[(n/a)log z]

log z = log r +

7r

0

0

h IIF(x0+h0y0 )

-

F(x0)II < KIIyOII

0

or

IIF(xo+hoyo) This contradicts (1).

-

F(x0)II < Kllhoyo II

Therefore

F'(x)

is regular every-

where.

F(x) = F(Z)

is injective.

F

Since

once by virtue of (1).

F'(x)

implies

x = :

at

is always regular, it

follows from the implicit function theorem that the inverse map

Q

is continuously differentiable and that

open mapping. F(Rn)

It maps open sets to open sets.

F

In particular,

is surjective.

be an arbitrary but fixed vector.

IIF(x)

-

F(0)II _ KIIxII

IIF(x)-x011 + Ilxo-F(0)II For all

is an

is an open set.

We must still show that x0 e]Rn

F

x

KIIxII

outside the ball

E = {x a mn

I

Ilxll _ 2IIx0-F(0)II/K}

we have

d(x) = IIF(x)-xoll > IIF(0)-x011

Let

By (1) we have

13.

Difference methods

237

Therefore there exists an

with

x1 e E

d(x1) < d(x), x c 1R'.

On the other hand,

d(x1) = Since

F(Rn)

inf

n II y-xo II

ycF(R )

is open, it follows that

is surjective.

Thus

x0 a F(1Rn).

F

It also follows from (1) that

IIx-RII = IIF(Q(x))-F(Q(R))II

KIIQ(x)-Q(R) II

This completes the proof of (2). Proof that (2) implies (1):

x,R a Rn.

Let

It follows by

virtue of (2) that

IIx-RII =

IIQ(F(x))-Q(F(X))II_

Theorem 13.6:

Let

KIF(x)-F(R) II

0

be a consistent and stable difference

D

method for Problem 13.2 and let

m, jo

constants as in Definition 13.3.

we define the lattice functions

K > 0

IN, and

For arbitrary

be

u e C2(G, IR)

wj, j = jo(l)m, to be the

solutions of the difference equations

F- (V ,wj) = Rj (r (q)). Here

= rr(u)

and

q = Lu.

Then we have:

Ilrj(u) wjIImJIFJ('Y,rj(u))-Rj(rj(q))II.,

(1)

j = jo(1)m. If

(2)

u e Cm(G, R), then

lim IIrj (u) -wjIIm = 0. j +00

Proof:

$

depends only on

13.5, the maps

F- (*,-) 3

We have

u

and not on

j.

By Theorem

have differentiable inverses

Q3.

238


11.

rj (u) = Qj (Fj

(u)) )

wi = Qj(Rj(rj(q)))

"jrj (u) -w.'m

we have the inequalities

II rj (u) -wjll
0. A-1 > 0.

(2) (3)

Theorem 13.9:

A = D - B

Let

MAT(n,n,IR), where A

D

and

B

be a splitting of

A c

satisfy 13.8(1) and (2).

is an M-matrix if and only if

p(D- IB) < 1

Then

(p = spectral

radius).

Proof:

Then the series

p(D-1B) < 1.

Let

(D-'B)"'

S

V=0

converges and

S > 0.

Obviously,

(I-D 1B)S = S(I-D-1B) = I, A-1 > 0

Conversely, let D-1B

with

x

A-1

and let

=

A

SD-I

> 0.

be an eigenvalue of

the corresponding eigenvector.

Then we have

the following inequalities: ID- IBxI < D-1BIxl

lXIlxi =

(I-D-1B)lxl < (l-lal)ixI (D-B)Ixl < (l-lal)Dlxi < (1-IXI)A-l Dlxl.

lxi

Since

x # 0, A-I > 0, and

plies that

IA!

0, the last inequality imp(D-1B) < 1. D-1B

o

can be estimated with the

help of Gershgorin circles (cf. Stoer-Bulirsch 1980).

For

Difference methods

13.

243

this let A = {aij

i = l(1)n, j = l(1)n}.

I

One obtains the following sufficient conditions for P(D-1B) < 1:

Condition 13.10:

A

is diagonal dominant, i.e.

n Jai'j

E

j=1 j#i

Condition 13.11:

A

i = l(1)n.

Iaiil,

0

v=0

converges, [(D+E)-IB]v > 0

T = 0

certainly converges.

The elements in the series are cer-

tainly no greater than the elements in preceding series. Therefore, I

-

(D+E)

1B

is regular, and [I-(D+E)-1B]-I

D+E-B

is also an M-matrix.

and this holds for all inverse monotone.

= T > 0.

We have

x,i c1Rn.

x < i

for

This shows that

F(x) F

< F(R), is

246

BOUNDARY VALUE PROBLEM

II.

In addition we have

ilx-RII ° II (D+E-B)1(F(x)-F(R)]

or

II

II

Ilg(x)-FcR)II

IIT(D+E)-III

The row sum norm of the matrix T(D+E)-1

((D+E)-1131'}(D+E)-

{

'=0

is obviously no greater than the norm of SD

1

=

{ Z (D 1B]'}D 1 = A 1. '=0

This implies that

IIT(D+E)-1II. < IIA-1II-

IIx-xII

IIF(x) Theorem 13.15:

IIA-1IIm

Hypotheses:

(a)

A E MAT(n,n,]R)

(b)

F: ]Rn -r IRn F(x)

o

.

is an

M-matrix

is diagonal and isotonic,

= Ax + F(x)

(c)

v EIRn, v > 0, Av > z = (1,...,1) EIRm

(d)

we]Rn,IIF(w)II_ 0

< v.

lwl

it follows from

Av > z

that

llxll, z

llx-xlim

11vil

x,x a IRn ,

For the proof of (2) we need to remember that tonic and

F(-x)

is antitonic.

-z < F(w)
2.

= 1(1)-}

is

consistent with respect to this problem, we have

lim

0.

j-

We now choose

j

F

)

and

q > 1

Rj

0

so large that for all

(rj (v)) -F3 )

j

> jo

we have

-Rj (rj (q)) II, < I.

are linear and isotonic.

For

i >

1

and

we have

R(r(q)) > Since we actually have

* > 2

and

q > 2, it follows that

Rj(rj(q)) > (2,...,2) and hence that

13.

Difference methods

253

Remark 13.21:

Instead of

tually proved

v e CW(G,1k)

(1,...,1).

o

v e C°(G,1k)

and

and

v >

v > 0, we ac-

However, the condi-

2.

Since one

tions of the lemma are sufficient for the sequel.

is again interested in the smallest possible functions of this type, constructions other than the one of our proof These other methods need only yield a continu-

could be used.

ous function

v > 0.

o

We choose

Proof of Theorem 13.16:

Then we can apply Theorem 13.15.

v

as in Lemma 13.20.

The quantities are related

as follows:

Theorem 13.15

Theorem 13.16

Fc1)

A

F(2)

F

J

J

rj (v) F(l)

v +

FJ2)

J

F

J

w

0

For

j

it follows from Theorem 13.15(1) that:

> jo

(w j)-Fj1) >

lIvIi,

Ilwj-wjII, I1rj(v)II-

>

IIwj -w,II

-

IIvIi

does not depend on

ity in 13.3(5) with

equivalent to

II R3 II

(i .)IIm

This proves the first inequal-

j.

K = 1/IIvII,.

< K.

wj,wj e C (Mj, D2).

The second inequality is

o

In view of the last proof, one may choose in Definition 13.3(5).

Here

v

K = 1/IIvII,,

is an otherwise arbitrary

function satisfying the properties given in Lemma 13.20.

254

II.


Conclusion (1) of Theorem 13.6 yields the error estimate

Ilrj(u)-wjII,, ` IIvII. IIFj(p,rj(u))-Rj(rj(q))II0.

j

= jo(l)o.

Here is the exact solution of the boundary value problem

u wj

is the solution of the difference equation

F(l,r.(u)) - Rj(rj(q))

is the local error

is a bounding function (which depends only on

v

The inequality can be sharpened to a pointwise estimate with the help of conclusion (2) of Theorem 13.15. points

(x,y)

and

c Mj

j

= j0(1)-

For all lattice

we have

Iu(x,Y)-wj (x,Y) I : v(x,Y)IIFj (*,rj (u))-Rj (rj (q))II_. In many important special cases, e.g., the model problem (Example 13.4), Rj

is the identity.

A straightforward

modification of the proof of Lemma 13.20 then leads to the following result: s > 0

and

exists a

e >

let

Ls(x,y) > 1 jl c 1N

0

and let

s

e Cm(G,]R)

(cf. Lemma 13.18).

with

Then there

such that

Iu(x,Y) -wj (x,Y) I < (1+e) S (x,Y)II Fj (,P,rj (u)) -rj (q))11_,

j = jl(1)-. In the model problem s(x,y) = 4 x(1-x) + 4 y(1-y) is such a function.

independently of

c.

Here one can actually choose

jl = 1,

It therefore follows that

Iu(x,y) -w3 (x,y) I < s ( x , y ) I I F j (,P,rj (u) ) -rj (q)II_,

j

= l(l)-.

We will now construct several concrete difference methods. Let

Difference methods

13.

e(1)

(11,

_

e(2)

(O),

=

0

255

if

Ih X

V

v = 1(1)4

let:

(x,y)+ae(v)

e G

Now for

_

(0). `

0

v = 1(1)4

with

for all

a

1

we associate

(x,y) c G

Nv(x,y,h) c G

four neighboring points

e(4)

`(-11,

With each point

(cf. Figure 13.22).

h > 0.

e(3) _

1

c

and

[0,h]

=

min {A >

(x,y)+xe(v)

0

c r}

otherwise

Nv(x,y,h) = (x,y) + ave(v)

dv(x,y,h) = II (x,Y) - Nv(x,Y,h)II2 = AvIIe(v)II2. Obviously we have 0 < dv(x,y,h) < h,

v = 1(1)4.

By Definition 13.1, the standard lattice with mesh size

h

is

Mh = {(x,y) e G

I

x = yh, y = vh where

u,v e 2Z},

0 < h < ho. For

(x,y)

long to

all the neighboring points

c Mh

Mh

or

Nv(x,y,h)

P.

Lip(2)(G,IR).

For brevity, we introduce the notation This is a subspace of every

be-

C&(G,IR)

f e Lip(Q)(G,IR) a''+Vf

there exists an

au+Vf ayv(x,Y)

-

axu ay v

ax u

(x,y)

E G,

defined as follows:

(x,y)

L >

0

0,

H(x,y,0) = 0,

a2(x,y) > 0

(x,y) c G,

z

e IR.

Hz(x,y,z) > 0

Lattice: 2-(0+£),

h. = Mj:

A point

j

= 1(l)-, t

sufficiently large, but fixed

standard lattice with mesh size

(x,y) e Mj

boring points

h..

is called boundary-distant if all neigh-

Nv(x,y,h

belong to

G; otherwise it is

called boundary-close.

Derivation of the difference equations:

At the boundary-

distant lattice points, the first two terms of

Lu(x,y)

are

Difference methods

13.

259

replaced, one at a time, with the aid of Lemma 13.23. hi, wi, and

abbreviate

ni, merely writing

{al(x+Zh,Y)[w(x,Y)

h, w, and

n:

- w(x+h,y)]

al(x-?h,Y)[w(x,Y)

- w(x-h,y)]

+ a2(x,Y+Zh)[w(x,Y)

- w(x,y+h)]

+ a2(x,Y-Zh)[w(x,Y)

- w(x,Y-h)])

+

We

+ H(x,Y,w(x,Y)) = Q(x,Y) If one replaces

by the exact solution

w

O(h2).

of the boundary

u c Lip(3)(G,]R), the local error will

value problem, where be

u

An analogous procedure at the boundary-close

lattice points yields E1KV(x,Y)[w(x,Y) - w(NV(x,Y,h))]

+ E2KV(x,Y)w(x,Y) + H(x,y,w(x,y))

= q(x,y) + E2KV(x,Y)p(NV(x,Y,h)) where

2a(XV,YV) KV(x,Y) =

du x,Y,h +

dv(x,Y,

u=

1

v = 1,3

2

v = 2,4

I

+2 (x, y, )l

(x,Y) + -11-dV(x,Y,h)e(v)

In the sums

El

and

E2, v

runs through the subsets of

{1,2,3,4}: El:

all

v

with

NV(x,y,h) c G

E2:

all

v

with

NV(x,y,h) e r.

260


II.

Formally, the equations for the boundary-distant points are special cases of the equations for the boundary-close points. However, they differ substantially with respect to the local error.

In applying Lemma 13.23 at the boundary-close points,

one must choose h1 = dl(x,y,h)

for the first summand of

h2 = d3(x,y,h)

Lu(x,y), and

h1 = d2(x,y,h) for the second.

and

and

h2 = d4(x,y,h)

The local error contains the remainder

R

and also the additional term hi-h

[4a(0)u"'(0) + 6a'(0)u"(0) + 3a"(0)u'(0)].

12

Altogether there results an error of may be reduced to

O(h3)

O(h).

However, this

by a trick (cf. Gorenflo 1973).

Divide the difference equations at the boundary-close points by

b(x,y) = E2KV(x,Y) The new equations now satisfy the normalization condition (4) of Theorem 13.16, since for

p > 1

and

q > 1

it is ob-

viously true that [q(x,Y) + E2Kv(x,Y)'U(x,Y)]/E2Kv(x,Y) > 1.

At the boundary-distant points such an "optical" improvement of the local error is not possible.

is

O(h2)

Therefore the maximum

.

We can now formally define (cf. Theorem 13.16) the difference operators:

13.

Difference methods

261

1

whenever

(x,y)

is boundary-distant

E2Kv(x,y)

whenever

(x,y)

is boundary-close

b(x,y) ll

4

Kv(x,Y)w(x,Y) V=1

E2Kv(x,Y)w(Nv(x,Y,h))]/b(x,Y)

-

H(x,y,w(x,y))/b(x,y)

Ri (ri (q))(x,Y) = q(x,y)/b(x,y).

there is a matrix

For

B-1A; B

is a diagonal matrix

b(x,y), whereas the particular

with diagonal elements

A

naturally also depends on the enumeration of the lattice points.

In practice, there are two methods of enumeration

which have proven themselves to be of value: (1)

Enumeration by columns and rows:

(x,y)

precedes

(z,y)

if one of the following conditions is satisfied: x < x,

(a)

(b)

x = z

and

With this enumeration, the matrix

A

y < y.

becomes block tridia-

gone1: D1

-S1

1

D2

-S2

2

D3

A =

-Sk

The matrices

Du

are quadratic and tridiagonal.

Their dia-

gonal is positive, and all other elements are nonpositive. The matrices

S11

and

SP

are nonnegative.

262

(2)

II.


Enumeration by the checkerboard pattern:

lattice

Divide the

into two disjoint subsets (the white and black

Mj

squares of a checkerboard): Mil) = {(uh,vh) c M.

u+v

even}

{(uh,vh) a Mj

u+v

odd}.

The elements of

Mil)

In each of these subsets, we use the column

second.

of

are enumerated first, and the elements

and row ordering of (1). D1

The result is a matrix of the form -S

A = -9

D1

and

are quadratic diagonal matrices with positive

D2

diagonals.

D2

S

and

S

are nonnegative matrices.

In Figures 13.26 and 13.27 we have an example of the two enumerations.

Figure 13.26.

Enumeration by columns and rows

13.

Difference methods

Figure 13.27.

263

Enumeration on the checkerboard pattern

We will now show that

B-1A

and

A

are M-matrices.

It is

obvious that: 4

(1)

app =

p = 1(1)n,

Kv(x,y) > 0,

(x,y)

c

V=1 (2)

apa = -Kv(x,y)

I

lapa1,

p = 1(1)n.

a=1 a+p (4)

For each row

(ap1,...,apn), belonging to a boundary-

close point, n a pp >

E

IapaI.

a=1

a#p (5)

apa = A

In case

0

implies

aap =

matrix

for

p,a = 1(1)n.

is irreducible, it is even irreducible diagonal

dominant, by (1) through (4)

wise, A

0

(cf. condition 13.11).

Other-

is reducible, and by (5) there exists a permutation P

such that

264


II.

A PAP

1

1

A2

=

l®

1

Av, v = 1(1)L

The matrices Each matrix

A

are quadratic and irreducible.

has at least one row which belongs to a

AV

boundary-close point.

Hence all of these matrices are ir-

reducible diagonal dominant, and thus quently, A

Conse-

is also an M-matrix.

For certain G = (0,1)

M-matrices.

x (0,1)

h

or

and certain simple regions (e.g.

G= {(x,y) E (0,1) x (0,1)

h = 1/m] it will be the case that

dv(x,y,h) = h.

x+y < 1},

I

When this

condition is met, we have the additional results: (6)

Kv(x,y,h) = Ku(Nv(x,y,h),h)

where

u-1 = (v+1)mod 4, (x,y)

(7)

apo = aop

(8)

A

(9)

B-IA

for

c M..

p,a = 1(1)n.

is positive definite. B-1/2AB-1/2

is similar to

and therefore has

positive eigenvalues only.

Of the conditions of Theorem 13.16 we have shown so far that (2)

(B- 1

A

is an M-matrix), (4) (normalization condition),

and (5) (consistency) are satisfied. H(x,y,w(x,y))/b(x,y) is trivially diagonal and isotonic. also satisfied.

Thus condition (3) is

Therefore, the method is stable.

In the following examples we restrict ourselves to the region

G = (0,1) x (0,1); for the lattice

M

we always

choose the standard lattice with mesh width h = h.

= 2-j.

In

13.

Difference methods

265

this way we avoid all special problems related to proximity In principle, however, they could be

to the boundary.

solved with methods similar to those in Example 13.25.

For

brevity's sake, we also consider only linear differential operators without the summand

Then the sumWhen

drops out of the difference operator.

mand (x,y)

H(x,y,u(x,y)).

c

w(x,y)

P, we use

for

Differential operator:

Example 13.28:

Lu = -a11uxx

a22uyy -

b u 1

x

- b2uy.

Coefficients as in Problem 13.2. Difference equations:

h2{[all(x,Y)+ch][2w(x,Y)-w(x+h,y)-w(x-h,Y)] [a22(x,Y)+ch][2w(x,Y)-w(x,y+h)-w(x,y-h)]}

+

Zh{bl(x,Y)[w(x+h,Y)-w(x-h,Y)]

-

+ b2(x,Y)[w(x,y+h)-w(x,y-h)]}

= q(x,Y) Here

When when

c

> 0

is an arbitrary, but fixed, constant.

u E Lip(3)(G,IR), we obtain a local error of

c = 0,

and

can be given by

0(h)

when

an M-matrix.

c > 0.

For small

h,

The necessary and sufficient

conditions for this are flbl(x,Y)l < all(x,Y) + ch,

(x,Y)

e M

Zjb2(x,Y)l < a22(x,Y) + ch,

(x,Y)

a M

which is equivalent to

0(h2)

266


II.

2[Ibl(x,Y)I-2c]

E Mj

(x,y)

< all(x,Y),

2[Ib2(x,y)I-2c] < a22(x,y),

(x,y) E M3.

If one of the above conditions is not met, the matrix may possibly be singular.

