Analytical and Numerical Approaches to Asymptotic Problems in Analysis: Conference Proceedings (North-Holland mathematics studies)

ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS IN ANALYSIS

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NORTH-HOLLAND MATHEMATICS STUDIES

47

Analytical and Numerical Approaches to Asymptotic Problems in Analysis Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems Universityof Nijmegen,The Netherlands, June 9-13,1980

Edited b y

0.AXELSSON L. S. FRANK Mathematicallnstitute The University of Nqmegen TheNetherlands and

A. VAN DER SLUE Mathematical institute The University of Utrecht TheNetherlands

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

North-Holland Publishing Company, I980

All rights rrscwerl. No purt of rhispublicurioti mu,v he, rrprvducecl. siori>clin N retrievtrlsy.stett~. or trunsmirtrd, in uny form or by uny nwuns, c+c~trotiic, mc~chanicd. photocopying. rcw)rtling or othrrwisc.. without the prior pertnissioti of the, copvrighr owtwr.

ISBN: 0 444 86131 9

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N O R T H - H O L L A N D PUBL.ISHING C O M P A N Y A M S T E R D A M O N E W YORK * O X F O R D S o l r di\rribiiror\for /tic U . . S . A . r r i r t l ~ ' u i r u d ~ r : ELSEVIER N O R T H - H O L L A N D . INC. 5 2 V A N D E R B I L T A V E N U E . NEW YORK. N . Y . 10017

Library of Congress Cataloging in Publication Data Conference on Analytical and Numerical Approaches to Asymptotic Problems, University of Nijmegen, 1980. Analytical and numerical approaches to asymptotic problems in analysis. (North-Holland mathematics studies ; 47) Differential equations--Asymptotic theory-Congresses. 2. Differential equations--Numerical solutions-Congresses. I. kcelsson, Axel, 1934-11. Frank, Leonid S., 1934- 111. Sluis, Abraham van der. IV. Title. &A370.C63 1980 515.3'5' 80-26580 ISBN 0-444-86131-9 1.

P R I N T E D IN T H E N E T H E R L A N D S

PREFACE

A n International Conference on Analytical and Numerical Approaches to Asymptotic

Problems was held in the Faculty of Science, University of Nijmegen, The Netherlaads from June 9th through June 13th, 1980 The development of the analytical asymptotic theory has been essentially stimulated by needs of applied sciences such as fluid dynamics, diffraction theory, reactiondiffusion processes, elasticity theory, quantum mechanics and so on.

On the other

hand, pure aesthetics have also been a strong attraction for mathematicians and theoretical physicists in the elaboration of this beautiful mathematical tool.

The

introduction of numerical methods during the last decades became a new turning point in the further development of asymptotic analysis. The main motivation in organizing this Conference was the idea to bring together people working in asymptotic theory and to try to survey achievements in this field of applied mathematics, due to the combination and synthesis of analytical and numerical approaches. The attendance at the Conference and discussions during and after its sessions were an explicit indication of a growing interest amongst mathematicians, numerical analysts and theoretical physicists for new aspects and methods in asymptotic analysis. The Proceedings of the Conference contain the full text of 17 invited addresses and

11 contributed papers. We are grateful to the participants, especially to the speakers who were instrumental in making this meeting a fruitful and enjoyable scientific event.

We are also most

indebted to the Administration of the Faculty of Science, University of Nijmegen, whose support was an indispensable contribution to the success of this Conference.

0. AXELSSON

L.S. FRANK A. VAN DER SLUIS Editors V

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ACKNOWLEDGEMENTS

The Conference on ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N ANALYSIS w a s sponsored by t h e U n i v e r s i t y of Nijmegen, t h e Dutch M i n i s t r y o f E d u c a t i o n , t h e Dutch Mathematical S o c i e t y , t h e I n t e r n a t i o n a l B u s i n e s s Machines C o r p o r a t i o n , t h e O f f i c e of t h e Naval Research (London), and t h e U n i t e d S t a t e s Army Research and Development Group ( E u r o p e ) .

We are d e e p l y i n d e b t e d t o a l l s p o n s o r s f o r t h e i r s u p p o r t of t h i s Conference.