Therefore these inequalities must be

satisfied in every case.

local error, and for h c (0,h0]. lb2I

For

For

c = 0, one obtains the smaller

c > 0, the larger stability interval

In the problems of fluid dynamics, Ib1I

are often substantially larger than

all

and

or a22.

c > 0, we introduce a numerical viscosity (as with the

Friedrichs method, cf. Ch. 6). in many other ways as well.

This could be accomplished

One can then improve the

global error by extrapolation.

o

Differential operator:

Example 13.29:

as in Example 13.28.

Difference equations: h2{all(x,Y)(2w(x,Y)-w(x+h,Y)-w(x-h,y)l

+ -

Here

D1

and

D2

a22(x,y)[2w(x,y)-w(x,y+h)-w(x,y-h)]}

h{D1(x,y) + D2(x,Y)} = q(x,y).

are defined as follows, where

(x,y)

`bl(x,Y) [w(x+h,Y)-w(x,Y)]

for

b 1(x,y) > 0

bl(x,y) [w(x,y)-w(x-h,y)J

for

bl(x,y)

1b2(x,y) [w(x,y+h)-w(x,y)]

for

b2(x,y) > 0

b2(x,y) [w(x,y)-w(x,y-h)]

for

b2(x,y)
0, a(x,y)2

-

(X,y)

CG

b(x,y)2 > 0.

Difference equations:

{a(x,y)[2w(x,y)-w(x+h,y+h)-w(x-h,y-h)] 2h

+ a(x,Y)[2w(x,Y)-w(x+h,y-h)-w(x-h,y+h)l - b(x,Y)[w(x+h,Y+h)-w(x-h,y+h)-w(x+h,y-h)+w(x-h,y-h)]} = q(x,y).

When Ib(x,Y)I < a(x,y) ,

(x,Y) c Mi

one obtains an M-matrix independent of

h.

However, the dif-

ferential operator is uniformly elliptic only for Ib(x,Y)I < a(x,Y)

When

b(x,y)

__

,

(x,Y) e G.

0, the system of difference equations splits

into two linear systems of equations, namely for the points (ph,vh)

where

p + v

is even

(ph,vh)

where

p + v

is odd.

and


[I.

268

One can then restrict oneself to solving one of the systems. The local error is of order

0(h2)

for

u e Lip(3)(U,IR).

o

MuZtipZace method.

Example 13.31:

Differential operator: Lu(x,y) = -Au(x,y).

Difference equations: {5w(x,Y)-[w(x+h,Y)+w(x,y+h)+w(x-h,Y)+w(x,Y-h)] h -

4[w(x+h,Y+h)+w(x-h,y+h)+w(x-h,Y-h)+w(x+h,Y-h)]}

= q(x,y) + S[q(x+h,y)+q(x,y+h)+q(x-h,y)+q(x,y-h)].

The local error is

0(h4)

for

13.16 is applicable because

u c Lip(5)(G, Ill).

Theorem

always has an M-matrix.

The natural generalization to more general regions leads to a method with a local error of

0(h3).

More on other methods

of similar type may be found in Collatz 1966.

o

So far we have only considered boundary value problems of the first type, i.e., the functional values on

t

were

Nevertheless, the method also works with certain

given.

other boundary value problems. Boundary value problem:

Example 13.32:

-Eu(x,Y) = q(x,y),

(x,Y)

u(x,y) = P(x,y),

(x,y)c r

u(0,Y)

where fixed.

ii

-

and

0'ux(0,Y)

4

E G = (0,1) x (0,1) and

x +

0

= 0(y), y E (0,1)

are continuous and bounded and

a > 0

is

13.

Difference methods

269

Lattice: A.:

the standard lattice

with mesh width h=hi2

M3 .

3

(0,µh), p = 1(1)2j-l.

combined with the points Difference equations:

For the points in

M. n (0,1) x (0,1), we use the same equa-

tions as for the model problem (see Example 13.4). u e Lip(3)(G,IR)

y = ph, u = 1(1)2j-1, and

For

we have

u(h,y) = u(O,Y) + hux(0,Y) + 1h2uxx(0,Y) + 0(h3) u(O,Y) + hux(0,Y) -h2uyy(O,y)

If we replace

-

Zh2[q(O,Y)+uyy(0,Y)] + 0(h3).

by

2u(O,Y) - u(0,y+h)

- u(0,y-h) + 0(h3)

we obtain u(h,y) = 2u(0,y)

Zu(0,y+h)

-

+ hu x(0,Y)

-

-

Zu(0,y-h)

Zh2q(O,Y) + 0(h3)

u x(O,Y) =

2h[2uCh,Y)+u(O,y+h)+u(O,Y-h)-4u(O,Y)]

+ Zhq(O,Y) + 0(h2).

This leads to the difference equation - a[2u(h,Y)+u(O,Y+h)+u(O,Y-h)l}

h{(2h+4a)u(O,Y)

(y)

Since

+

Zhq(0,Y)

a > 0, the corresponding matrix is an M-matrix.

theorem similar to Theorem 13.16 holds true. converges like tion by possible.

0(h2).

The method

If one multiplies the difference equa-

1/a, the passage to the limit o

A

a - -

is immediately

270

14.

II.


Variational methods

We consider the variational problem I[u]

= min{I[wl

I

w e W},

(14.1)

where I[w]

= fi [a1w2 + a2wy + 2Q(x,y,w)Idxdy. G

Here

G

is to be a bounded region in

integral theorem is applicable, and

Q F C2(G x ]R, ]R)

to which the Gauss

al,a2 a C1(G,IR), and

where

al(x,y) > a > 0,

a2(x,y) > a > 0,

0 < QzZ(x,y,z) < d,

The function space below.

IR2

W

(x,y)

e G,

z aIR.

will be characterized more closely

The connection with boundary value problems is es-

tablished by the following theorem (cf., e.g., GilbargTrudinger 1977, Ch. 10.5).

Theorem 14.2:

is a solu-

A function u e C2(G, IR) fl C°(G, ]R)

tion of the boundary value problem -[alux]x -

(a2uyly + Qz(x,y,u) = 0,

(x,y) e G

(14.3)

u(x,y) = 0,

(x,y)

e DG

if and only if it satisfies condition (14.1) with

W = {w a C2(G, IR)

fl

C°(-a, IR)

I

w(x,y) = 0 for all (x,y) e 8G}.

In searching for the minimum of the functional

I[w],

it has turned out to be useful to admit functions which are not everywhere twice continuously differentiable.

In practice

one approximates the twice continuously differentiable solutions of the boundary value problem (14.3) with piecewise once

14.

Variational methods

271

continuously differentiable functions, e.g. piecewise polyThen one only has to make sure that the functions

nomials.

are continuous across the boundary points.

We will now focus on the space in which the functional I[w]

will be considered. K(G,IR)

Let

Definition 14.4:

w e C°(G,IR)

functions

such that:

(1)

w(x,y) = 0,

(2)

w

(x,y) e aG.

is absolutely continuous, both as a function with

x

of

with

y

y

held fixed, and as a function of

held fixed.

x

w. e L2(G, ]R).

wx,

(3)

be the vector space of all

We define the following norm (the Sobolev norm) on

K(G,]R):

2

1IwIIH =

[If (w2 + wx + wy )dxdy]l/2 G

We denote the closure of the space H(G,]R).

this norm by

We can extend setting

plies that

w

with respect to

a

continuously over all of

w

w(x,y) = 0

K(G,]R)

outside of

G.

]R2

by

Then condition (2) im-

is almost everywhere partially differentiable, (a,b) c]R2

and that for arbitrary

(cf. Natanson 1961, Ch.

IX) : rx

wx(t,y)dt

w(x,y) = J

a

(x, Y)

e IR2

rY =

J

wy(x,t)dt.

The following remark shows that variational problem (14.1) can also be considered in the space H(G,]R).

II.

272

Remark 14.5:

Let


u e C2(G, IR) n C°(G,IR)

be a solution of

Then we have

problem (14.3).

= min{I[w]

I[u]

When the boundary

3G

w e H(G, IR)}.

I

is sufficiently smooth, the converse

For example, it is enough that

also holds.

be piece-

2G

wise continuously differentiable and all the internal angles of the corners of the region be less than

2n.

o

The natural numerical method for a successive approximation of the minimum of the functional

I[w]

is the

Ritz method: Choose

linearly independent functions

n

v = 1(1)n, from the space

K(G, IR).

n-dimensional vector space

Vn.

minimum of the functionals

I[w]

I[v]

Each the

V

I

These will span an

Then determine

v e Vn, the

in V:

w e Vn}.

can be represented as a linear combination of

w e Vn f

= min{I[w]

fv,

:

n

w(x,y) =

I

Bvfv(x,Y)

v=1

In particular, we have n

v(x,Y) =

I

cvfv(x,Y),

v=1 I[w]

= I(Sl,...,8n).

From the necessary conditions

2c

(cl,...,cn)

= 0,

v = 1(1)n

v

one obtains a system of equations for the coefficients

cv:

Variational methods

14.

fG[a,(fv)x

'I c(fx

u=1

273

E cu(fu)Y (14.6) + a2(fv)Y u=1 n E cufu)]dxdy = 0, v = 1(1)n. fvQz(x,y,

+

p=1

Whenever the solution

of the boundary value problem

u

(14.3) has a "good" approximation by functions in can expect the error

to be "small" also.

u - v

Vn, one

Thus the

effectiveness of the method depends very decidedly on a suitable choice for the space

Vn.

These relationships will be

investigated carefully in a later part of the chapter.

Now

we will consider the practical problems which arise in solvIt will turn out

ing the system of equations numerically.

that the choice of a special basis for

Vn

is also important.

In the following we will generally assume that is of the special form

Q(x,y,z)

Q(x,Y,z) = 2 a(x,Y)z2 - q(x,y)z, where

a(x,y) >

0

for

(x,y)

e G.

In this case, the system

of equations (14.6) and the differential equation (14.3) are The system of equations has the form

linear.

A c = d where

A = (auv), c = (c1,...,cn)1, and

d = (dl,...,dn)T

with

auv = Gf[al(fu)x(fv)x + a2(fu)y(fv)y + afufv]dxdY, du = If qfu dxdy. G

A

is symmetric and positive semidefinite.

tions

fv

definite.

are linearly independent, A Therefore, v

Since the func-

is even positive

is uniquely determined.

We begin with four classic choices of basis functions

274

II.


fV, which are all of demonstrated utility for particular problems: (1) (2)

xkyR

monomials

products of orthogonal polynomials

gk(x)gZ(y)

I sin(kx) sin(Ry) (3)

sin(kx)cos(iy)

trigonometric monomials

:

Icos(kx)cos(iy) (4)

Bk(x)BR(y)

products of cardinal splines.

If the functions chosen above do not vanish on

8G, they

must be multiplied by a function which does vanish on and is never zero on

G.

It is preferable to choose basis

functions at the onset which are zero on if

aG

G.

For example,

G = (0,1)2, one could choose

x(1-x)Y(1-y),

x2(1-x)y(l-y),

x(1-x)y2(1-y),

x2(1-x)y2(1-y),

or sin(Trx) sin(Try) ,

sin(2rrx) sin(Try) ,

sin(rx) sin(2iry) , sin(2nx)sin(2rry)

For

G = {(r cos ¢, r sin 0)

1

r e

[0,1),

a good

c

choice is: r2-1,

(r2-1)sin ,

(r2-1)cos 0,

(r2-1)sin 20, (r2-1)cos 20.

Usually choice (2)

is better than (1), since one ob-

tains smaller numbers off of the main diagonal

of

A.

The

system of equations is then numerically more stable.

For

periodic solutions, however, one prefers choice (3).

Choice

(4)

is particularly to be recommended when choices (1)-(3)

give a poor approximation to the solution.

14.

Variational methods

27S

A shared disadvantage of choices (l)-(4) is that the A

matrix compute tions.

is almost always dense. n(n+3)/2

As a result, we have to

integrals in setting up the system of equa-

The solution then requires tedious general methods The com-

such as the Gauss algorithm or the Cholesky method.

putational effort thus generally grows in direct proportion with

n3.

One usually chooses

n < 100.

The effort just described can be reduced by choosing initial functions with smaller support. fufvo

(f11 )x(fv)x.

The products

(fu)y(fv)y

will differ from zero only when the supports of have nonempty intersection. are zero.

A

fu

In all other cases, the

fv

and auv

In this case, specialized, faster

is sparse.

methods are available to solve the system of equations. Estimates of this type are called finite element methods. The expression "finite element" refers to the support of the initial functions.

In the sequel we present a few simple

examples.

Example 14.7:

Linear polynomials on a triangulated region.

We assume that the boundary of our region is a polygonal line. Then we may represent

as the union of

G

AP, as in Figure 14.8.

N

closed triangles

It is required that the intersection

of two arbitrary distinct triangles be either empty or consist of exactly one vertex or exactly one side. tices of the triangles be denoted by

&v.

which do not belong to

Let them be enumerated from

We then define functions rules:

AP

Those ver-

1

2G, will to

n.

fv, v = 1(1)n, by the following

276

Triangulation of a region

Figure 14.8.

(1)

fv e C°(G, IR)

(2)

fv

restricted to

nomial in

IR2,

(3)

fvW')

(4)

fv(x,y) = 0

The functions (4).


11.

is a first degree poly-

Op

p = 1(1)N.

dvu for

(x,y)

c 3G.

are uniquely determined by properties (1)-

fv

They belong to the space

fv

vanishes

which does not contain vertex

AP

on every triangle

K(G,IR), and

CV.

If the triangulation is such that each vertex v belongs to at most

k

triangles, then each row and column of

contain at most

k +

1

A

will

elements different from zero.

In the special case v = (rvh, svh)T,

rv,sv eZZ

we can give formulas for the basis functions

fv.

The func-

tions are given in the various triangles in Illustration 14.9.

The coefficients for matrix

this.

We will demonstrate this for the special differential

A

equation -Du(x,y) = q(x,y)

Thus we have

a1 = a2 = 1, a

=_

0, and

can be computed from

(

Variational methods

14.

0

0

0

277

/0 /0 0

0

0

1-rv+sv

0

l+rv x

+h 1-rv+

0

h

l+s

0

x

l+rv-sv-

0

0

0

V V0 Figure 14.9.

svh

v

1 sv+h

0

0

0

0

0

0

r h v

Initial functions triangulation

fv for a special

auv = It [(fu)x(fv)x + (fu)y(fv)y]dxdy Since

(fo)x

and

(fa)y

are

1/h, -1/h, or

0,

depending

on the triangle, it follows that 4

for p = v

-1

for

sv = su

and

rv = ru+1

or

rv = ru-1

-1

for

rv = ru

and

sv = su+1

or

S. = su-

0

otherwise.

In this way we obtain the following "five point difference

278

II.


operator" which is often also called a difference star: 0

-1

0

-1

4

-1

0

-1

0

Tk,i

are the translation operators from Chapter

Here the

= 41

-

(Th,l + T_ h,1 + Th,2 + T_ h,2)'

10.

The left side of the system of equations is thus the same for this finite element method as for the simplest difference method (cf. Ch. 13).

On the right side here, how-

ever, we have the integrals

du = If gfudxdy while in the difference method we had h2q(r11 h, suh)

In practice, the integrals will be evaluated by a sufficiently accurate quadrature formula.

In the case at hand the follow-

ing formula, which is exact for first degree polynomials (cf., e.g. Witsch 1978, Theorem 5.2), is adequate: If g(x,y)dxdy z h2 [6g(0,0) + 6g(h,0) + 1 (O,h) where

is the triangle with vertices

A

Since the

fu

(0,0), (h,0), (O,h).

will be zero on at least two of the three

vertices, it follows that 2

du

a

6 (+l+l+l+l+l+l)q(ruh,suh) = h2q(ruh,suh).

Example 14.10:

Linear product approach on a rectangular is the union of

N

closed

rectangles with sides parallel to the axes, so that

may

subdivision.

We assume that

G

14.

Variational methods

279

Figure 14.11.

Subdivision into rectangles

be subdivided as in Figure 14.11.

We require that the inter-

section of two arbitrary, distinct rectangles be either empty or consist of exactly one vertex or exactly one side. denote by

CV

(v = 1(1)n)

We

those vertices of the rectangles

p which do not belong to

Then we define functions

G.

fv

by the following rule:

(1)

fv e G°(G, IR)

(2)

fv

restricted to

op

is the product of two

first degree polynomials in the independent variables and

x

y. (3)

fv(u)

(4)

fv(x,y) =

As in Example

svv 0

for

(x,y)

c

G.

14.7, the functions

fv

are uniquely

determined by properties (1)-(4), and belong to the space K(G,IR).

Each

fv

with common vertex

vanishes except on the four rectangles

v.

Thus each row and column of

at most nine elements which differ from zero. In the special case Ev = (rvh, svh)T,

rv,sv C 2Z

A

has

H.

280


we can again provide formulas for the basis functions

fv,

namely: (1-Ifi -rvI)(1-Ih -svI)

for Ih-rvl

0

otherwise.

< 1,

< 1

Ih-svI

fv

We can compute the partial derivatives of the

fv

on the

interiors of the rectangles: -1Fi(1

I

(1

I

S

for

sv

for -1
2) + Since

1,

IIull1 =

A2<w,w>1.

is the minimum of the variation integral, the ex-

u

pression in the parentheses in the last equality must be Otherwise, the difference

zero.

sign as

with

A

changes sign.

I[u+Aw]

- I[u]

will change

The second conclusion follows

A = 1.

It is also possible to derive equation (14.16) directly from the differential equation (14.3).

For

286


II.

a(x,Y)z2 - q(x,y)z

Q(x,Y,z) =

Z we multiply (14.3) by an arbitrary function (test function) and integrate over

w e K(G,]R)

G:

(a2uy)y + au-qlw dxdy = 0.

ff[-(alux)x G

It follows from the Gauss integral theorem that ff[aluxwx + a2uywy + auw]dxdy = This is equation (14.16).

Gf qw dxdy.

It is called the weak form of dif-

ferential equation (14.3).

With the aid of the Gauss inte-

gral theorem, it can also be derived immediately from similar differential equations which are not Euler solutions of a variational problem.

Ac = d

The system of equations

can also be obtained

This process is called the GaZerkin

by discretizing (14.16). method: Let

fv, v = l(1)n, be the basis of a finite dimen-

sional subspace

Vn

We want to find an approxi-

K(G,]R).

of

mation n

v(x,Y) =

E cvfv(x,Y)

v=1 such that

2,

u = 1(1)n.

As in the Ritz method it follows that auv = I

and

du =

2

A derivation of this type has the advantage of being applicable to more general differential equations.

We prefer to

proceed via variational methods because the error estimates follow directly from (14.17).

14.

Variational methods

Theorem 14.18:

Let

K(G, IR) .

be an n-dimensional subspace of

Vn

Let

287

u e H(G, ]R)

IM = min{I [w]

I

v c Vn be such that

and

w e H(G, IR) },

I (v] = min(I [w]

I

w e Vn}.

Then it is true that

Here

(1)

IN] < I[VI

(2)

(Iu-v112 < Y211u-viII < Y2 min

11u-v*11I

is the positive constant from Theorem 14.14.

Y2

Proof:

v*eVn

Inequality (1) is trivial.

It follows from this,

with the help of Theorem 14.15, that for every

Ii u-v (j 2 = I [v]

I (U]

-

t

for

x < t.

= 0

For fixed

and

we denote the Hermite interpolation polynomial gm(x,t).

We set

Gm(x,t) = g(x,t) - gm(x,t).

Then

al'Gm/axu

is called the Peano kernel of

Au.

e

The coefficients of the Hermite interpolation polygm(x,t)

nomial

are functions of

t

which can be repreTherefore,

sented explicitly with the aid of Cramer's Rule. gm e C

2m-2

tion in

Since

([a,b] x [a,b],IR). C2m-2([a,bl

g(x,t)

is also a func-

x [a,bl,]R), the same is true for

Gm(x,t) . Theorem 15.4:

Let

f

e

C2m([a,b],]R)

Hermite interpolation polynomial for x

e [a,b]

f(p) (x)

and let

fm

be the

Then for all

f.

we have the representation: - fmp) (x) =

rb m

T

1a

f(2m) (t)

all

m(x,t)dt,

ax

p = 0(1)2m-1.

296


II.

We begin by showing that

Proof:

b

m(x)

=

f

J

(2m)

(t)Gm(x,t)dt

a

is a solution of the following boundary value problem: 0(2m)(x)

f(2m)(X)

= (2m-l)!

(15.5)

0(v)(a)

=

(u)(b) = 0,

µ = 0(1)m-l.

will then be the Green's function for the

Gm(x,t)/(2m-1)!

boundary value problem (cf., e.g. Coddington-Levinson 1955).

Since

Gm a

C2m-2

([a,bJ

x

[a,b],]R), it follows that

(2m-2)-times continuously differentiable on

is

[a,b].

We have o

(2m-2) (x) =

b r

2m 2

f(2m)(t)a

m (x,t)dt

ax

a

2m-2

f(2m) (t) axZm-Z G M(x,t)dt Jxa rb +

2m-2

f(2m)(t)Gm(x,t)dt.

x

For

x # t, g(x,t), gm(x,t), and hence

It follows that

arbitrarily often differentiable. 10

e

C2m-1([a,b],]R).

(2m-1)(x) _

Differentiation yields

(a f(2m)(t)a2(x,t)dt m + f

respect to

m(x,x-0)

a

m-7Gm(x,t)dt

(t) a

ax -

Since the

(x)

2m-1

b

Jx f(2m)

+

a2m-2

(21m)

Z

J

+

f

(2m)

2m-2 (x)

:xzmm(x,x+0).

(2m-2)-th partial derivative of x

is continuous in

integral terms remain.

are all

Gm(x,t)

x

and

Gm(x,t)

with

t, only the two

As above, it follows that

15.

0

297

Hermite interpolation and the Ritz method

e C2m([a,b],]R)

(2m)

and

m(x,t)dt + f(2m) (x)

ax f(2m)a

(x) =

ax

fax

-Gm(x,t)dt + Jb f(2m)(t)a-ax

a 2Zm=iGm(x,x-0) ax

f(2m)(x)3 2m2m1lGm(x,x+0).

-

axr

x

We have a2m-1 ax

J(2m-l)!

IM--_79 (x,t) = 0

a2m 2 g(x,t) =

ugm(x,t) ax

and

x >

for

x < t

is continuous in

and

x

t

for

u = 0(1)2m-1,

Combining all this, we obtain

= 0.

mm(x,t)

ax

t

x # t

am

all

and

for

0

for

a2m 1

a2m-1

-0)

-

a 2m l-m(x'x+0) _ (2m-1):

a

a2m

a XTM_Gm

(x,t) = 0,

x # t.

From this it follows that (2m)

(x) = (2m-l)!

In addition, it follows for tion of

Gm(x,t)

m

u = 0(1)m-1

from the construc-

that

0(u) (a) Thus

f(2m)(X)

=

0(u) (b) = 0.

is a solution of boundary value problem (15.5).

The

function (2m-l)![f(x)

- fm(x)]

is obviously also a solution of (15.5).

Since the boundary

value problem has a unique solution, it follows that f(x)

-

fm(x) _ (2m-l)! m(x) _

1

(2m-1

fb f(2m) (t)Gm(x,t)dt. Ja

298


II.

Differentiating this and substituting the derivatives of O(x)

obtained farther above yields the conclusion.

Example 15.6:

g, gm, and

Gm

for

a

m = 1,2,3.

Case m = 1:

x-b-ab-t

gl(x,t) -

g(x,t) = (x-t)+,

(b-x4 t-a

for

x > t

(b-t)(x-a) b-a

for

x < t

G1(x,t) -

t-a

for x > t

b-t

for

Glx(x,t) _

GIx(x,x+0) _ X-a

Glx(x,x-0)

E --a

x < t

+ F = 1.

Case m = 2:

g(x,t) = (x-t)+ (b-t

2

x-a

2

(2(b-t)(x-a) + 3(b-a)(t-x)]

92(x,t)

(b-a)

b-x 2 t-a

2

[2(b-x)(t-a)+3(b-a)(x-t)] for x>t

(b-a)

G2(x,t) _

(b t)2 x- a 2

[2(b-t)(x-a)+3(b-a)(t-x)] for x t

({2(x-a)[2(b-t)(x-a)+3(b-a)(t-x)]

(b-a) +

(x-a)2(3a-b-2t)}

for x < t

15.

299


(t-a) 2{2[2(b-x)(t-a)+3(b-a)(x-t)] (b-a)

4(b-x)(3b-a-2t)} for x > t G2xx(x,t)

(b-t 2{2(2(b-t)(x-a)+3(b-a)(t-x)] (b-a)

3

+ 4(x-a)(3a-b-2t)} for x < (t-a 2 6(3b-a-2t)

for

x >

for

x < t

t

t

(b-a)

G2xxv(x,t) = b-t

2

6(3a-b-2t)

(ba) G2xxx(x,x-0)

- G2xxx(x,x+0) = 6.

Case m = 3: g(x,t) = (x-t)+ (b-t)3(x-a 3 {5(b-a)(t-x)[2(b-a)(t-x)+3(x-a)(b-t)]

g3(x,t)

(b-a)

+ 6(x-a)2(b-t)2}

G3(x, t)

=

J (x-t)

5

- g3(x,t)

for

x > t

- g3(x,t)

for

x < t.

Theorem 15.4 immediately yields an estimate for the interpolation error with respect to the norm Theorem 15.7:

Let

cmu holds true.

2

r rl rl

m-1

Here

o

Gm(x,t)

tion 15.3 for the interval computed for small by Lehmann 1975.

Then the inequality

< cmu(b-a) 2m-u IIf(2m)I12,

IIf(1j)-fmu)II2

where

f e C2m([a,b],]R).

11-112-

m.

all in [-i

ax

u - 0(1)2m-1

t) ] 2 dx dt I

1/2

M1

is the function (0,1].

Gm

The constants

from Definicmu

can be

The values in Table 15.8 were obtained

2.24457822314E-5 2.77638992969E-4

4.27311575545E-1

4.24705992865E-3 5.37215309350E-2 1

4.21294644506E-11 5.08920680460E-3 5.92874650749E-2

4.45212852385E-4

4.87950036474E-2

4.14039335605E-1

2

3

4

5

6

7

2.38010180208E-6

3.19767674247E-7

6.56734371321E-5

7.27392967453E-3

4.08248290464E-1

7.175679561o6E-8

m=4

1

1.63169843917E-5

m=3

m = 1,2,3,4.

2.01633313311E-3

for

1.05409255339E-1

m=2

cmu

o

m=1

TABLE 15.8.

15.

301


Proof:

From Theorem 15.4 and an application of the Cauchy-

Schwarz Inequality we obtain

f(u) (x)

-

fmu) 1

(x)

_7

((2m-1)!)

12

(

b

b If(2m)(t)]2dt J [auGm(x,t)]2dt.

a

a ax

By integration, this becomes < cmu(b-a) Ja [f(2m)(t)32dt

Jalf(u)(x)-fmu)(x)12dx where cmu(b-a) =

T_ L

b b u 1/2 {J J (a uGm(x,t)]2dt dx} a a 2x

2m-1

Every interval can be mapped onto transformation.

by an affine

(0,1]

With that substitution, we get (b-a)2m-u

cm11(b-a) _

cm11(1)

Letting 1

1

1

cmu = amu(1)

1/2

u

(2m-1)! {Jo1o [aa u(x,t)]2dx dt}

yields the desired conclusion.

a

The polynomials of degree less than or equal to form a 2m-dimensional vector space. (l,x,...,x

2m-1 )

2m-1

The canonical basis

is very impractical for actual computations

with Hermite interpolation polynomials.

Therefore we will

define a new basis which is better suited to our purposes. Definition 15.9: space.

Basis of the 2m-dimensional polynomial

The conditions S(11),m(0) a,X

= 6112

6a6

(a,6 = 0,1,

u,R = 0(1)m-l)

302


II.

define a basis

{SI

I

(x)

R = 0(l)m-1}

a = 0,1;

of the 2m-dimensional space of polynomials of degree less than or equal to

2m-1.

o

It is easily checked that the Hermite interpolation polynomial

f e Cm-1([a,bl,]R)

for a function

fm

has the

following representation: M-1

fm(x) =

I

(b-a)t[f(-')(a)SO,1,m(b-a) (15.10)

Z=0

This corresponds to the Lagrange interpolation formula for ordinary polynomial interpolation. Sa ,

R , m(x)

explicitly for

Table 15.11 gives the

m = 1,2,3.

In order to attain

great precision it is necessary to use Hermite interpolation formulas of high degree.

This can lead to the creation of

numerical instabilities.

To avoid this, we pass from global

interpolation over the interval

[a,b]

to piecewise inter-

polation with polynomials of lower degree. partitioning

[a,b]

intermediate points.

into

n

We do this by

subintervals, introducing

n-1

The interpolation function is pieced

together from the Hermite interpolation polynomials for each subinterval.

Theorem 15.12:

Let

m,n c IN,

f

e

Cm-1([a,b],]R)

and let

a = xo < x1 < ...... < xn-1 < xn - b be a partition of the interval and

i = 0(1)n-l

we define

[a,b].

For

x

E [xi,xi+1]


15.

TABLE 15.11:

S

m = 1,2,3.

for

(x)

m

Sa

m(x)

a

k

0

0

1

1-x

1

0

1

x

0

0

2

1

0

1

2

x - 2x2 + x3

1

0

2

1

1

2

0

0

3

1

0

1

3

x - 6x3 + 8x4 - 3x5

0

2

3

2

1

0

3

10x3

1

1

3

-

1

2

3

-x3

t

+ 2x3

3x2

-

3x2

1x2

2x3

-

x2 + x3

-

-

l0x3

+

-

3x3 + 3x4

_

6x5

1x5

$

Y

2

15x4 + 6x5

-

4x3 + 7x4 -

15x4

-

3x5

x4 + 2x5

M-1

x-x i

xi) [f (t) (xi)SO,R,m(

fm(x) =

303

kI0(xi+1

l

xi+1 xil

x-x. + f( )(xi+1)S1,L,m(x1+11xi)]

Then

fm

is the Hermite interpolation polynomial for

each subinterval

[xi,xi+l]

(cf. Theorem 15.1(1)).

(m-1)-times continuously differentiable on f

[a,b].

f

fm

is

Whenever

is actually 2m-times continuously differentiable on

the following inequalities hold:

on

[a,b],

304

II

f(u) _ fmu)II*

1

whenever

(b,b+h)

c DG

or

(b,b-h)

c 8G

> 1

whenever

(b+h,b)

a 8G

or

(b-h,b)

E 8G.

P.

The basis functions belong to

Cm-1(G,IR).

They vanish on

m = 1, we obtain the basis already discussed in

For

3G.

k,t = 0(1)m-1, but

Example 14.10, if the subdivision assumed there agrees with the one prescribed here.

Thus piecewise Hermite interpola-

tion supplies a generalization of this method to The matrix

A = (auv)

m > 1.

of the Ritz method for the

basis chosen here can be given explicitly for the special case

Q(x,y,z) = q(x,y)z + 1-a(x,y)z2, a(x,y) > 0 for (x,y)

c G

by:

auv =

[al(x,Y)Tb,k'm(x)Tb,R,m(Y)Tb*k*,m(x)Tb*'R*,m(Y) +

a(x,Y)Tb,k,m(x)Tb,.,,m(Y)Tb*,k*,m(x)Tb*,R*'m(Y)]dxdy. In this case

Tb,k,m(x)Tb,t,m(Y) is the

u-th basis function, and

316

II.


Tb*,k*,m(x)Tb*,IC*,m(Y)

v-th basis function.

is the

11 (b,b)

-

The integrals vanish whenever

(b*,b*)II 2

>

Therefore, each row and column of matrix elements which differ from zero.

9m2

A

has at most

At most four squares

contribute to the integrals.

Theorem 14.18 supplies the inequalities

Y211u-wl'I < Y2I!u-w*III.

11u-w112

Here we have u

solution of the variation problem in space

w

Ritz approximation from space

W

V u

w*

arbitrary functions from

:

Vu.

We have the additional inequality: II u-w*II 2

max

C° (r, IR) that

(x,y) e

0,

all >

0,

g > 0.

The execution of the method presupposes that we are given: (1) j

n

= 1(1)n, from

linearly independent basis functions C2(G,IR).

vj,

318

n

(2)

different collocation points

of these, the first ing


II.

n2 = n - n1

are to belong to

n1

are to lie in

The solution

(xk,yk) e G;

G, and the remain-

r.

of boundary value problem (16.1)

u

will now be approximated by a linear combination functions

w

of the

vj, where we impose the following conditions on w: k = 1(1)n1

Lw(xk,yk) = q(xk,yk),

(16.2)

k = nl + 1(1)n.

w(xk,yk) _ (xk,yk), In view of the fact that n

w(x,y) =

cj e IR

E cjv.(x,y), j=1

the substitute problem (16.2) is concerned with the system of linear equations: n

k = 1(1)n

8k,

akj =

Lvj(xk,yk)

for

k < nl

vj(xk,yk)

for

k > nl

q(xk,yk)

for

k < nl

(16.3)

$I,

for k > n

(xk'yk)

In many actual applications, the system of equations can be simplified considerably by a judicious choice of the

vj.

It is often possible to arrange matters so that either the

differential equation or the boundary conditions are satisfied exactly by the functions (A)

Boundary collocation: Lvj(x,y) = 0,

All

(xk,yk)

must lie in

We distinguish:

vj.

We have

j

q _ 0

and

= 1(1)n, (x,y) E

r, i.e. nl = 0, n2 = n.

G.

16. Collocation methods and boundary integral methods

(B)

Interior collocation:

We have

j

vj (x, Y) = 0 , All

(xk,yk)

must lie in

i ° 0

319

and

1 (1) n, (x,y) c r.

=

G, i.e. nl = n, n2 = 0.

The system of equations (16.3) does not always have a unique solution. rarily large.

When it does, the solution can be arbit-

A priori conclusions about the error

u-w

only be drawn on the basis of very special hypotheses. is the weakness of collocation methods.

can

This

It is therefore

essential to estimate the error a posteriori.

Nevertheless,

collocation methods with a posteriori error estimation frequently are superior to all other methods with respect to effort and accuracy.

Error estimates can be carried out in the norm as explained in Section 14 (cf. Theorem 14.19).

However,

this seems unduly complicated in comparison with the simplicity of collocation methods.

Therefore, one usually premonotone principles.

fers to estimate errors with the aid of

We wish to explain these estimates for the cases of boundary collocation and interior collocation. c = u-w,

r = q-Lw,

To this end, let = iy-w.

Then we have Lc(x,y) = r(x,y),

(x,y)

c G

c(x,Y) = O(x,Y),

(x,Y)

c r.

(16.4)

(A)

For boundary collocation, we have

r(x,y) = 0.

It

follows from the maximum-minimum principle (cf. Theorem 12.3) that for all

(x,y) c G:

320


II.

min max {q(x,y),0} < e(x,y) < (x,y)Er (x,y)er

Thus it suffices to derive estimates for . We assume that r

consists of only finitely many twice continuously dif-

ferentiable curves

rR.

Each arc then has a parametric

representation

t E [0,1].

(x,y) = [c1(t),F,2(t) ], We set

fi(t)

= wl(t),E2(t))

finitely many points h = 1/m.

tj

= jh,

and compute j

= 0(1)m, where

Then it is obviously true for all < h max - 2 te[0,1]

min j=0(1)m

for the

;

t e

m c 1N [0,1]

and that

'(t)

can be interpolated linearly between the points

tj.

The

interpolation error will be at most h2 4

max te[0,1]

"(t)

Combining this and letting dl = Zh

max tE[0,1)

we have, either for min

m(x,y) =

For small

or for

min

fi(t)

v = 2, that >

ta[0,1]

c(x,y) =

max

;(t)

are given weights and

0

m

w(x,y) = jIlcjvj(x,Y) Because of these conditions, the coefficients

cj,

j

= 1(1)m,

can be computed as usual with balancing calculations (cf. StoerBulirsch 1980, Chapter 4.8).

Only with an explicit case at

hand is it possible to decide if the additional effort (relative to simple collocation) is worthwhile.

For

n = m, one

simply obtains the old procedure.

Occasionally there have been attempts to replace condition (16.12) with max{

6kILw(xk,Yk) max k=1(1)n1

- q(xk,Yk)I,

max 6klw(xk,Yk) k=n1+1(1)n

- *(xk,Yk)I} = Min!

(minimization in the Chebyshev sense).

Experience has demon-

strated that this increases the computational effort tremenConsequently, any advantages with respect to the pre-

dously.

cision attainable become relatively minor.

We next discuss a boundary integral method for solving Problem (16.1), with region

G

unit disk

L = A, q = 0, and

0 e C1(r,IR).

The

is to be a simply-connected subset of the closed IzI

Consider the trial function

(2n u(z) =

z

J

(16.13)

c G.

0

If

p

is continuous, u c C0(G,IR)

(cf. e.g. Kellog 1929).

By differentiating, one shows in addition that monic in

G.

is har-

u

The boundary condition yields 2n

p(t)logjz-C(t)jdt = (z),

z

e F.

(16.14)

0

This is a linear Fredholm integral equation of the first kind with a weakly singular kernel. determined solution

p

There exists a uniquely The numeri-

(cf. e.g. Jaswon 1963).

cal method uses (16.14) to obtain first an approximation of

p

at the discrete points

tj

= 27r(j-1)/n,

j

Next (16.13) is used to obtain an approximation u(z)

for arbitrary

z

u

= 1(1)n. u(z)

of

c G.

The algorithm can be split into two parts, one dependent only on

r

and

E, and the other only on .

(A)

Boundary dependent part:

(1)

Computation of the weight matrix

W = (wjk)

quadrature formulas JTrf(t)loglzj-C(t)Idt °

zj = C(tj),

= k11wjkf(tk) + R(f)

j = 1(1)n

for

n

0.

330


II.

R(fv) = 0

The matrix

fv(t) =

for

11

v = 1

cos(2 t)

v = 2(2)n

sin(t)

v = 3(2)n.

Therefore

is regular.

(fv(tj))

W

is uniquely

determined.