THE ORGANIZING COMMITTEE

vii

INTERNATIONAL CONFERENCE ON ANALYTICAL AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N ANALYSIS

JUNE 9-13,

1980, FACULTY OF SCIENCE, UNIVERSITY OF NIJMEGEN, THE NETHERLANDS

Photograph taken of s o m e of t h e p a r t i c i p a n t s b e f o r e an excursion

CONTRIBUTORS

B. Aulbach, Mathematical Institute, The University of Wiirzburg, Germany 0. Axelsson, Mathematical Institute, The University of Nijmegen, The Netherlands

M. Bertsch, The Mathematical Centre, Amsterdam, The Netherlands D. Caillerie, Laboratoire de mCcanique theorique, Universite Pierre et Marie Curie, Paris, France S . Dobrokhotov, Institute of Machinebuiding, Moscow, USSR J.J. Duistermaat, Mathematical Institute, The University of Utrecht, The Netherlands P.C. Fife, Department of Mathematics, The University of Arizona, Tucson, U.S.A. L.S. Frank, Mathematical Institute, The University of Nijmegen, The Netherlands R. Geel, Ubbo Emmius Institute, Groningen, The Netherlands D.F. Griffiths, Department of Mathematics, University of Dundee, Scotland E.W.C. van Groesen, Mathematical Institute, The University of Nijmegen, The Netherlands I. Gustafsson, Mathematical Institute, The University of Nijmegen, The Netherlands F.C. Hoppensteadt, Department of Mathematics, University of Utah, Salt Lake, U.S.A. D. Huet, U.E.R. de Mathematiques, Universite de Nancy I, France A. Iserlies, Department of Applied Mathematics, University of Cambridge, England E.M. de Jager, Mathematical Institute, The University of Amsterdam, The Netherlands C. Johnson, Chalmers Institute of Technology, GGteborg, Sweden Ya. Kannai, The Weizmann Institute of Science, Rehovot, Israel R.B. Kellogg, Department of Mathematics, The University of Maryland, College Park, U.S.A. J . Lorentz, Mathematical Institute, The University of Konstanz, Germany V.P. Maslov, Institute of Machinebuilding, Moscow, USSR R.M.M. Mattheij, Mathematical Institute, The University of Nijmegen, The Netherlands A. Meiring, Department of Mathematics, University of Dundee, Scotland A.R. Mitchell, Department of Mathematics, University of Dundee, Scotland S. Osher, Department of Mathematics, The University of California, Los Angeles, U.S.A.

J.P. Pauwelussen, The Mathematical Centre, Amsterdam, The Netherlands L . A . Peletier, Mathematical Institute, The University of Leiden, The Netherlands H.4. Reinhardt, Mathematical Institute, The J.W. Goethe University, Frankfurt, Germany M. Schatzman, Universite Pierre et Marie Curie, Paris, France K. Soni, Mathematics Department, The University of Tennessee, Knoxville, U.S.A. R.P. Soni, Mathematics Department, The University of Tennessee, Knoxville, U.S.A. M. Tabata, Kyoto University, Japan and Universit6 Pierre et Marie Curie, Paris, France M.E. Taylor, Department of Mathematics, The Rice University, Houston, U.S.A. R. Temam, Laboratoire d'Analyse Numerique, Universite Paris-Sud, Orsay, France

ix

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TABLE OF CONTENTS

PREFACE

V

vi

ACKNOWLEDGEMENTS

vii

CONTRIBUTORS

xiii

OPENING ADDRESS PART I

:

INVITED LECTURES

Finite gap almost periodic solutions in asymptotical expansions S.Yu. DOBROKHOTOV and V.P. MASLOV Periodic solutions near equilibrium pqints of Hamiltonian systems

1

n . DUISTERMAAT

27

Asymptotics for elementary spherical functions n. DUISTERMAAT

35

On the question of the existence and nature of homogeneous-center target patterns in the Belousov-Zhabotinskii reagent P.C. FIFE

45

Singular perturbations of hyperbolic type E.M. DE JAGER and R. GEEL

57

Computation by extrapolation of solutions of singular perturbation problems F.C. HOPPENSTEADT and W.L. MIRANKER

73

Proper approximation of a normed space and singular perturbations D. HUET

a7

An analysis of some finite element methods for advection-diffusion problems C. JOHNSON and U. NAVERT