Most of the computation is devoted to determin-

ing the

integrals

n2 f2s

fv(t)log;z;-E(t)ldt,

v,j = 1(1)n.

1-

Triangulation of

(2)

algorithm or into

W

into

W = QR

W = LU

using the Gauss

using the Householder transforma-

tions. (B)

Boundary value dependent part:

(1)

Computation of

u(tk)

from the system of equations

n

wjku(tk) = (zj),

kIl

Since

W = LU

W = QR, only

or

j

O(n2)

= l(1)n.

operations are re-

quired for this.

Computation of

(2)

u(z)

integrand is a continuous

for

z e G

from (16.13).

2n-periodic function.

The

It seems

natural to use a simple inscribed trapezoid rule with partition points

tj,

j

u(z) =

= 1(1)n:

2n

(16.15)

1u(tk)log1z-E(tk)I.

k= If

z

does not lie in the vicinity of

yields good approximations for For boundary-close

z,

r,

(16.15) actually

u(z).

-loglz - g(t)I

extremely large on a small part of the interval (16.15) is useless.

becomes [0,27T].

Then

The following procedure improves the re-

sults by several decimal places in many cases.

But even this

16. Collocation methods and boundary integral methods

331

approach fails when the distances from the boundary are very small. Let

A(t)

boundary values uc(z) = c +

be that function Then, for

i ° 1.

which results from

u(t)

c e1R,

n

2n

(16.16)

I

k=1

are also approximations to

u(z).

It is best to choose

c

so that u(tR)

whenever a(t)

- ca(tR) = 0

is minimal.

Since the computation of

can proceed independently of the boundary values ,

the effort in (16.15) is about the same as in (16.16). each functional value operations.

one needs

u(z)

0(n)

For

arithmetic

The method is thus economical when only a few

functional values

are to be computed.

u(z)

In the following example, we present some numerical results: ,P(z)

= Re[exp(z)] = exp(x)cos(y)

al(t) = 0.2 cos(t) + 0.3 cos(2t)

-

0.3

E2(t) = 0.7[0.5 sin(t-0.2)+0.2 sin(2t)-0.l sin(4t)] + 0.1.

The region in question is the asymmetrically concave one shown in Figure 16.17.

The approximation

u

was computed on

the rays 1, 2, and 3 leading from the origin to the points E(0), l;(n), and

points.

E(5ii/3).

R

is the distance to the named

Table 16.18 contains the absolute error resulting

from the use of formula (16.15) (without boundary correc-

tion); Table 16.19 gives the corresponding values obtained

332

II.


from formula (16.16) (with boundary correction).

We note

that the method has no definitive convergence order.

FIgure 16.17.

Asymetrically concave region

n

n

1.9E-2 3.3E-3

8.3E-4

2.SE-3

1.9E-7

1.1E-10

96

5.0E-10 2.2E-7

3.3E-6

9.8E-5

2.4E-5

1.1E-6

1.7E-12

4.7E-12

1.3E-12

4.0E-5

4.4E-8

2.6E-7

1.2E-7

2.8E-4

9.6E-4

2.0E-3 3.4E-4

5.1E-4

4.6E-3

7.4E-3 1.3E-2

9.4E-3 7.5E-5

1/128 1/32

2.4E-5

1/8

1.4E-2

1/128

1.SE-4

1/32

Ray 3

2.1E-2

1/8

Ray 2

Absolute error when computing with boundary correction

3.SE-6

4.6E-12

96

2.4E-9

5.4E-7

48

4.0E-4

4.7E-3

4.3E-5

1.6E-4

1.1E-4

24

1/128

1.9E-6

3.9E-3

3.1E-3

12

Ray 1

1/32

TABLE 16.19:

2.4E-12

Absolute error when computing without boundary correction

1/8

R

TABLE 16.18:

6.7E-2

1.4E-4

2.2E-3

1.9E-7

7.0E-6

1.8E-2

3.0E-5

5.5E-7

48

1.0E-6

1.1E-2

S.SE-3

1.9E-1

6.9E-2

2.5E-2

7.0E-3

2.2E-4

1.3E-2

1.2E-4

8.1E-2

4.3E-3

7.SE-4 2.2E-5

9.8E-3

2.8E-1

5.5E-2

2.6E-2

24

1/128

12

1/32

1/8

1/128

1/32

1/8

1/128

1/32

R

1/8

Ray 3

Ray 2

Ray 1

U4 LA

w

PART III. SOLVING SYSTEMS OF EQUATIONS

17.

Iterative methods for solving systems of linear and nonlinear equations When we discretize boundary value problems for linear

(nonlinear) elliptic differential equations, we usually ob-

tain systems of linear (nonlinear) equations with a great many unknowns.

The same holds true for the implicit discreti-

zation of initial boundary value problems for parabolic differential equations.

For all practical purposes, the utility

of such a discretization is highly dependent on the effectiveness of the methods for solving systems of equations. In the case of systems of linear equations, one distinguishes between direct and iterative methods.

Aside from

rounding errors, the direct methods lead to an exact solution in finitely many steps (e.g. Gauss algorithm, Cholesky method, reduction method).

Iterative methods construct a

sequence of approximations, which converge to the exact solution (e.g. total step method, single step method, overrelaxation method).

These are ordinarily much simpler to

program than the direct methods.

334

In addition, rounding errors

17.

335

Iterative methods

play almost no role.

However, in contrast to direct methods

fitted to the problem (e.g. reduction methods), they require so much computing time that their use can only be defended when the demands for precision are quite modest.

When using

direct methods, one must remain alert to the fact that minimally different variants of a method can have entirely different susceptibilities to rounding errors.

We have only iterative methods for solving systems of non-linear equations.

Newton's method (together with a few

variants) occupies a special position. only a few iterations.

It usually requires

At each stage, we have to solve a

system of linear equations.

Experience shows that a quick

direct method for solving the linear system is a. necessary adjunct to Newton's method.

An iterative method for solving

the linear equations arising in a Newton's method is not to be recommended.

It is preferable instead to apply an itera-

tive method directly to the original non-linear system.

The

Newton's method/direct method combination stands to nonlinear'systems as direct methods to linear systems.

However,

the application is limited by the fact that frequently the linear systems arising at the steps of Newton's method are too complicated for the fast direct methods.

This section will serve as an introduction to the general theory of nonlinear iterative methods.

A complete treat-

ment may be found, e.g., in Ortega-Rheinboldt 1970. In the following two sections, we examine overrelaxation methods (SOR) for systems of linear and nonlinear equations.

After that, we consider direct methods. Let

F : G c

to find a zero

1n y 1n

x* e G

of

be a continuous function.

We want

F, i.e. a solution of the equation

336

SOLVING SYSTE'1S OF EQUATIONS

111.

F(x)

lying in

=

(17.1)

0

In functional analysis, one obtains a number of

G.

sufficient conditions for the existence of such a zero. Therefore, we will frequently assume that a zero

x* E G

exists, and that there exists a neighborhood of F

We further demand that

has no other zeros.

in which

x* G

be an

open set.

Iterative methods for determining a zero of

F

are

based on a reformulation of (17.1) as an equivalent fixed point problem, x = T(x),

so that

x*

point of

is a zero of

T.

T(x(v-1)),

=

One expects the sequence if the initial point

proximation to case.

exactly when

F

is a fixed

x*

Then we set up the following iteration: x(")

x*

(17.2)

x*.

{x(v)

x(0)

I

v = 1(1)-. v = 0(1)oo}

(17.3)

to converge to

is a sufficiently close ap-

But this is by no means true in every

In addition to the question of convergence of the

sequence, we naturally must give due consideration to the speed of the convergence, and to the simplicity, or lack thereof, of computing

T.

Before we begin a closer theoreti-

cal examination of these matters, we want to transform Equation (17.1) into the equivalent fixed point problem for a special case which frequently arises in practice. Suppose that the mapping

Example 17.4:

into a sum, F(x) = R(x) + S(x), in which dependent on

x

and

R

can be split

F S

is only "weakly"

is constructively invertible.

By

the latter we mean that there exists an algorithm which is

337

Iterative methods

17.

realizable with respect to computing time, memory storage demand, and rounding error sensitivity, and for which the R(y) = b

equation

neighborhood of R

can be solved for all

-S(x*).

in a certain

b

Such is the case, for example, when

is a linear map given by a nonsingular diagonal matrix or

by a tridiagonal symmetric and positive definite matrix. we set

then equation

T = R- lo(-S)

to the fixed point problem fore

F(x) = 0

x = T(x).

When

is equivalent S, and there-

also, depends only weakly on the point, one can ex-

T

pect the iterative method (17.3) to converge to

x*

ficiently close approximations

o

Definition 1 7 . 5 :

Let

x(0)

T : G c IRn -' 1R n

a fixed point of

x* e G

sequence (17.3) for

x(0)

The fixed point

x*.

an interior point of

x*.

I(T,x*)

of

II

II

T

in

1R'

A point

y e G

x*, if the and converges

G

is called attractive if it is

x*

The iteration (17.3) is

I(T,x*). x*

The mapping

is attractive. a e

is called contracting if there exists an

norm

for suf-

be a mapping and

remains in

= y

called locally convergent if T

of

T, i.e. T(x*) = x*.

belongs to the attractive region

to

If

[0,1)

and a

such that

IIT(x) - T(Y)IIT < allx-YIIT

x,y e G.

0

Every contraction mapping is obviously continuous. Theorem 17.6:

Let

T:G c IRn

-

iRn

be a contraction mapping.

Then it is true that: (1)

T

has at most one fixed point

x* c G.

x*

is

attractive. (2)

x*.

In case

G =]R

,

there is exactly one fixed point

n Its attractive region is all of ]R.

338

SOLVING SYSTEMS OF EQUATIONS

III.

Proof of (1): Since

Let

and

x*

be two fixed points of

y*

is contracting, there is an

T

a e

T.

such that

[0,1)

IIx* Y*IIT = IIT(x*) T(y*)IIT < allx*-y*IIT. It follows that

x* = y*.

We now choose

r eIR+

so small

that the closed ball KT

lies entirely in

r = {y e Itn

I

IIx*-yIIT < r}

It follows for all

G.

z

e KT r

that

IIT(z)-T(x*)IIT < allz-x*IIT < r. Therefore

T

maps the ball x(v)

is defined for

KT r

T(x(v-1)),

and satisfies the inequality

IIx(v)-x*IIT < avllx(0)-x*IIT < avr, Ix(v)

It follows that the sequence to

I

v = 1(1)00.

v = 0(1)00}

converges

x*.

Proof of (2): T

The sequence

v = 1(1)00

=

a KT r

x(0)

into itself.

Let

x(0) e]Rn

is contracting there is an

v = 0(1)00

be chosen arbitrarily. a e

Since

so that for

[0,1)

it is true that

Iix(v+l) -x(v)IIT = IIT(x(v))

-T(x(v-1)

allx(v)-x(v-1)IIT
0

satisfying

I

- J(x*)F'(x*).

there exists a IIY112

6

>

0

so that for

< d, it is true that

IIF(x*+Y)-F(x*)-F' (x*)YII2 =IIF(x*+Y)-F'(x*)Y112 av+1 Then the last lemma implies that

av+1 < a

v+l

-

1

2

_

(cf. Lemma 17.15) we finally

obtain (2v+2v-1)

1 \1(2

nv+l - 2(laa/

nv

< 2- v

nv -

nv-1 < ..

(2-1) v

a

1-a

PV = 2nv/(i+

PV < r12-v(1aa)

The sequence

(10-10

x(v)

no

)

0. 2nv+l/[(1+/&)(1-6d))

P =

pv+l = PV = 0.

nv+1 < 06VnV,

6V = 6v+1'

PV = nv/(1-BdV),

Case 3:

and

x(v)

=

nv + 66VPV = PV. Let Sd)

=

ay

Lemma 17.20 implies that

p > PV+l.

nv+1 i.

wb, all eigenvalues of m have magnitude Proof:

For

w - 1.

We derive the proof from a series of intermediate

conclusions: (i)

All eigenvalues of

B

are real.

If condition

(a) does not hold, then by (b) all the matrices are symmetric and positive definite. $ = D-1/2(D-A)D-1/2

=

Then

D-1/2(R+S)D-1/2

A

and

D

374

III.


is also symmetric and hence has only real eigenvalues. B

and

are similar, B

B

If

(ii)

Since

too has only real eigenvalues.

is an eigenvalue of

p

has the same eigenvalues as For arbitrary

(iii)

and ±,a (L+U) clear for

-p.

(-1)-1U

z =

B.

the matrices

z,w e 1

have the same eigenvaZues. 0

or

w = 0, for then

upper or lower triangular matrix. So now let

z # 0

and

zL + wU

The assertion is

zL + wU

is a strictly

Its eigenvalues are all

w # 0.

zL + wU = Y/'z-w[(z/w)1/2L +

Since

B, then so is

is consistently ordered,

B

-B = -L +

zero.

Since

Then we can rearrange

(z/w)-1/2U].

is consistently ordered, the square-bracketed ex-

B

pression has the same eigenvalues as

L + U.

In view of (ii),

the conclusion follows. (iv)

It is true for arbitrary

z,w,y e

that:

det(yI-zL-wU) = det(yI±I(L+U)). The determinant of a matrix is equal to the product of its eigenvalues. (v)

w e (0,2)

For

and

A c ¢

it is true that:

det((A+w-1)I±w I). It follows from the representation =

(I-wL)-1[(1-w)I+wU]

that

det(AI-5) = det(AI-(I-wL)-1[(l-w)I+wU]) =

det((I-wL)-1(AI-awL-(1-w)I-wU)).

375

Overrelaxation methods for linear systems

18.

it further follows that

det(I-wL) = 1

Since

det(AI-.) = det(AI-AwL-(l-w)I-wU) = det((A+w-1)I-AwL-wU).

This, together with (iv) yields the conclusion. B = p(B) = 0

(vi)

implies that for all

w c (0,2),

Since the determinant of a matrix is the product of its eigenvalues, it follows from (v) that for

p(B) = 0,

n II

(A-Ar) = ()L+w-1)n

r=1

Here the

i = l(1)n, are the eigenvalues of .V.

Ai,

The

conclusion follows immediately. (vii)

Let

w e (0,2), µ e IR

and

A

c 4, A

Further

0.

let (A+W-l)2 Then

p

is an eigenvalue of

value of W.

=

B

Aw2u2.

exactly when

is an eigen-

A

The assertion follows with the aid of (v):

det(AI-.) = det(±wµTI±wTB) _ (wT)ndet(±uI±B) . We are now ready to establish conclusions (1) By (vii), u # 0

Proof of (1):

and only if (a) implies

p2 S2

-

(4):

is an eigenvalue of

is an eigenvalue of .l.

Thus

S2

B

if

= p("5).

< 1, and (b), by Theorem 18.4(2), implies

P(Y1) < 1. Proof of (2):

The conclusion p(.f) > p(. )

follows from

b

considering the graph of the real valued function p(-W), defined in (3), over the interval Remark 18.13).

(0,2)

f(w) _

(cf. also


III.

376

We solve the equation

Proof of (3) and (4): (a+w-1)2

-

given in (vii) for

Aw2p2 = 0

W2112)

A2-2A(1-w+

+ (w-1)2 =

x:

0

2 A

w2µ2 + wu(1-w

= 1-w +

For

w2p2)1/2.

+ 4

2

[wb,2), the element under the radical is non-positive

w c

for all eigenvalues

of

p

B:

W2s2

w2p2 < 1-w +

1-w + 4

< 0.

4

Therefore it is true for all eigenvalues 2 1 .2112) 2 2 2 +

1

w p (1-w +

2

that

of

A

4

2

wp

2

(w-1)

2

= w-1. It follows that p(`.) W We now consider the case too there can exist eigenvalues

w e (0,wb). of

p

B

In this case

for which the ex-

pression inside the above radical is non-positive. corresponding eigenvalues JAl

= w-1.

of S we again have

However, there is at least one eigenvalue of

B

p = $) for which the expression under the radical is

(namely

positive. B.

A

For the

We consider the set of all of these eigenvalues of

The corresponding eigenvalues

are real.

of

A

positive root gives the greater eigenvalue.

The

For

u > 2(lw-11)1/2/w

the function 1-w +

1 _I

grows monotonically with p = a.

W2,12)1/2

w2 p2 + wp(1-w +

It follows that

4

p.

The maximum is thus obtained for


18.

P(yw) = 1-w + 1 w2s2 + ws[1 w +

377

w2s2]1/2, 4

p(.

W)

is an eigenvalue of

also implies that whenever

is a simple eigenvalue of

p( W)

a = p(B)

The monotonicity

by (vii).

.


other eigenvalues of ./

are smaller.

o

In the literature, the matrix

Remark 18.12:

2-cyclic whenever

A

is called

is weakly cyclic of index 2.

B

allows matrices other than the true diagonal of matrix

D, then

B

All of the

B.

depends not only on

particular choice of

If one

for the

A

A, but also on the

Therefore it seemed preferable to

D.

us to impose the hypotheses directly on matrix

a

B.

Conclusion (1) of Young's Theorem means that

Remark 18.13:

the Gauss-Seidel method converges asymptotically twice as fast as the Jacobi method. convergence for

w = wb

ally greater than for

For the SOR method, the speed of

in many important cases is substantiw = 1

(cf. Table 18.20 in Example

18.15). In (3) the course of the function exactly for

w c (0,2).

is described

A consideration of the graph shows

that the function decreases as the variable increases from to

wb.

On the interval Figure 18.14). known.

w + wb

The limit of the derivative as

-

0

is

0

(wb,2), the function increases linearly (see wb

is easily computed when

S = p(B)

is

However that situation arises only in exceptional

cases at the beginning of the iteration.

As a rule, wb

will

be determined approximately in the course of the iteration. We start the iteration with an initial value

X(V) _ Y x(v-l) + 0

wo(D-woR)-lb.

wo a

[l,wb):

378


III.

1

i W wb

1

Figure 18.14.

For a solution

2

Typical behavior of p(.VW)

of the system of equations we have

x*

x* = - x* + W0 (D-w0R)

lb.

0

It follows that x(v)-x*

(x(v-1)-x*)

= W

= W

O

By (4), ao = p(W ) 8 = p(B)

)

is an eigenvalue of

0

one if

)v-1(x(1)-x* 0

W 0,


and a simple This occurs,

B.

by a theorem of Perron-Frobenius (cf. Varga 1962), whenever, e.g., the elements of irreducible.

B

are non-negative and

We now assume that )'0

of W with eigenvector

e.

B

is

is a simple eigenvalue

Then the power method can be

0

used to compute an approximation of For sufficiently large x(v) x* z aye, It follows that

v

p(B).

it holds that:

x(v+l)

x* z X0ave,

x(v+2)-x* = aoave.

18.


379

x(v+2)-x(v+l) z (a0-1)A0ae x(v+I)-x(v) z (ao-1)ave x(v+2)-x(v+1)112

a. o 2

(v+1) -X(V)112

11x

The equation (ao+wo-1)2

aowo62

=

makes it possible to determine an approximate value 82.

Next compute

wb

82

for

from the formula

wb = 2/[1+(1-52)l/2]

and then continue the iteration with The initial value

w0

wb.

must be distinctly less than

wb, for otherwise the values of the eigenvalues of Yw

will 0

be too close together (cf. the formula in 18.11(3)) and the power method described here will converge only very But it is preferable to round up

slowly.

function w < wb

p(.9)

grows more slowly for

(cf. Figure 18.14).

difference

2-1b

wb, since the

w > wb

than for

It is worthwhile to reduce the

by about ten percent.

o

In the following example we compare in an important special case the speed of convergence of the Jacobi, GaussSeidel, and SOR methods for Example 18.15:

w = wb.

Sample Problem.

The five-point discretization

of the problem tu(X,y) = q(x,y), u(x,y) = V+(x,y),

(X,y) C G = (0,1)2 (x,y) c 3G.

380


III.

leads to a linear system of equations with coefficient matrix 1

A=

A

I

I.

A,

e MAT(N2,N2, ]R). I

Here we have (

-4

1

11,

-4

e MAT(N,N, IR) 4

1

N+l = 1/h.

A, as we know from Section 13, are

The eigenvalues of

Let

where

A

avu = -2(2-cos vhn - cos phn),

v,y = 1(1)N.

be partitioned triangularly into

A = D - R - S,

D = -41.

The iteration matrix D-1(R+S) = D-1(D-A)

of the Jacobi method has eigenvalues p(D-1(R+S))

I-D-1A

=

1

+

4

= cos hn = 1- 2 h2n2

Therefore

A

.

+

O(h4).

By Theorem 18.11 (Young) we further obtain

P()

2

=

cos2hn

=

1-h2n2

+ O(h4)

wb = 2/(l+./l-$ ) = 2/(l+sin hn) P(mob )

_ Wb-1 = 1-2h7T + O(h2).

Table 18.16 contains step sizes.

g, p(S), urb, and

p( mob)

for different

18.

381


Spectral radii and

TABLE 18.16:

h

a

)

P(`Sz

wb.

P( W )

Wb

b

1/8

0.92388

0.85355

1.4465

0.44646

1/16

0.98079

0.96194

1.6735

0.67351

1/32

0.99519

0.99039

1.8215

0.82147

1/64

0.99880

0.99759

1.9065

0.90650

1/128

0.99970

0.99940

1.9521

0.95209

1/256

0.99993

0.99985

1.9758

0.97575

Now let

e(v)

=

be the absolute error of the

x(v)-x*

v-th

approximation of an iterative method x(v+l)

Here let Since

Mx(v) + C.

be an arbitrary matrix and let

M

e(v)

=

=

Mve(°)

and

lim

IIMVIIl/v

'V-.W

there is for each

IIMvII

n

.

x* = Mx* + c.

>

0

a

v

0

= P(M), eIN, such that

(P(M)+n)V, V > v

-

1

o

k(V)II < (P(M)+n)" IIe(°)II.

The condition 11e(m)II

< elle(°)II

thus leads to the approximation formula

m=

to

e

log P(M)

(18.17)

which is sufficiently accurate for practical purposes. In summary we obtain the following relations for the iteration numbers of the methods considered above:

382


III.

mJ c lo

Jacobi:

C

-h Tr /2 e

ml Z 1o

Gauss-Seidel (w=1):

(18.18)

-h it

SOR (w=wb

mw

=

1

b

Here the exact formulas for the spectral radii were replaced by the approximations given above.

The Jacobi method thus

requires twice as many iterations as the Gauss-Seidel method in order to obtain the same degree of accuracy.

one frequently requires that

In practice,

Since

1/1000.

log 1/1000 = -6.91, we get ml z

6.91 h-2 = 0.7/h2 Tr

m

wb

ml/mw

z 0.64/h.

Table 18.20 contains

ml, h2ml,

various step sizes.

(18.19)

6.91 h-1 - 1.1/h

mwb, hmwb , and

ml/mwb

for

These values were computed using Formula

(18.17) and exactly computed spectral radii.

One sees that

the approximate formulas (18.18) and (18.19) are also accurate enough. TABLE 18.20:

h

Step sizes for reducing the error to

m1

2

h ml

mw

hmw

b

1/1000.

m1/m b

b

43

0.682

8

1.071

5

178

0.695

17

1.092

10

1/32

715

0.699

35

1.098

20

1/64

2865

0.700

70

1.099

40

1/128

11466

0.700

140

1.099

81

/256

45867

0.700

281

1.099

162

1/8

1/16

19.

Overrelaxation methods for nonlinear systems

383

For each iterative step, the Jacobi and Gauss-Seidel methods require

4N2

The SOR method, in

floating point operations.

contrast, requires

From (18.19) we get

operations.

7N2

as the total number of operations involved (e = 1/1000): Jacobi:

1.4.4N4 z 6N4

Gauss-Seidel (w=1):

0.7.4N4 z 3N4

SOR (w=wb):

1.1.7N3 = 8N3.

The sample problem is particularly suited to a theoretical comparison of the three iterative methods.

Practical experi-

ence demonstrates that these relations do not change signifiHowever, there exist sub-

cantly in more complex situations.

stantially faster direct methods for solving the sample problem (cf. Sections 21, 22).

SOR is primarily recommended,

therefore, for non-rectangular regions, for differential equations with variable coefficients, and for certain nonlinear differential equations. 19.

o

Overrelaxation methods for systems of nonlinear equations In this chapter we extend SOR methods to systems of non-

linear equations.

The main result is a generalization of

Ostrowski's theorem, which assures the global convergence of SOR methods and some variants thereof. In the following we let

G

denote an open subset of

In Definition 19.1: tions.

Let

SOR method for nonlinear systems of equa-

F c C1(G,IRn), and let

an invertible diagonal

D(x).

F

have a Jacobian with

Then we define the SOR method

384


III.

for solving the nonlinear equation

F(x) = 0

by generalizing

the method in Definition 18.1:

X(O) e G x(v-1)-wD-1(x(v-1)/x(v))F(x(v-1)/x(v))

X(V)

t(x(v-1)

=

w e (0,2),

v = l(1)-.

(19.2)

Ortega-Rheinholdt 1970 calls this the single-step SOR Newton method. If

of

T.

has a zero

F

x* E G, then

is a fixed point

x*

This immediately raises the following questions: attractive?

(1)

When is

(2)

How should the relaxation parameter

(3)

Under which conditions is the convergence of the method

x*

w

be chosen?

global, i.e., when does it converge for all initial values (4)

x(0) a G?

To what extent can the substantial task of computing the partial derivatives of

(5)

be avoided?

F

Do there exist similar methods for cases where

F

is

not differentiable?

The first and second questions can be answered immediately with the help of Theorems 17.8 and 17.25. Theorem 19.3:

Let the Jacobian of

F

at the point

x*

be

partitioned triangularly (cf. Definition 18.1) into F'(x*) = D* matrix.

-

R*

- S*, where

D*

is an (invertible) diagonal

Then p(I-w[D*-wR*1-1F'(x*)) < 1,

implies that

x*

is attractive.

19.


Proof:

385

By Theorem 17.25 we have I-[I-w(D*)-1R*]-lw(D*)-1F'(x*)

T'(x*) =

I-w[D*-wR*]-1F'(x*).

=

The conclusion then follows from Theorem 17.8.

o

The SOR method for nonlinear equations has the same convergence properties locally as the SOR method for linear equations.

The matrix - w[D*-wR*]

I

1F'(x*)

indeed corresponds to the matrix ..

of Lemma 18.2.

Thus

the theorems of Ostrowski and Young (Theorems 18.4, 18.11), with respect to local convergence at least, carry over to the nonlinear case.

The speed of convergence corresponds asymp-

totically, i.e. for the linear case.

the optimal

v 4 -, to the rate of convergence for

Subject to the corresponding hypotheses, can be determined as in Remark 18.13.

w

sufficiently accurate initial value

x(O)

If a

is available for

the iteration, the situation is practically the same as for linear systems.

This also holds true for the easily modified

method (cf. Remark 17.26) X(V)

wD-1(x(v-l))F(x(v-1)/x(v)).

x(v-l)

=

-

The following considerations are aimed at a generalization of Ostrowski's theorem.

Here convergence will be estab-

lished independently of Theorem 17.8.

The method (19.2) will be generalized one more time, so that it will no longer be necessary to compute the diagonal of the Jacobian

F'(x).

The hypothesis

"F

differenti-

386


III.

able" can then be replaced by a Lipschitz condition.

Then

questions (4) and (5) will also have a positive answer.

In

an important special case, one even obtains global convergence. Definition 19.4:

A mapping

F e C°(G,)Rn)

gradient mapping if there exists a F(x)T = '(x), x c G.

We write

e

is called a such that

C1(G,IR1)

F = grad

o

In the special case of a simply connected region

G,

the gradient mappings may be characterized with the aid of a well-known theorem of Poincare (cf. Loomis-Steinberg 1968, Ch. 11.5).

Theorem 19.5:

Let

G

be a simply connected region

Then

F

is a gradient mapping if and

Poincare.

and let

F e C1(G,IRn).

only if

F'(x)

is always symmetric.

Our interest here is only in open and convex subsets of

IRn, and these are always simply connected.

then, we always presuppose that

set of

a c (0,1)

Let

4

0: G +]R1

and for all

x,y E G

(1-a)4(y)

- (ax + (1-a)y).

is called, respectively, a

convex function

if

r(x,y,a)

strictly convex function

if

r(x,y,a) > 0,

uniformly convex function

if

r(x,y,a) > ca(l-a)jjx-y,j

for all c

and

let

r(x,y,a) = Then

is an open, convex sub-

G

IRn.

Definition 19.6: all

In the sequel

x,y e G

with

>

x # y, and for all

0,

a e (0,1).

is a positive constant which depends only on

4.

2,

Here 0

19.


387

The following theorem characterizes the convexity properties of

with the aid of the second partial deriva-

tives.

Theorem 19.7:

A function 41 EC2(G,1R1)

is convex, strictly

convex, or uniformly convex, if and only if the matrix of the second partial derivatives of ing inequalities for all

x e G

0

satisfies the follow-

and all nonzero

z e]Rn

respectively,

zTA(x)z > 0

(positive semidefinite)

zTA(x)z >

(positive definite)

0

zTA(x)z > czTz

Here Proof:

(uniformly positive definite in x and z).

depends only on

c > 0

A, not on

x

or

z.

x,y e G, x # y, we define

For

p(t) = r(x,x+t(y-x),a),

t e [0,1).

Then we have P(t) = a41(x)

and

+ (l-a)$(x+t(y-x))

p(O) = 0, p'(0) = 0.

PM =

- 41(x+t(1-a)(y-x))

It follows that

(1

(1-s)p"(s)ds

J

0

PM = (1-a)J (1-s)(y-X) TA(x+s(y-x))(y-x)ds 0

(1-s)(y-x) TA(x+s(1-a)(y-x))(y-x)ds.

(1-a) 2 0

In the second integral, we can make the substitution s = (1-a)s, and then call integrals:

s'

again

A(x)

s, and combine the

III.

388


1

P(1) = j T(s)(Y-x)TA(x+s(Y-x))(Y-x)ds 0

where

Jas (1-a)(1-s)

for

0 < s < 1-a

for

1-a < s < 1.

The mean value theorem for integrals then provides a suitable 6

for which

c (0,1)

a(1-a)(Y-x)TA(x+6(Y-x))(Y-x).

r(x,Y,(x) = P(1) =

2 The conclusion of the theorem now follows easily from Definition 19.6.

o

is only once continuously differentiable, the

c

If

convexity properties can be checked with the aid of the first derivative. Theorem 19.8:

c c C1(G,IR1), F = grad 4, and

Let

p(x,y) = [F(Y)-F(x)]T(Y-x). Then

0

is convex, strictly convex, or uniformly convex, if

and only if

p(x,y)

satisfies the following inequalities,

respectively, p(x,Y) > 0,

p(x,Y) > 0,

p(x,Y) > c*IIY-x112 Here

c* > 0

Proof: t e

depends only on

Again, let

[0,1].

F.

p(t) = r(x,x+t(y-x),a), x,y c G, x # y,

Then we have

p(t) = a1(x) + (1-a)4(x+t(Y-x)) - (x+t(1-a)(Y-x)) and


19.

389

(1

p(1) = r(x,y,a) =

p'(t)dt

J

0 1

[F(x+t(y-x))-F(x+t(1-a)(y-x))] T(y-x)dt.

_ (1-a) 0

It remains to prove that the inequalities in Theorem 19.8 and Definition 19.6 are equivalent.

We content ourselves

with a consideration of the inequalities related to uniform convexity.

Suppose first that always

P(x,Y) > c*IIY-x112. Then it follows that P(x+t(1-a)(Y-x),x+t(Y-x)) > c*a2t2IIY xll2 IF(x+t(y-x))-F(x+t(1-a)(y-x))] T(y-x) > c*atIIY-xll2

r(x,y,a) >

c*a(1-a)IIY-x442. 2

The quantity here.

c

in Definition 19.6 thus corresponds to

1-2c

Now suppose that always

r(x,Y,a) > ca(1-a)IIY-xI12

Then it follows that aO(x)+(1-a)0(Y) > (x+

a)(Y-x))+ca(1-a)I1Y-x12

O(x+(1-(x)(ya-x))-4(x) + callY-xll2. Since this inequality holds for all the limit

a -

a c (0,1), by passing to

1, we obtain

m(Y) fi(x) > F(x)T(Y-x) + clly-x112. Analogously, we naturally also obtain

390


III.

(x)-4(y) > F(Y)T(x-Y) + cIIy-xlI2.

Adding these two inequalities yields 0 > -[F(Y)-F(x))T(Y-x) + 2cjjY-x,12.

0

The following theorem characterizes the solution set F(x) = 0

of the equation

for the case where

is the

F

gradient of a convex map. Theorem 19.9: F = grad 0.

Let

m E C1(G,1R1)

Then:

The level sets

(1)

convex for all

is a zero of

global minimum at If

(3)

one zero (4)

x*

N(y,q) = {x e G

O(x) < yl

I

are

y eIR.

x*

(2)

be convex and let

x*.

in

assumes its

0

The set of all zeros of

is strictly convex, then

0

If

exactly when

F

F

is convex.

F

has at most

G.

is uniformly convex and

0

has exactly one zero

G =IRn, then

F

x*, and the inequality

c*IIx*112 < !IF(0)112. is valid, where Proof of (1):

is the constant from Theorem 19.8.

c*

Let

a c (0,1)

follows from the convexity of

y,x c N(y,1).

and 0

Then it

that

4,(ax+(1-a)y) < x (x)+(1-a)0(Y) < ay+(l-a)y = Y.

Thus

ax + (1-a)y

Proof of (2):

Let

also belongs to x*

N(y,o).

be a zero of

F

and let

x E G,

x # x*, be arbitrary.

By the mean value theorem of differ-

entiation, there is a

A e (0,1)

such that

19.


O(x) _ O(x*)

391

[F(x*+A(x-x*))] T(x-x*).

+

It follows from Theorem 19.8 that p(x*,x*+X(x-x*)) = [F(x*+X(x-x*))] TX(x-x*) > 0.

Thus we obtain cp(x)

Therefore

o(x*) > 0.

-

is a global minimum of

x*

sion is trivial, since the set of all zeros of

particular zero of

If

The reverse conclu-

It follows from (1) that

is convex, for if

F

is a

x*o

F, then

{x* e G

Proof of (3):

is open.

G

4>.

F(x*) = 0} = N(4>(xo),4>).

is a strictly convex function, then in

4>

the above proof we have the stronger inequality p(x*,x+A(x-x*)) =

[F(x*+X(x-x*))] TX(x-x*) > 0.

It follows that 4>(x)

Therefore

> (x*) .

is the only point at which

x*

can assume the

global minimum.

By (3), F

Proof of (4): to show that examine

0

x = bt,

F

has at most one zero.

has at least one zero.

Thus we need

To this end we

along the lines

t e ]R,

b c IRn

fixed with

JjblJ 2 =

1.

There it is true that t

4>(bt)

= (0) + F(0)Tbt +

[F(bs)-F(0)]Tbds.

i

0

392


III.

Since

[F(bs)-F(0)]Tbs > c*s2 it follows that 0(bt)

> 0(0) + F(0)Tbt

c*t2.

+ 2

with

x = bt

For all

Iti

=

11x112

> 211F(0)112/c* ¢(x) > 4(0).

this inequality means that

Therefore

0

as-

sumes its minimum in the ball

{x a ]Rn

1

11x112

0, either

Aj

# 0

In the first case we define

fi (y(J-l)) = 0.

t e [0,1]

YO-l) - tai e(J) y(t) = p(t) = O(Y(t))

This leads to p'(t) = -ai fj(Y(t))

p'(0) = -Aifi (Y(0)) _ -qjfj(Y(0))2
0

and

y(t) c G*,

this hypothesis implies:

2ta.

(Y(O)) -fj Y t)

q]
1/(ajj+6).

ajj + d

ii

ajj

contains the factor

ajj >> 6.

h, therefore,

For small

1/h2.

Every sequence arbitrary

w(0)

w(v+1)

=

w(v) - Q(Aw(v-1)/w(v)

+

H(w(v-1)

converges, if 0

The condition strictive.

< qj

< 2/(ajj+d).

Hz(x,y,u(x,y))
r, set

mj = width of g = kj

and

return to (B). < p, increase

(E)

If

(F)

Choose set

j

j

e {1,...,p}

h = kj.

j

by 1 and repeat (D). so that

mj

is minimal.

Then

412

III.


The first part of the algorithm thus determines two knots and

h

g

and the corresponding level structures R(h) = (M1,M2,...,M5).

R(g) _ (L1,L2,...,Lr), Lemma 20.9:

The following are true:

(1)

s = r.

(2)

h e Lr.

(3)

g e Mr.

(4)

Li c U Mr+l_j = Ai,

i

i = 1(1)r.

j=1 (5)

Mi c U Lr+l-j = Bi,

i = 1(1)r.

j=1

Proof:

For

Conclusion (2) is trivial.

the empty set.

in

Mi

Mi+l

All knots in

(cf. Theorem 20.8(2)).

Mi

s

let

M1 = {h} c Lr.

be

M.

is a subset of

are connected with knots

Lu_1, Lu, and

Bi, we have

Conclusion (5) implies that (cf. Step (D)), s < r.

Now the

Since the elements of

are connected only to elements in since

>

We then prove Conclusion (5) by induction.

By (2), the induction begins with induction step.

i

Lu

Lu+l, and

Mi+l c Bi+l.

s > r.

By construction

This proves Conclusion (1).

From (5)

we obtain M. c B. i

i

r-l i Ul

r-l

M. c

B.

Ul

i

=

Br-1

G-L

L1 C Mr g e Mr.

This is conclusion (3).

The proof of (4) is analogous to the

proof of (5), in view of (3).

13

20.

Band width reduction for sparse matrices

413

The conclusions of the lemma need to be supplemented In most practical applications it turns out

by experience. that:

The depth

(6)

and

R(h)

r

of the two level structures

R(g)

is either the maximal depth which can be achieved

for a level structure of

(G,-)

or a good approximation

thereto. (7)

Li n Mr+l_i, i = 1(1)n, in many cases

The sets

contain most of the elements of

L.

U

Mr+l_i.