99

Short time asymptotic behavior for parabolic equations Y. KA"A1

117

Difference approximation for a singular perturbation problem with turning points R.B. KELLOGG

133

Stability and consistency analysis of difference methods for singular perturbation problems J. LORENZ

141

Finite element Galerkin methods for convection-diffusion and reaction-diffusion A.R. MITCHELL, D.F. GRIFFITHS and A. MEIRING

157

xi

xii

CONTENTS

Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws S. OSHER

179

Nonstationary filtration in partially saturated media M. BERTSCH and L.A. PELETIER

205

A-Posteriori error estimates and adaptive finite element computations for singularly perturbed one space dimensional parabolic equations H.-J. REINHARDT

213

Diffraction of waves by cones and polyhedra M.E. TAYLOR

2 35

Some asymptotic problems in mechanics R. TEMAM

249

PART I1

:

CONTRIBUTED PAPERS

Asymptotic amplitude and phase for isochronic familie..of periodic solutions B. AULBACH

265

Quasioptimal finite element approximations of first order hyperbolic and of convection-dominated convection-diffusion equations 0. AXELSSON and I. GUSTAFSSON

273

Homogenization of the equation of stationary diffusion in cylindrical domains D. CAILLERIE

281

Singular perturbations of an elliptic operation with discontinuous nonlinearity L . S . FRANK and E.W. VAN GROESEN

289

Coercive singular perturbations: asymptotics and reduction to regularly perturbed boundary value problems L.S. FRANK and W.D. WENDT

305

Efficient two-step numerical methods for parabolic differential equations A. ISERLES

319

Estimating the discretization error in three point difference schemes for second order linear singularly perturbed BVP R.M.M. MATTHEIJ

327

Failure of nerve impulse propagation for nonuniform nerve axons J.P. PAUWELUSSEN

339

The penalty method for the vibrating string with an obstacle M. SCHATZMAN

345

On the asymptotic behavior of the solution of a nonlinear Volterra integral equation K. SONI and R.P. SONI

359

Conservative upwind finite element approximation and fts applications M. TRBATA

369

OPENING ADDRESS bY L.S. FRANK

Ladies and Gentlemen, Distinguished Guests and Participants of the Conference, Dear Colleagues and Friends It is a privilege to welcome you at the Opening'of the Conference on "Analytical and numerical approaches to asymptotic problems in analysis", which is being held in the Faculty of Science of the University of Nijmegen. The main motivation in organizing such a conference has been the idea to bring together mathematicans and theoretical physicists working in asymptotic analysis and to try to survey recent achievements in this field. Considerable progress in the treatment of applied problems affected by the presence of small or large parameters has resulted from the combination and synthesis of analytical and numerical approaches in asymptotic analysis. The fundamental rdle played by asymptotics in the applied sciences is not only stressed in mathematical research but is also acknowledged by our colleagues in physics, chemistry, biology and other disciplines of the natural sciences. Perhaps one of the most striking illustrations of the efficient use of asymptotic methods is provided by fluid dynamics, that has been particularly inspiring for the development of other mathematical branches as well. It is well known that the Euler-Lagranye equations for nonviscous fluids gave rise to several paradoxes, which could not be explained in the framework of these equations. One can mention here some of them:

lo. The reversibility paradox connected with the drag and lift acting on a solid in steady translation. 2'.

Paradox of D'Alambert leading to an absurd conclusion that the dray and

the lift are both zero. 3'.

Paradox of Jukowski's ideal flow (the drag is zero and the lift is defi-

ned in an artificial way). 4'.

The development of different kinds of shocks, which contradicts Leibnitz' xiii

xiv

OPENING ADDRESS

numerical approaches to asymptotic problems. This combination turns out to be an efficient mathematical tool to investigate asymptotic phenomena in applied scien-

"Factwn e s t " (it is done), a second small parameter h took its place next to

ces.

the biq brother

E.

The name of Gottfried Silbermann is not widely spread in mathematical circles. Indeed, this gentleman was never involved in any business connected with asymptotics. He is known amongst professional musicians as a creator of the piano, one of the wonders of the musical world. Of course, his claim to the piano paternity was disputed by some other European colleagues of him, a situation which might make mathematicians feel a little more at home when talking about Gottfried Silbermann . Unfortunately, the time-limitation does not allow me to go into detailed analysis of Gottfried Silbermann's criativity. But, the wonderful idea of the piano, as combination and synthesis of a clavichord and an harpsichord can be compared with the idea of putting together analytical and numerical aspects leading to asymptotic problems affected by the presence of two parameters

E

and h.

Ladies and Gentlemen, a very old-fashioned principle in criminology (which one should consider nowadays as a very applied science) says (in French): "Chey-

cher la fermne".

This principle, of course, is not any longer fashionable amongst

criminologists. In asymptotic analysis of different phenomena in natural sciences one of the oldest (but not old-fashioned) principles: "look for a srnaZZ p a r ~ m c t e ~ ~ " turns out to be as vital and strong, as the thirst of human beings for knowledge. Indeed, this small thing, the proverbial parameter

E

provides the mathematical

world with such beauty, that one would be tempted to compare its r6le with that played in men's lives by our charming female companions. Of course, one should keep in mind that the beauty is always in the eye of the beholder. I feel that I should stop here, lest I be accused by the Women's Liberation

movement of discriminating against the better half of human kind and before I start talking on behalf of men's liberation. Anyhow, let me express the hope that the conference on "Analytical and numerical approaches to asymptotic problems in analysis", being held in Nijmegen, will be an enjoyable meeting of people who placed their yearnings for beauty in the wonderful world of the small parameter. One last word. We are, certainly, aware, all of us, of a possible misuse of

our knowledge. Namely, some political systems in the Twentieth century, are still making full use of scientific research for the preparation of war, the grave limitation of freedom and the presecution of dissidents and non-conformists. May we hope that our research will be exploited for the protection of our freedom at least as efficiently, as it has been misused for some evil purposes,

OPENING ADDRESS

xv

famous maxim: "natura non f a c i t saZtus" (the nature does not tolerate jumps). In the book by Garrett Birkhoff: "Hydrodynamics, a study in logic, fact and similitude", one comes across of the following witticism, due to Sir Cyril Hinshelwood and quoted previously by M.J. Lighthill: in the last century " f l u i d

dynamists were divided i n t o hydroZic engineers who observed what could not be explained, and muthernaticians who explained things t h a t could not be observed". This witticism reflects the paradoxial state of fluid dynamics in the past.

A

following comment by Birkhoff, himself, is not very reassuring with respect to the modern times: "It is my impression", he says, "that many survivors of both species are still with us".

Shouldn't one recall in this case Carlyle's famous epigram

(to be found in "Heroes and Hero-Worship"): their s i l e n c e is more eloquent than

words If? The Navier-Poisson equations, fully justified later on by the use of the continuum-approach in the works of San-Venan and Stokes, seemed to be an adequate mathematical model in order to save theoretical fluid dynamics and to remove some of its paradoxes, at least in the cases when relativistic, quantum and some other effects could be neglected. The Navier-Stokes equations, thanks to Reinold's number had become one of the most striking examples of singular perturbations, leading to the idea of boundary layer, due to Prandtl. Prandtl's theory (developed later on in a more rigorous and consistent way) provides a beautiful mathematical tool for the investigation of practical hydrodynamical problems. Still, it is unable to explain some other phenomena, such as, for instance, the appearence of turbulent flows (a well-known boundary layer paradox).

Italian is probably a suitable language to describe this situation: se non

& vero 6 bene t r ovato (if it is not true, still it is a good discovery). Further developnents of the small parameter method led to its efficient use in other fields of applied mathematics. Here, one can mention, for instance:

lo. Von Karman's model of thin elastic plates, that stimulated the development of the elliptic singular perturbation theory. 2O.

Diffraction theory, that gave birth to the Fourier integral operators,

and so on. The problems to be treated by means of asymptotic analysis and analysis itself are becoming progressively, more and more, complicated.

')Ad augusta per angusta"(to

august results by narrow paths) says a latin aphorism used also as a catch-word in Victor Hugo's "Hernani". Or, in other words, one achieves a triumph only by overcoming difficulties.