The two level

structures thus are very similar, through seldom identical. (8)

The return from step (D) of the algorithm to step

(B) occurs at all in only a few examples, and then no more frequently than once per graph.

This observation naturally

is of great significance for computing times.

In the next part of the algorithm, the two level structures

and

R(g)

R(h)

are used to construct a level

structure S = (Sl,S2,....sr)

of the same depth and smallest possible width. k e G

cess, every knot

is assigned to a level

In the proSi

by one

of the following rules: Rule 1:

If

k c Li, let

Rule 2:

If

k e Mr+l-i, let

For the elements of

k c Si. k c Si.

Li n Mr+l-i, the two rules have the

So in any case, L. n Mr+1_i e Si.

same result.

The set

r

V=G splits into

p

-

U (Li n Mr +1 i)

i=1

connected components,

V1,V2,...,Vp.


III.

414

Unfortunately it is not possible to use one of the Rules 1 and 2 independently of each other for all the elements of

V.

Such an approach would generally not lead to a level structure

S.

But there is

Lemma 20.10:

the same

V

If in each connected component of

rule (either always 1 or always 2) is applied constantly, then

We leave the proof to the

is a level structure.

S

reader.

In the following we use elements of the set K2

the width of

Let

T.

to denote the number of

ITI

K1

be the width of

The second part of the algorithm

R(h).

consists of four separate steps for determining (G)

Compute

Si = L. n Mr+l-i,

and determine

S:

K2, set

and

KI

V.

V

If

i = 1(1)r

is the empty set, this part of

the algorithm is complete (and continue at (K)). wise split Order the (H)

Set v = 1.

(I)

Expand all from

Vv

into connected components

V

so that always

Vi

Si

Si,

to

by rule 1.

IVj+1I

a 7(0) 3 5

z.

P2

3

P 1(0) ,P 2(0) ,P 3 (0) ,P 4(0) ,P 5(0) ,P 6(0) ,P 7(0)

3

initialization: p(.0 )=0, q(0)=

computes

7

x (1) 6

6

{

,

P 2(1)

,2 a (1)

2

,3 P (0)

,a (0) 3

3

,

6

,a S(0) ,a (1) (1) 4 6

5

P 4(2) ,P 5(0) ,P 6(1)

, 0.

4

,

,

P (0) 7

a (0) 7

P7

w

1'

w

2'

w

3'

w

4'

w S

'w61w7

7

,g2 1) ,g 30) ,w4,q 50) ,q 61) ,q 0)

q1(0) ,w2,g3(0) ,w 4,g5(0) ,w6, g7(0)

qi 0)

a 1(0) ,a 2(1) ,a (0) ,a (2) ,a 5(0) ,q 6(1) ,a 7(0) 3 4

P 1(0)

a 1(0)

1

memory content at end of step

Stages of the computation for the reduction phase (upper part) and the solution phase (lower part)

requires

Table 21.11:

Buneman Algorithm

21.

425

The Buneman algorithm described here thus requires twice as many memory locations as there are discretization points.

There exists a modification, (see Buzbee-Golub-

Nielson 1970), which uses the same number of locations.

How-

ever, the modification requires more extensive computation. We next determine the number of arithmetic operations in the Buneman algorithm for the case of a square p = n.

G, i.e.

Solving a single tridiagonal system requires

6n-5

operations when a specialized Gaussian elimination is used. For the reduction phase, noting that p = n =

2k+l

k+l = log2(n+1),

1,

-

we find the number of operations to be k

[6n+2r 1(6n-5)]

I

I

r=l j eMr where 2r(2r)2k+l-2r}

Mr =

{j

2k+l-r

and cardinality(Mr) = k

k+1 r

2(2 1)[6n+2

=

-

r -l

1.

Therefore,

(6n-5)] = 3n1og2(n+l)

r=1

+ 0[n log2(n+l)].

Similarly, for the solution phase we have k

J_ [3n+2r(6n-5)]

I

r=0 j eMr where Mr =

{i

and cardinality(:Nr) = 2k k 1

r=0

2k r[3n+2r(6n-5)]

=

r

=

2r(2r+l)2k+1-2r} Therefore,

3n21og2(n+l)+3n2+0[n log2(n+l)].

426


III.

Altogether, we get that the number of operations in the Buneman algorithm is 6n21og2(n+l) + 3n2 + O[n log2(n+l)].

At the beginning of the method, it is necessary to compute values of cosine, but with the use of the appropriate recursion formula, this requires only

0(n)

operations, and thus

can be ignored.

Finally, Appendix 6 contains a FORTRAN program for the Buneman algorithm. 22.

The Schroder-Trottenberg reduction method We begin by way of introduction with the ordinary dif-

ferential equation

-u"(x) = q(x). The standard discretization is 2u(x)-u(x+h)-u(x-h)

= q(x)

h2

for which there is the alternative notation (2I-Th-ThI)u(x) Here

I

=

denotes the identity and

h2q(x).

Th

the translation opera-

tor defined by Thu(x) = u(x+h).

Multiplying equation (22.1) by (2I+Th+Th1)

yields (2I-Th-Th2)u(x)

=

(22.1)

h2(21+Th+Th1)q(x)

The Schroder-Trottenberg reduction method

22.

427

Set

q0(x) = q(x),

ql(x) = (21+Th+Th1)go(x)

and use the relations Th = Tjh,

ThJ

=

This simple notational change leads to the simply reduced equation (2I-T2h-T-h1 )u(x)

=

h2g1(x).

This process can be repeated arbitrarily often.

The m-fold

reduced equations are (2I-Tkh-Tkh)u(x) = h2gm(x)

where

k = 2m

(22.2)

and

qm(x) _ (21+TRh+Tth)gm-1(x),

R =

2m-1

The boundary value problem -u"(x) = q(x),

u(O) = A,

can now be solved as follows. into

2n

u(l) = B

Divide the interval

subintervals of length

h = 1/2n.

Then compute

sequentially ql(x)

at the points

2h, 4h, ..., 1-2h

q2(x)

at the points

4h, 8h, ..., 1-4h

qn-1(x) The

at the point

2n-lh = 1/2.

(n-l)-fold reduced equation

[0,1]

428


III.

(2I-T n 1 -T n-1 )u(1/2) = h2gn 1(1/2)

h

2

to be determined immediately, since the

u(1/2)

allows

values of

u

h

2

are known at the endpoints

and

0

Simi-

1.

larly, we immediately obtain

u(1/4)

(n-2)-fold reduced equation.

Continuing with the method

and

u(3/4)

from the

described leads successively to the functional values of at the lattice points

jh,

u

= 1(1)2n-1.

j

In the following we generalize the one dimensional reduction method to differential or difference equations with constant coefficients. Definition 22.3:

Gh = {vh

Let

I

v e a}, h > 0.

Then we call

r

av Th

Sh =

av c R

v=-r

a one-dimensional difference star on the lattice called symmetric if a2v =

odd (even) if

a v = av 0

for

v = 1(1)r.

(a2v+1 = 0) for all

-r < 2v < r

(-r < 2v+l < r).

the sum for

Sh

v

Gh.

Sh

Sh

is

is called

with

In Schroder-Trottenberg 1973,

is simply abbreviated to [a-ra-r+l .....* ar-lar]h.

o

Each difference star obviously can be split into a sum

Sh = Ph + Qh, where Since

Ph T

2hv

is an even difference star, and

Qh

is an odd one.

v

= T 2h, the even part can also be regarded as the

difference star of the lattice

G2h = {20

I

v e22}.

The reduction step now looks like this:


22.

Shu(x) = (Ph+Qh)u(x)

=

429

h2q(x)

(Ph-Qh)(Ph+Qh)u(x) = (Ph-Qh)u(x) = h2(Ph-Qh)q(x) Ph

Since

obviously is an even difference star, it can

Q2

-

be represented as a difference star on the lattice denote it by

We

G2h.

We further define

R2h.

ql(x) = (Ph-Qh)q(x) Thus one obtains the simply reduced equation R2hu(x) = h2g1(x).

In the special case Sh

1Th1 + a o Th + a1Th

one obtains 2 o

2

Ph = aoTh 2

Q2 = a2 Th2 + 2a-la1Th + a T R2h =

2-T-1 2h +

2_

2

- aIT2h.

The reduction process can now be carried on in analogy with (22.2).

In general, the difference star will change from

step to step.

The number of summands, however, remains con-

stant.

The reduction method described also surpasses the Gauss algorithm in one dimension in numerical stability. main concern however is with two dimensional problems.

Our

We

begin once again with a simple example.

The standard discretization of the differential equation

430


III.

-Au(x) = q(x),

x c IR 2

is

4u(x)-u(x+he1)-u(x-hel)-u(x+he2)-u(x-he2) = h2q(x).

The translation operators Tv,hu(x) = u(x+hev)

lead to the notation (4I-Tl,h-T- 1h-T2,h-T-1 )u(x) = h2q(x) I, 2,h

Multiplying the equation by (41+T1,h+T11h+TZ

T_

1

yields T_

{16I

1

T2,h

-

-

T12h-T2

h2(4I+Tl,h+T11h+T2

h-T2_

h+T21h)q(x)

We set

ql(x) = (41+Tl,h+T11h+T2,h+T21h)q(x)

and obtain the simply reduced equations {121-2(T

l,hT2,h+Tl,hT2,h+TllhT2,h+Tl,hT2,h)

h+T12h+T2 h+T22h))u(x)

=

h2g1(x),

(T121

In contrast to the reduction process for a one dimensional difference equation, here the number of summands has increased from

5

to

spread farther apart.

9.

The related lattice points have The new difference star is an even

polynomial in the four translation operators

Tl,h' Tllh'


22.

®

431

0 8

Figure 22.4:

Related lattice points after one reduction step.

T2 h,

and

The related lattice points are shown in

TZlh.

Figure 22.4 with an "x".

Such an even polynomial can be

rewritten as a polynomial in two 'larger' translation operators.

Let e3 = (el + e2)/./-2-

e 4 = (e2

(22.5)

- el)/17

be the unit vectors rotated by

it/4

counterclockwise, and

let

k = h"T T3,ku(x) = u(x+ke3) = u(x+hel+he2) T4 ku(x) = u(x+ke4) = u(x+he2-hel). Then we have T3,k =

Tl,h T2,h -1

-1

-1

(22.6)

-1

Tl,h T2,h.

The simply reduced equation therefore can be written in the following form:

432


III.

{12I-2(T3,k+T4,k+T31k+T41k)-(T3,kT4'k+T3'kT-1k+T-1kT4,k + T3IkT41k)u(x) = h2g1(x).

The difference star on the left side once again can be split into an even part, 121

(T3,kT4,k+T3,kT41k+T31kT4,k+T31kT41k)

and an odd part,

2(T+TT 3,k 4,k+ and reduced anew.

3,1k+T- 1

4,k)

One then obtains a polynomial in the

translation operators

T1,2h, T1,2h' T2,2h, T2,2h.

Thus, beginning with a polynomial in -1

-1

Tl,h' Tl,h' T2,h' T2,h we have obtained, after one reduction step, a polynomial in

-1

T3,h/' T3,hr' T4,h 2

-1 T4,hV2

and after two reduction steps, a polynomial in -1

1

T1,2h' T1,2h' T2,2h' T2,2h In particular, this means that the and

e2 -> e4

r/4

rotation

e1

e3

has been undone after the second reduction.

This process as described can be repeated arbitrarily often. The amount of computation required, however, grows substantially with the repetitions.

For this reason we have not

carried the second reduction step through explicitly. We now discuss the general case of two-dimensional reduction methods for differential and difference equations


22.

with constant coefficients.

433

We preface this with some gen-

eral results on polynomials and difference stars. Definition 22.7: p

variables

be a real polynomial in

P(xl,...,xp)

Let

xi,...,xp.

is called even if

P

P(-x1,...,-xp) = P(xl,...,xp and

is called odd if

P

P(-xl,...,-xp) _ -P(xl,...,xp It is obvious that the product of two even or of two odd polynomials is even, and that the product of an even polyFurther, every polynomial can

nomial with an odd one is odd.

be written as the sum of one even and one odd polynomial. Lemma 22.8:

Let

P(xl,x2,x3,x4)

there exist polynomials

and

P

P(x,x 1,y,Y 1) = P(xy,(xy) P(xY,(xY)

.,-

i+j

l,yx l,xy 1)

=

P

Then

with the properties:

l,yx l,xy 1)

P(x2,x 2,y2,y-2).

We have

Proof:

Since

be an even polynomial.

P

--1 --

-

S

iE7l

aijx iY J

aij

,

E

pl

is even, we need only sum over those i,j For such

even.

x1Y3

---11

=

i,j

(xy)r(yx-1)s,

E 7l

with

we have r = (i+j)/2,

s

= (j-i)/2.

We obtain P(x,x-1,Y,Y-1)

I

r,sE ZZ

r+s(xY)r(Yx-1)s

ar-s ,

and therefore, the first part of the conclusion.

Similarly,

434


III.

we obtain 1)

l,yx-l,xy

P(xy,(xy)

=

a..(xy)1(yx

X

1)j

i.,jE ZZ

(xy)1(yx-1 ) j

=

P(xy,(xy) l,yx

x2ry2s, l,xy 1)

ar+s,s

rx2ry2s 0

r,sE ZZ h > 0, define the lattice

For fixed

Definition 22.9:

s = (i+j)/2

r = (i-j)/2,

Gv

by

{xe7R2Ix=2v/2h(iel+je2); i,jeZZ}

when v is even

{xe ]R2Ix=2v/2h(ie3+je4); i,jeZl}

when v is odd.

Gv = .

A difference star Ti

k,

on the lattice

Sv

Gv

T2-1k

for even

T4-1k

for odd

Tllk, T2 k,

is a polynomial in v

or in

T3 k, T31k, T4 k, Here

k = 2v/2h.

Sv

is called even (odd) if the correspond-

ing polynomial is even (odd). Go = Gl = G2 D ...

p

O

.

0

It follows that

Figure 22.10 shows

@

p

Go, G1, G2, and

Go

:

G1

:

G2

:

G3

Figure 22.10:

v.

.

0 X

D

A representation of lattices Go, Gl, G2 and G3.

G3.

22.

The Schr6der-Trottenberg reduction method

Theorem 22.11: Gv.

be partitioned into a sum

Sv

Let

S

where

be a difference star on the lattice

Sv

Let

v = Pv

Qv

is an even difference star and

PV

435

ference star.

Qv

is an odd dif-

Then

Sv+1 = RV - Qv)Sv may be regarded as a difference star on the coarser lattice Gv+l c Gv. Proof:

The difference star S

is even since

P2

v+l

2

= P2v

and

- Qv

are even.

Q2

Now let

By Lemma 22.8, there exists a polynomial Sv+l(Tl,k,T-

1

k,T2,k,T-

Sv+l

v

be even.

with

1

2, k)

l,

1,T2,kT1'k,Tl,kT2,k). Sv+1(Tl,kT2,k,(Tl,kT2,k) Since

1

T l,k

one obtains, where

_ T4,k,

m = k/ = 2(v+l)/2h,

Sv+l(Tl,k,Tl1k,T2,k,T2 k) = 9v+l(T3,m,T3,mT4,m,T4,m).

The right side of this equation is a difference star on the lattice

Gv+l.

For odd

taking note of (22.6):

v, one obtains, correspondingly and


III.

436

-1

-1

Sv+l(Tl,RT2,L,(Tl,kT2,k)

-1

-1

-1

,T1,kT2,k,Tl,kT2,k)

Sv+1(T1,k,T1 2 k,T2,k,T2,2) = Sv+l(Tl,m,Tl,m,T2,m,T2,m) 2

k = k//i,

m = 2k = k/ = 2(v+l)/2h.

a

We may summarize these results as follows: Reduction step for equations with constant coefficients: initial equation:

Svu(x) = h2gv(x),

partition:

Sv = pv + Qv,

compute:

qv+l(x) _ (PV-Qv)gv(x),

x E Gv+l

reduced equation:

Sv+lu(x) = h2gv+1(x),

x e Gv+1'

Pv

x E Gv even, Q.

odd

The reduction process can be repeated arbitrarily often. The difference stars then depend only on the differential equation and not on its right side.

Thus, for a particular

type of differential equation, the stars can be computed once for all time and then stored.

Our representation of the difference stars becomes rather involved for large

f

v.

aiiT iI.

Instead of

k . Tj2.k

for

v

even

for

v

odd

Sv =

i

j

1i T 3,k T 4,k i,3EZZ a

k = 2v/2h

we can use the Schroder-Trottenberg abbreviation and write

22.


-1,1

437

......

a0,1

a1,1

a0,-1

al,-1 ......

Sv

a-1,-1

V

With the differential equation

-Au(x) = q(x), the standard discretization leads to the star

So =

0

-1

0

-1

4

-1

0

-1

0 0

The first three reductions are:

S1 =

-1

12

-2

-1

-2

-1 1

1

0

1

0

0

0

-2

-32

-2

0

1

-32

132

-32

1

0

-2

-32

-2

0

0

1

0

0

S3 =

-2

-2

0

S2 =

r

-1

0J 2

l

-4

-4

-752

-2584

-752

6

-2584

13348

-2584

6

-4

-752

-2584

-752

-4

1

-4

6

-4

The even components

6

PV

-4

of the stars

1 "1 -4

1J3 SV

are:

438


III.

PO =

P2

0

0

0

0

4

0

0

0

0

0

0

0

1

0

-2

0

-2

0

0

132

0

1

0

-2

0

-2

0

0

0

1

0

0

0

1

6 0

1

0

-752 0

-752 0

6 0

13348 0

6

-1

0

-1

0

12

0

-1

0

-1

1

0

0

0 3

,

1

1

P

P1

-752

2

0

0

6

-752

0

0

1

3

When the reduction method is applied to boundary value problems for elliptic differential equations, a difficulty arises in that the formulas are valid only on a restricted region whenever the lattice point under consideration is sufficiently far from the boundary.

In certain special cases, however, one

is able to evade this constraint. Let the unit square G = (0,1) x (0,1) be given.

Let the boundary conditions either be periodic, i.e. u(O,x2) = u(1,x2) u(x1,0) = u(xl,l) x1,x2 e [0,1] alu(0,x2) = a1u(1,x2) a2u(xl,o) = a2u(x1,l),

or homogeneous, i.e.

u(xl,x2) =

0

for

(xl,x2) e

DG.

Let

the difference star approximating the differential equation be a symmetric nine point formula:

22.


SO

FY

9

3

a

Y

439

Y S

Y

10'

The extension of the domain of definition of the difference equation S0u(x) = q0(x) to

x E Go

is accomplished in these special cases by a con-

tinuation of the lattice function

to all of

u(x)

the case of periodic boundary conditions, u(x)

are continued periodically on boundary conditions, u(x)

and

G0.

and

Go.

In

q0(x)

In the case of homogeneous

q0(x)

are continued anti-

symmetrically with respect to the boundaries of the unit square:

u(-xl,x2) _ -u(xl,x2),

g0(-xl,x2) = -g0(x1,x2)

u(x1,-x2) = -u(xl,x2),

go(x1,-x2) = -g0(x1,x2)

u(l+xl,x2) _ -u(1-xl)x2), q0(1+x1,x2) = -q0(l-xl,x2) u(xl,l+x2) = -u(xl,l-x2), 90(x1,1+x2) = -go(xl,l-x2) g0(x1,x2) =

0

for

(xl,x2) e @G.

This leads to doubly periodic functions.

Figure 22.12 shows

their relationship.

Figure 22.12:

Antisymmetric continuation.

The homogeneous boundary conditions and the symmetry of the difference star assure the validity of the extended difference equations at the boundary points of

G, and therefore,

440


III.

on all of

An analogous extension of the exact solution

Go.

of the differential equation, however, is not normally possible, since the resulting function will not be twice differentiable.

We present an example in which we carry out the reduction method in the case of a homogeneous boundary condition and for

h = 1/n

and

After three reduction steps,

n = 4.

we obtain a star, S3, which is a polynomial in the translation operators 1 -1 T3,k,T3,k,T4,k,T4,k,

k=

2

3/2

h = 2hI.

The corresponding difference equation holds on the lattice G3

(cf. Figure 22.10).

Because of the special nature of the

u, it only remains to satisfy the equation

extension of

S3u(1/2,1/2) = 83(1/2,1/2).

By appealing to periodicity and symmetry, we can determine all summands from

u(1/2,1/2), u(0,0), u(1,0), u(0,1), and

But the values of u(1/2,1/2)

u

on the boundary are zero.

can be computed immediately.

sulting from the difference star

S2

u(1,1).

Thus

The equations re-

on the lattice

longer contain new unknowns (cf. Figure 22.10).

G2

no

The equations

S1u(1/4,1/4) = gl(1/4,1/4),

Slu(3/4,1/4) = gl(3/4,1/4),

S1u(1/4,3/4) = g1(1/4,3/4),

Slu(3/4,3/4) = g1(3/4,3/4),

for the values of

u

at the lattice points

(3/4,1/4), (1/4,3/4), and

(3/4,3/4)

otherwise only the boundary values of determined value

u(1/2,1/2)

(1/4,1/4),

are still coupled; u

and the by now

are involved.

Thus the

4 x 4

The Schroder-Trottenberz reduction method

22.

system can be solved.

441

As it is strictly diagonally dominant,

it can be solved, for example, in a few steps (about 3 to 5) with SOR.

All remaining unknowns are then determined from

S0u(1/2,1/4) = q0(1/2,1/4), S0u(1/4,1/2) = q0(1/4,1/2), S0u(3/4,1/2) = 80(3/4,1/2), S0u(1/2,3/4) = g0(1/2,3/4).

In all cases arising in practice, this system too is strictly diagonally dominant.

The method of solution described can

generally always be carried out when say

n = 2m, m e 1N, h = 1/n.

n

is a power of

2,

The (2m-l)-times reduction

equation S2m-lu(1/2,1/2)

= q2m-1(1/2,1/2) u(1/2,1/2).

is then simply an equation for

The values of

u

at the remaining lattice points then follow one after another from strictly diagonally dominant systems of equations. By looking at the difference stars

S1, S2, S3

formed

from -1

0

-1

4

-1

0

-1

0

0

S

0

=

one can see that the number of coefficients differing from In addition, the coefficients differ

zero increases rapidly.

greatly in order of magnitude.

This phenomenon generally can

be observed with all following

SV, and it is independent of

the initial star in all practical cases.

As a result, one

typically does not work with a complete difference star SV, but rather with an appropriately truncated star. a truncation parameter

a

Thus, after

has been specified, all coeffici-

442


III.

of

ents

with

SV

IaijI

< a1aool

are replaced by

aiJ

zeros.

For sufficiently small

this has no great influ-

a

ence on the accuracy of the computed approximation values of 10-g

u.

As one example, the choice of

for the case of

a =

the initial star

S

=

0

0

1

1

4

-1

0

-1

0

with

leads to the discarding of all coefficients a1J

Iii

+

Iii

> 4,

lii

>

3,

Iii

> 3.

In conclusion, we would like to compare the number of operations required for the Gauss-Seidel method (single-step method), the SOR interation, the Buneman algorithm, and the reduction method.

We arrived at Table 22.13 by restricting

ourselves to the Poisson equation on the unit square (model problem) together with the classic five point difference formula.

All lower order terms are discarded.

N2

denotes the

number of interior lattice points, respectively the number of unknowns in the system.

The computational effort for the

iterative method depends additionally on the factor which the initial error is to be reduced.

e

by

For the reduction

method, we assume that all stars are truncated by

o = 10

8.

More exact comparisons are undertaken in Dorr 1970 and Schroder-Trottenberg 1976. marks on computing times.

Appendices 4 and 6 contain reIn looking at this comparison, note

that iterative methods are also applicable to more general problems.

22.


Gauss-Seidel

443

2 N4Ilog ej Tr

SOR

21

N3Ilog e

Buneman

[61og2(N+1)+3]N2

Reduction

36 N2

Table 22.13:

The number of operations for the model problem

APPENDICES: FORTRAN PROGRAMS

Appendix 0:

Introduction.

The path from the mathematical formulation of an algorithm to its realization as an effective program is often difficult.

We want to illustrate this propostion with six

typical examples from our field.

The selections are intended

to indicate the multiplicity of methods and to provide the reader with some insight into the technical details involved. Each appendix emphasizes a different perspective:

computa-

tion of characteristics (Appendix 1), problems in nonlinear implicit difference methods (Appendix 2), storage for more than two independent variables (Appendix 3), description of arbitrary regions (Appendix 4), graph theoretical aids (Appendix 5), and a comprehensive program for a fast direct method (Appendix 6).

Some especially difficult questions,

such as step size control, cannot be discussed here.

As an aid to readability we have divided each problem into a greater number of subroutines than is usual.

This is

an approach we generally recommend, since it greatly simplifies the development and debugging of programs. 444

Those who

Appendix 0:

Introduction

445

are intent on saving computing time can always create a less structured formulation afterwards, since it will only be necessary to integrate the smaller subroutines.

However, with

each modification, one should start anew with the highly structured original program.

An alternative approach to re-

duced computing time is to rewrite frequently called subroutines in assembly language.

This will not affect the

readability of the total program so long as programs equivalent in content are available in a higher language.

The choice of FORTRAN as the programming language was a hard one for us, since we prefer to use PL/l or PASCAL. However, FORTRAN is still the most widespread language in the technical and scientific domain.

The appropriate compiler

is resident in practically all installations.

Further,

FORTRAN programs generally run much faster than programs in the more readable languages, which is a fact of considerable significance in the solution of partial differential equations.

The programs presented here were debugged on the CDCCYBER 76 installation at the University of Koln and in part on the IBM 370/168 installed at the nuclear research station Julich GmbH.

They should run on other installations without

any great changes.

We have been careful with all nested loops, to avoid any unnecessary interchange of variables in machines with virtual memory or buffered core memory. for example, the loop DO 10 I = 1,100

DO 10 J = 1,100 10

A(J,I) =

0

In such installations,

APPENDICES

446

is substantially faster than DO 10 I = 1,100

DO 10 J = 1,100

10

A(I,J) = 0.

This is so because when FORTRAN is used, the elements of

A

appear in memory in the following order:

A(1,l), A(2,1), ..., A(100,1), A(1,2), A(2,2),

..

For most other programming languages, the order is the contrary one:

A(l,l), A(1,2), ..., A(1,100), A(2,1), A(2,2),

...

.

For this reason, a translation of our programs into ALGOL, PASCAL, or PL/l requires an interchange of all indices. There is no measure of computing time which is independent of the machine or the compiler.

If one measures the

time for very similar programs running on different installations, one finds quite substantial differences.

We have ob-

served differences with ratios as great as 1:3, without any plausible explanations to account for this.

It is often a

pure coincidence when the given compiler produces the optimal translation for the most important loops..

Therefore, we use

the number of equally weighted floating point operations as a measure of computing time in these appendices.

This count

is more or less on target only with the large installations. On the smaller machines, multiplication and division consume substantially more time than addition and subtraction.

Appendix 1:

Method of Massau

Appendix 1:

Method of Massau

447

This method is described in detail in Section 3.

It

deals with an initial value problem for a quasilinear hyperbolic system of two equations on a region

where

in

G

uy = A(x,y,u)ux + g(x,y)u),

(x,y)

e G

u(x,0) _ ip(x,0),

(x,0)

e G

A e CI(G x G, MAT(2,2,]R)), and

is an arbitrary subset of

at the equidistant points belong to

g e C1(G x G,IR2).

1R2.

(x) _ (i1(x),Iy2(x))

The initial values

1R2:

are given

(xi,0), insofar as these points

G:

+ 2h(i-nl),

xi = xn

i = nl(1)n2.

1

The lattice points in this interval which do not belong to G

must be marked by

B(I) = .FALSE..

Throughout the complete computation, U(*,I)

contains

the following four components: ul(xi,yi),

u2(xi,yi),

xi,

yi.

The corresponding characteristic coordinates (cf. Figure 3.10) are:

SIGMA = SIGMAO + 2hi,

TAU.

At the start, the COMMON block MASS has to be filled:

Ni = n1 N2 = n2

H2 = 2h SIGMAO = xn

-

1

TAU = 0.

2hn1

APPENDICES

448

For those

for which

i

B(I) =

belongs to

(xi,0)

G, we set

TRUE.

U(1,I) = 01(xi) U(2,I) = 02(xi)

U(3,I) = xi U(4,I) = 0,

and otherwise, simply set B(I) =

FALSE.

Each time MASSAU is called, a new level of the characteristic lattice (TAU = h, TAU = 2h, ...) is computed. also alters N1, N2, and SIGMAO.

The program

The number of lattice points

in each level reduces each time by at least one. N2 < Ni.

At the end,

The results for a level can be printed out between

two calls of MASSAU.

To describe

A

and

g, it is necessary in each con-

crete case to write two subroutines, MATRIX and QUELL. initial parameter is a vector u1(x,Y),

The

of length 4, with components

U

u2(x,Y),

x, Y

Program MATRIX sets one more logical variable, L, as follows: L =

tion

FALSE. if G x G

lies outside the common domain of defini-

U

of

A

and

g; L =

TRUE. otherwise.

To determine the eigenvalues and eigenvectors of MASSAU calls the subroutine EIGEN.

A,

Both programs contain a

number of checks: (1)

Are

(2)

Are the eigenvalues of

(3)

Is the orientation of the (x,y)-coordinate system and of the

A

and

g

defined? A

(o,2)-system the same?

real and distinct?

Appendix 1:

Method of Massau

449

The lattice point under consideration is deleted as necessary B(I) = .FALSE.).

(by setting

Consider the example

2u2

u1

A=

,

g = 0, * =

10exsin(2lrx) ,

ul

u2/2

G = (0,2) x]R.

1

Figures 1 and 2 show the characteristic net in the system and the (x,y)-system.

For

(o,T)-

2h = 1/32, we have the

sector

0.656250 < o < 1.406250 0.312500 < T < 0.703125

20th to 45th level.

In the (x,y)-coordinate system, different characteristics of the type

a + T = constant will intersect each other, so in

this region the computation cannot be carried out.

If one

drops the above checks, one obtains results for the complete In the region where the solution

"determinancy triangle".

exists, there cannot be any intersection of characteristics in the same family.

The method of Massau is particularly well suited to global extrapolation, as shown with the following example: 0

A =

0

1

2 2

1-u2 1

2u1u

,

g =

4ule

2x

G = (0,1) x IR. The exact solution is u(x,y) = 2ex cos y

The corresponding program and the subroutine MATRIX and QUELL are listed below.

0.35

0.40

0. 50

0 .60

0 .70

0.75

0.80

0.7

0.8

Figure 1:

0'.9

1 .0

1.1

Characteristic net in the (x,y)-coordinate system

1.2

1.3

452

APPENDICES

Table 3 contains the results for

(a,T) = (1/2,1/2).

The discrete solution has an asymptotic series of the form T0(x,Y) + T1(x,Y)h + T2(x,Y)h2 +

The results after the first and second extrapolation are found in Tables 4 and S.

The errors

Du1

and

Au2

in all

three tables were computed from the values in a column: Dul = 2exsin y - ul,

Au2 = 2excos y - u2.

H2

1/32

1/64

x

1.551750

1.582640

1.598355

1.606317

y

1.122827

1.138263

1.147540

1.152624

Ul

8.804323

9.004039

9.104452

9.154850

U2

3.361230

3.664041

3.838782

3.932681

AU1 -0.296282

-0.165040

-0.087380

-0.044999

0.727335

0.416843

0.223264

0.115574

AU2

Table 3:

1/128

1/256

Results for (o,T) = (1/2,1/2).

1.614279 1.157708

1.613530 1.153699

1.614070

y Ul

9.203755

9.204865

U2

3.966852

4.013523

9.205248 4.026580

tU1

-0.023579

-0.007085

-0.001948

AU2

0.100843

0.027710

0.007299

x

Table 4:

1.156817

Results after the first extrapolation

Appendix 1:

Method of Nassau

453

x

1,614250

1.614349

y

1.157856

1.158005

U1

9.205235

9.205376

U2

4.029080

4.030932

LU1

-0.001605

-0.000234

AU2

0.003320

0.000496

Table 5:

Results after the second extrapolation

SUBROUTINE MASSAU C

C C C C

C C C C C

VARIABLES OF THE COMMON BLOCK U(1,I) AND U(2,I) COMPONENTS OF U, U(3,I)=X, U(4,I)=Y, WHERE MI.LE.I.LE.N2. B(I)=.FALSE. MEANS THAT THE POINT DOES NOT BELONG TO THE GRID. THE CHARACTERISTIC COORDINATES OF THE POINT (U(3,I),U(4,I)) ARE (SIGMAO+I*H2,TAU). THE BLOCK MUST BE INITIALIZED BY THE MAIN PROGRAMME.

C

REAL U(4,500),SIGMAO,TAU,H2 INTEGER N1,N2 LOGICAL B(500) COMMON /MASS/ U,SIGMAO,TAU,H2,N1,N2,B C

REAL E1(2,2),E2(2,2),LAMBI(2),LAMB2(2),G1(2),G2(2) REAL CI,C2,D,XO,X1,X2,YO,Y1,Y2,RD REAL DXOI,DX21,DY01,DY02,DY21,UI1,U12 INTEGER N3,N4,I LOGICAL L,LL C

N3=N2-N1 10 IF(N3.LE.0) RETURN IF(B(N1)) GOTO 20 11 N1=N1+1 N3=N3-1 GOTO 10 20 LL=.FALSE. N4=N2-1 C C

BEGINNING OF THE MAIN LOOP

C

DO 100 I=N1,N4 IF(.NOT.B(I+1)) GOTO 90 IF(LL) GOTO 30 IF(.NOT.B(I)) GOTO 90 CALL EIGEN(U(1,I),E1,LAMBI,L) IF(.NOT.L) GOTO 90 CALL OUELL(U(1,I),G1)

APPENDICES

454

30 CALL EIGEN(U(1,I+1),E2,LAMB2,L) IF(.NOT.L) GOTO 90 CALL QUELL(U(1,I+1),G2) C C C

SOLUTION OF THE FOLLOWING EQUATIONS (XO-X1)+LAMBI(1)*(YO-Y1)=O (XO-X1)+LAMB2(2)*(YO-Y1)=(X2-X1)+LAMB2(2)*(Y2-Y1)

C

C

CI=LAMBI (I) C2=LAI4B2 (2) D=C2-C1 IF(D.LT.1.E-6*AMAX1(ABS(C1),ABS(C2))) GOTO 80 X1=U(3,I) X2=U(3,I+1)

Y1=U (4, I) Y2=U(4,I+1) DX21=X2-X1 DY21=Y2-Y1 RD=(DX21+C2*DY21)/D DXOI=-C1*RD DYOI=RD XO=XI+DXOI Y0=Y1+DYO1 DYOZ=YO-Y2 C

C C C

CHECK WHETHER THE TRANSFORMATION FROM (SIGMA,TAU) TO (X,Y) IS POSSIBLE IF((DX21*DYOI-DXOI*DYZI).LE.O.) GOTO 80 SOLUTION OF THE FOLLOWING EQUATIONS E1(1,1)*(U(1,I)-U11)+E1(1,2)*(U(2,I)-U12)= DYOI*(El(1,1)*G1(1)+E1(1,2)*G1(2)) E2(2,1)*(U(1,I)-U11)+E2(2,2)*(U(2,I)-U12)= E2(2,1)*(DY02*G2(1)+U21-U11)+E2(2,2)*(DYOZ*62(2)+U22-U12) U11=OLD U12=OLD U21=OLD U22=OLD

VALUE VALUE VALUE VALUE

OF OF OF OF

U(1,I) U(2,I) U(1,I+1) U(2,I+1)

D=E1(1,1)*E2(2,2)-E2(2,1)*E1(1,2) IF(ABS(D).LT.1.E-6) GOTO 80 U11=U(1,I) U12=U(2,I) C1=DYO1*(El(1,1)*G1(1)+El(1,2)*61(2)) C2=E2(2,1)*(DY02*62(1)+U(1,I+1)-UI1) + F E2(2,2)*(DY02*G2(2)+U(2,I+1)-U12) U(1,I)=U11+(C1*E2(2,2)-C2*E1(1,2))/D U(2,I)=U12+(E1(1,1)*C2-E2(2,1)*C1)/D C

C

U(3,I)=XO U(4,I)=YO

Appendix 1:

Method of Massau

455

70 LAMB1(1)=LAMBZ(1)

El (1,1)=E2(1,1) El (1,2)=E2(1,2) 61(1)=G2(l) G1 (2)=G2(2) LL=.TRUE.

GOTO 100 80 B(I)=.FALSE.

GOTO 70 90 B(I)=.FALSE. LL=.FALSE. 100 CONTINUE C C C

END OF THE MAIN LOOP B(N2)=.FALSE. 110 N2=N2-1 IF(.NOT.B(N2).AND.N2.GT.N1) GOTO 110 SIGMAO=SIGMAO+H2*0.5 TAU=TAU+H2*0.5 RETURN END

SUBROUTINE EIGEN(U,E,LAMBDA,L) C

REAL U(4),E(2,2),LAMBDA(2) LOGICAL L C C C C C C C C

INPUT PARAMETERS U CONTAINS U(1),U(2),X,Y OUTPUT PARAMETERS EIGENVALUES LAMBDA(1).LT.LAMBDA(2) MATRIX E (IN THE TEXT DENOTED BY E**-1) L=.FALSE. INDICATES THAT THE COMPUTATION IS NOT POSSIBLE

C

REAL A(2,2),C,D,C1,C2,C3,C4 LOGICAL SW L=.TRUE. CALL

IF(.NOT.L) RETURN C C

COMPUTATION OF THE EIGENVALUES OF A

APPENDICES

456

C

C=A(1,1)+A(2,2) D=A(1,1)-A(2,2) D=D*D+4.*A(1,2)*A(2,1) IF(D.LE.O) GO TO 101 D=SQRT(D) IF(D.LT.1.E-6*ABS(C)) GO TO 101 LAMBDA(1)=0.5*(C-D) LA14BDA(2)=0.5*(C+D) C C C C C C C

SOLUTION OF THE FOLLOWING HOMOGENEOUS EQUATIONS E(1,1)*(A(1,1)-LAMBDA(1))+E(1,2)*A(2,1)=O E(1,1)*A(1,2)+E(1,2)*(A(2,2)-LAMBDA(1))=0 E(2,1)*(A(1,1)-LAMBDA(2))+E(2,2)*A(2,1)=O E(2,1)*A(1,2)+E(2,2)*(A(2,2)-LAMBDA(2))=0 C=LAMBDA(1) SW=.FALSE. 10 C1=ABS(A(1,1)-C) C2=ABS(A(2,1)) C3=ABS(A(1,2)) C4=ABS(A(2,2)-C) IF(AMMAX1(C1,C2).LT.AMAXI(C3,C4)) GO TO 30 IF(C2.LT.C1) GO TO 20 C1=1. C2=(C-A(1,1))/A(2,1) GO TO 50 20 C2=1. C1=A(2,1)/(C-A(1,1)) GO TO 50 30 IF(C3.LT.C4) GO TO 40 C2=1.

C1= (C-A (2,2) )/A (1,2) 60 TO 50 40 C1=1. C2=A(1,2)/(C-A(2,2))

50 IF(Sl!) GO TO 60 E(1,1)=C1 E(1,2)=C2 C=LAMBDA(2) SU=.TRUE. GO TO 10 60 E(2,1)=C1 E(2,2)=C2 RETURN 101 L=.FALSE. RETURN END

Appendix 1:

Method of Massau

EXAMPLE (MENTIONED IN THE TEXT)

MAIN PROGRAMME: C C

DESCRIPTION OF THE COMMON BLOCK IN THE SUBROUTINE MASSAU REAL U(4,500),SIGMAO,TAU,H2 INTEGER N1,N2 LOGICAL B(500) COMMON /MASS/ U,SIGMAO,TAU,H2,NI,N2,B

C

REAL X,DUI,DU2,SIGMA INTEGER I,J C

C

INITIALIZATION OF THE COMMON BLOCK

C

TAU=O. NI=1 N2=65

/32.

P_

.--

_ .*ATAN(1.)

SIGMA.:?--=-H2

X=O. DO 10 I=1,N2

U(1,I)=0.1*SIN(2.*PI*X)*EXP(X) U(2,I)=1.

U(3,I)=X

U(4,I)=0. B(I)=.TRUE. 10 X=X+H2 C C C

LOOP FOR PRINTING AND EXECUTING THE SUBROUTINE DO 40 I=1,65 DO 39 J=N1,N2 IF(.NOT.B(J)) GOTO 39 SIGMA=SIGMA0+J*H2 WRITE(6,49) SIGMA,TAU,U(3,J),U(4,J),U(1,J),U(2,J) 39 CONTINUE WRITE(6,50)

C

IF(N2.LE.N1) STOP CALL. MASSAU

40 CONTINUE STOP C

49 FORMAT(IX,2F8.5,IX,6F13.9) 50 FORMAT(IHI) END

4S7

APPENDICES

458

SUBROUTINES:

SUBROUTINE QUELL(U,G) C C C C

INPUT PARAMETER U CONTAINS U(1),U(2),X,Y OUTPUT PARAMETERS ARE G(1),G(2)

REAL U(4),G(2) G(1)=0. G(2)=0. RE".` 'RN END

SUBROUTINE MATRIX(U,A,L) C C C C C

INPUT PARAMETER U CONTAINS U(1),U(2),X,Y OUTPUT PARAMETERS ARE THE MATRIX A AND L L=.TRUE. IF U BELONGS TO THE DOMAIN OF THE COEFFICIENT MATRIX A AND OF THE TERM G. OTHERWISE, L=.FALSE.

C

REAL U(4),A(2,2) LOGICAL L C

REAL U1,U2 C

U1=U (1) U2=U (2)

L=.TRUE. A(1,1)=-U1 A(1,2)=-2.*U2 A(2,1)=-O.S*U2 A(2,2)=-U1 RETURN END

Appendix 2:

Nonlinear implicit

Appendix 2:

Total implicit difference method for solving a

difference method

459

nonlinear parabolic differential equation.

The total implicit difference method has proven itself useful for strongly nonlinear parabolic equations.

With it

one avoids all the stability problems which so severely complicate the use of other methods.

In the case of one (space)

variable, the amount of effort required to solve the system of equations is often overestimated.

The following programs solve the problem ut = a(u)uxx - q(u),

x e (r,s), t > 0

u(x,0) = $(x),

x c [r,s]

u(r,t) _ ar(t), u(s,t) = 4s(t),

t > 0.

Associated with this is the difference method u(x,t+h)-u(x,t) = Xa(u(x,t+h))[u(x+&x,t+h)+u(x-Ax,t+h) -

where

2u(x,t+h)]

Ax > 0, h > 0, and

- hq(u(x,t+h))

A =

ox = (s-r)/(n+l),

x = r + jAx,

fixed

n e 1N

j

= l(l)n

this becomes a nonlinear system in solved with Newton's method.

When specialized to

h/(Ax)2.

n

unknowns. It is

For each iterative step, we

have to solve a linear system with a tridiagonal matrix. linear equations are

aljuj_1 + a2juj + a3juj+1 = a4j, where

j = l(1)n

The

APPENDICES

460

alj = -aa(uj)

aaI(uj)[uj+l+uj-l-2uj]+hgI(uj

a2j

= 1+2aa(uj

aij

= -Xa(uj)

a4j

= uj-[1+2Aa(uj)]uj+aa(uj)[uj+l+uj-1)-hq(uj)

uj = solution of the difference equation at the point (r+jAx,t+h).

uj = corresponding Newton approximation for u(r+jtx,t+h). When this sytem has been solved, the by

uj

+ uj; the

aij

are replaced

uj

are recomputed; etc., until there is

no noticeable improvement in the

uj.

Usually two to four

Newton steps suffice.

Since the subscript 0 is invalid in FORTRAN, the quantities

u(x+jAx,t)

are denoted in the programs by

For the same reason, the Newton approximation

uj

U(J+1).

is called

Ul(J+1).

The method consists of eight subroutines:

HEATTR, AIN, RIN, GAUBD3, ALPHA, DALPHA, QUELL, DQUELL. HEATTR is called once by the main program for each time increment.

Its name is an abbreviation for heat transfer.

other subroutines are used indirectly only.

The

The last four

subroutines must be rewritten for each concrete case.

They

are REAL FUNCTIONs with one scalar argument of REAL type, which describe the functions

a(u), a'(u), q(u), and

q'(u).

The other subroutines do not depend on the particulars of the problem.

AIN computes the coefficients

linear system of equations.

aij

of the

GAUBD3 solves the equations.

This program is described in detail along with the programs

Appendix 2:

Nonlinear implicit difference method

dealing with band matrices in Appendix S. alj, a2j, and

a3j

Newton's step.

461

The coefficients

are recomputed only at every third

In the intervening two steps, the old values

are reused, and the subroutine RIN is called instead of AIN. RIN only computes

a4j.

Afterwards, GAUBD3 runs in a simpli-

For this reason, the third variable is .TRUE..

fied form.

We call these iterative steps abbreviated Newton's steps. Before HEATTR can be called the first time, it is necessary to fill the COMMON block /HEAT/: N = n DX = Ax = (s-r)/(n+l) U(J+1)

4(r+jIx)

j

= 0(1)n+l

H = h. H

and

can be changed from one time step to another. u(s,t)

depend on

boundary values

If

u(r,t)

t, it is necessary to set the new

U(l) _ r(t+h)

and

U(N+2) = 0s(t+h)

be-

fore each call of HEATTR by the main program.

An abbreviated Newton's step uses approximately 60% of the floating point operations of a regular Newton's step: (1)

(2)

Regular Newton's step: n

calls of ALPHA, DALPHA, QUELL, DQUELL

21n+4

operations in AIN

8n-7

operations in GAUBD3

4n

operations in HEATTR.

Abbreviated Newton's step: n

calls of ALPHA, QUELL

10n+3

operations in RIN

5n-4

operations in GAUBD3

4n

operations in HEATTR.

APPENDICES

462

This sequence of different steps--a regular step followed by two abbreviated steps--naturally is not optimal in every Our

single case.

error test for a relative accuracy of If so desired, it suffices to

10-5 is also arbitrary.

make the necessary changes in HEATTR, namely at IF(AMAX.LE.0.00001*UMAX) GO TO 70 and

IF(ITERI.LT.3) GO TO 21.

As previously noted, two to four Newton's iterations usually suffice.

This corresponds to four to eight times

this effort with a naive explicit method.

If

u

and

a(u)

change substantially, the explicit method allows only extremely small incrementat.ions

h.

This can reach such extremes

that the method is useless from a practical standpoint. ever, if

q1(u)

one should have

How-

< 0, then even for the total implicit method hq'(u) > -1, i.e. h < 1/lq'(u)l.

For very large

n, to reduce the rounding error in

AIN and RIN we recommend the use of double precision when executing the instruction

A(4,J)=U(J+1)-(1.+LAMBD2*AJ)*UJ+LAMBDA*AJ* *

(U1(J+2)+U1(J))-H*QJ.

This is done by declaring DOUBLE PRECISION LAMBDA, LAMBD2, AJ, UJ, U12, U10 and replacing the instructions above by the following three instructions:

Appendix 2:


463

U12 = Ul(J+2) U10 = U1(J)

A(4,J) = U(J+1)-(1.+LAMBD2*AJ)*UJ +

+LAMBDA*AJ*(U12+UlO).

All remaining floating point variables remain REAL. other than AIN and RIN do not have to be changed.

Programs

APPENDICES

464

SUBROUTINE HEATTR(ITER) C C C C

U(I) VALUES OF U AT X=XO+(I-1)*DX, I=1(1)N+2 U(1), U(N+2) BOUNDARY VALUES H STEP SIZE WITH RESPECT THE TIME COORDINATE

C

REAL U(513),H,DX INTEGER N COMMON/HEAT/U, H,DX,N C

REAL U1(513),AJ,UJ,AMAX,UMAX,A(4,511) INTEGER ITER,I,ITERI,N1,N2,J N1=N+1 N2=N+2 C

C

FIRST STEP OF THE NEWTON ITERATION

C

CALL AIN(A,U) CALL GAUBD3(A,N,.FALSE.) DO 20 J=2,N1 20 U1(J)=U(J)+A(4,J-1)

UI (1)=U(1)

U1(N2)=U(N2) ITER=1 ITER1=1 C C C

STEP OF THE MODIFIED NEWTON ITERATION 21 CALL RIN(A,U1) CALL GAUBD3(A,N,.TRUE.) GO TO 30

C C C

STEP OF THE USUAL NEWTON ITERATION 25 CALL AIN(A,U1) CALL GAUBD3(A,N,.FALSE.) ITER1=0.

C C C

RESTORING AND CHECKING 30 AHAX=O. UHAX=O. DO 40 J=2,N1 AJ=A(4,J-1) UJ=U1(J)+AJ AJ=ABS(AJ) IF(AJ.GT.AMAX) AMAX=AJ U1(J)=UJ UJ=ABS(UJ) IF(UJ.GT.UMAX) UMAX=UJ 40 CONTINUE

C

Appendix 2:


465

C

ITER=ITER+1

ITERI=ITER1+1 IF(AMAX.LE.0.00001*UMAX) GO TO 70 IF(ITER.GT.20) GO TO 110 IF(ITERI.LT.3) GO TO 21 GO TO 25 C C

U=U1

C

70 DO 80 J=2,N1

80 U(J)=U1(J) RETURN C C

110 WRITE(6,111) 111 FO.RMAT(15H NO CONVERGENCE) STOP END

SUBROUTINE AIN(A,U1) C C C C

EVALUATION OF THE COEFFICIENTS AND OF THE RIGHT-HAND SIDE OF THE SYSTEM OF LINEAR EQUATIONS REAL A(4,511),U1(513)

C

C C

COMMON BLOCK COMPARE HEATTR REAL U(513),H)DX INTEGER N

C

REAL LAMBDA,LAMBD2,LAMBDM,UJ,AJ,DAJ,QJ,DQJ INTEGER J REAL Z LAMBDA=H/(DX*DX) LAHBD2=2.*LAMBDA

DO 10 J=1,N UJ=U1(J+1) AJ=ALPHA(UJ) DAJ=DALPHA(UJ) QJ=QUELL(UJ) DQJ=DQUELL(UJ) 2=LAi1BDH*AJ

466

APPENDICES

A(1,J)=Z *

A(2,J)=1.+LAMBD2*(AJ+DAJ*UJ)-LAMBDA*DAJ* (U1(J+2)+U1(J))+H*DQJ

A (3,J)=Z

A(4,J)=U(J+1)-(1.+LAMBD2*AJ)*UJ+LAMBDA*AJ* (U1(J+2)+U1(J))-H*QJ 10 CONTINUE RETURN *

END

SUBROUTINE RIN(A,U1) C

C C C

EVALUATION OF THE RIGHT-HAND SIDE OF THE LINEAR EQUATIONS REAL A(4,511),U1(513)

C C

COMMON BLOCK COMPARE HEATTR

C

REAL U(513),H,DX INTEGER N COMMON/HEAT/U,H,DX,N C

REAL LAMBDA LAMBDZ,UJ,AJ,QJ INTEGER J LAMBDA=H/(DX*DX) LAMB02=2.*LAMBDA DO 10 J=1,N UJ=U1(J+1) AJ=ALPHA(UJ) QJ=QUELL(UJ) A(4,J)=U(J+1)-(1.+LAMBD2*AJ)*UJ+LAMBDA*AJ* * (U1(J+2)+U1(J))-H*QJ 10 CONTINUE RETURN END

Appendix 2:


EXAMPLE

MAIN PROGRAMME:

REAL U(513),H,DX INTEGER N COMMON/HEAT/U,H,DX,N REAL PI,T INTEGER I,J,ITER PI=4.*ATAN(1.) C

N=7 DX=1 ./8. H=1 ./64.

DO 10 1=1,9 10 U(I)=SIN(PI*DX*FLOAT(I-1)) C

T=O.

DO 20 I=1,10 CALL HEATTR(ITER) WRITE(6,22) ITER T=T+H WRITE(6,21) T WRITE(6,21)(U(J),J=1,9) 20 CONTINUE STOP C

21 FORMAT(1X,9F12.9/1X,9F12.9) 22 FORMAT(6H ITER=,I2) END

467

468

SUBROUTINES:

REAL FUNCTION ALPHA(U) REAL U ALPHA=(1.-0.5*U)/(4.*ATAN(1.))**2 RETURN END

REAL FUNCTION DALPHA(U) REAL U DALPHA=-.5/(4.*ATAN(1.))**2 RETURN END

REAL FUNCTION QUELL(U) REAL U QUELL=U*U*0.5 RETURN END

REAL FUNCTION DQUELL(U) REAL U DQUELL= U RETURN END

APPENDICES

Appendix 3:

Lax-Wendroff-Richtmyer method

Appendix 3:

Lax-Wendroff-Richtmyer method for the case of

469

two space variables.

The subroutines presented here deal with the initial value problem ut = A 1 u

x

+ A2uy + Du + q,

u(x,y,0) = (X,y), Here

x,y e1R, t > 0

x,y e 1R.

A1,A2,D c C1(IR2,MAT(n,n,IR)), D(x,y) = diag(dii (x,y)),

q e C1(ii2 x [0,-),]R n).

properly posed in

We require that the problem be

L2(IR2,cn).

are always symmetric, and

that (1) A1(x,y), A2(x,y) (2) Al, A2, D, q

For this it suffices, e.g.,

have compact support.

Because of the terms

Du + q, the differential equation

in the problem considered here is a small generalization of Example 10.9.

There is one small difficulty in extending the

difference scheme to this slightly generalized differential equation.

One can take care of the terms

Du + q

in the

differential equation with the additional summand h[D(x,y)u(x,y,t) + q(x,y,t)]

or better yet, with h{ZD(x,y)[u(x,y,t) + u(x,y,t+h)] + q(x,y,t+ 2h)} This creates no new stability problems (cf. Theorem 5.13). Consistency is also preserved.

However, the order of consis-

tency is reduced from 2

The original consistency proof

to 1.

considered only solutions of the differential equation ("consistency in the class of solutions") and not arbitrary sufficiently smooth functions; here we have a different differential equation.

470

APPENDICES

We use the difference method u(x,y,t+h) = [I- ZD(x,Y)] 1[I+S(h)oK(h)+ + h(I- ZD(x,Y)] lq(x,Y,t+

D(x,Y)]u(x,Y,t)

-T)+2(I-ZD(x,Y)) 1S(h)q(x,Y,t)

where

K(h) = ZS(h) + (4I+8D(x,Y)) (TA,1+TA,2+TA1+T012)

S(h) = ZA[A1(x,Y)(TA'1-Tpl1) + A2(x,Y)(TA 2-TA12)1. For

D(x,y) = 0

and

method (r = 1). every case.

q(x,y,t) = 0, this is the original

But then the order of convergence is 2 in

Naturally, this presupposes that the coeffici-

ents, inhomogeneities, and solution are all sufficiently often differentiable.

The computation procedes in two steps. 1st Step

(SUBROUTINE STEP1):

v(x,y) = K(h)u(x,y,t) + 2nd Step

hq(x,y,t).

(SUBROUTINE STEP2):

u(x,y,t+h) = {[I+

[I

-

ZD(x,Y)]-1o

ZD(x,Y)lu(x,Y,t)+S(h)v(x,Y)+hq(x,Y,t+ 11)1.

The last instruction is somewhat less susceptible to rounding error in the following formulation:

u(x,y,t+h) = u(x,y,t) +

[I-

ZD(x,Y)]

to

{S(h)v(x,y)+h[D(x,y)u(x,y,t)+q(x,y,t+ 2)]}. If

u(x,y,t)

is given at the lattice points

(x,Y) = (uo,vt)

(p,v e2z, p+v

even)

Appendix 3:

then

Lax-Wendroff-Richtmeyer method

v(x,y)

can be computed at the following points: p,v e22, p+v

(x,Y) = (PA,vt),

From these values and the old values u(x,y,t+h)

471

odd.

u(x,y),

one obtains

at the points p,v c

(x,y) = (pt,vt),

p+v

,

even.

This completes the computation for one time increment. If steps 1 and 2 follow each other directly, then and

Therefore, we divide each time

have to be stored.

v

u

step into substeps, in which the u-values are computed only for the lattice points on a line x + y = 2aA = constant.

For this one only needs the v-values for x + y = (2a-1)A (as shown in Figure 1).

and

x + y = (2a+1)a

At first, only these v-values are

stored, and in the next substep, half of these are overThus we alternately compute

written. on a line. a line.

v

on a line and

SUBROUTINE STEP1 computes only the

STEP2 does compute all of

v

u

values on

u, but in passing from

one line to the next, STEP2 calls STEP1 to compute

v.

The program starts with the lattice points in the square t(x,y)

-1 < x+y

Numerical Solution of Partial Differential Equations (Applied Mathematical Sciences)

Numerical Solutions of Partial Differential Equations (Applied Mathematical Sciences)

Numerical solution of partial differential equations

Numerical solution of partial differential equations (MA3243)