A new turning point in the development of asymptotic analysis occurred with the introduction of numerical methods and the combination of both analytical and

xvi

OPENING ADDRESS

up t o now. May I r e c a l l h e r e , when t a l k i n g a b o u t t h e v a l u e o f p e r s o n a l freedom f o r human b e i n g s , t h e remarkable poem, t h a t was w r i t t e n by P a u l E l u a r d i n 1943 i n o c c u p i e d P a r i s and t h a t ends w i t h t h e f o l l o w i n g words E t p a r l e p o u v o i r d ' u n mot

J e recommence ma v i e . Je s u i s n6 pour t e c o n n a i t r e ,

Pour t e nommer: Liberte. L a d i e s and Gentlemen, t h a n k you f o r your a t t e n t i o n

ANALYTICAL AND NUMERICAL 4PPRCjACtiES TO ASYMPTOTIC PROBLEMS IN A N A L Y S I S 5 . A x e l s s o n , L . S . F r a n k , A . v a n d e r S l u i s (eds.) @ N o r t h - t i o l l a n d Publishing C o m p a n y , 1 9 6 1

FINITE GAP ALMOST PERIODIC SOLUTIONS IN ASYMPTOTICAL EXPANSIONS S.Yu. Dobrokhotov, V . P .

Maslov

Moscow Institute of Electronic Machinebuilding Moscow USSR

Recently discovered almost periodic solutions of some nonlinear equations are shown to be used in asymptotical expansions in the following way: firstly for generalization of the multiphase nonlinear WKB method (Whitham method) both with real and complex phases, secondly for quantization of these solutions being Lagrangian manifolds in quasiclassical approximation.

1.

INTRODUCTION

An important class of exact solutions which were called the almost periodic finite gap solutions was recently obtained for some nonlinear wave equations with partial derivatives. The methods of these solutions constructing became available after the founding paper by S.P. Novikov ilO3 had appeared and are nowadays being further developed and applied to practically all equations, for which the multisoliton solutions have been found earlier. The finite gap almost periodic solutions form the generalization of cnoidal waves, which are known in the shallow water theory, i.e. of solutions of the following form: U(x,t)

,

y(Ux t Vt)

=

x,t

€

R

2

,

(1.1)

where Y(T) is an elliptic function. Thus the finite gap almost periodic solutions are defined by the formula: U(X,t)

a.

y ( U1 x

=

t

v 1 t ,..., u kx +

(1.2)

Vet)

Here the parameters U . and V . as well as U, V in (1.1) may accept both real and 3

J

k

e ( T l,...,~Q),T

...,

similarly = (Tl, T k to elliptic functions, has the property of being 2k-periodic in k-dimensional com-

complex values, and the function y plex-valued space C

(T)

y

=

a. . Namely there exist

29. complex-valued E-dimensional vectors

(vector periods) a , = (ajl,...,a . 1 and bj = (bjl,...,b . 3

3k

3k

)

j = 1 ,...,k which are

linearly independent of the ring of integers, and such that II Q Q y (T+b.) = y fa ) = y (T) for all j,k = 1, . . . , k . Without loss of generality we 3 k

1

2

S

.Yu . DOBROKHOTOV and

V . P . MASLOV

consider the vectors a . being real-valued and a , = 0, if j # i. Thus for real U . 1 ii 7 and V . solution (1.2) represents an almost periodic function in x and in t. Being 3

a limit (degenerate) case the solutions of the form (1.2) contain also multisoliton solutions: then the lengths of P. periods of function y

a.

(7)

are equal to infi-

nity . Cnoidal waves (1.1) appear to be the nonlinear analogue of plane waves, i.e. of the linear wave equations solutions of the form: v

=

A cos(kx

+

wt)

(1.3)

and they form such a narrow class among all solutions of the equation under consideration as some plane waves among all solutions of wave equations. Moreover, it is well known, that solutions of the form (1.3) exist only for equations with constant coefficients except some rare cases. At the same time various problem of mathematical and theoretical physics lead to equations with a small parameter in the highest derivatives and the approximate solutions of such equations can be constructed by means of asymptotic methods. One of the basic linear mathematical physics methods of such type is the WKB method which represents the solution as a sum of "distorted" plane waves:

or in the exponential form: U = $(x,t) exp(iS(x,t)/h).

Here the phase

S

(1.5)

(x,t) and the amplitude A(x,t) (or @ (x,t)) are smooth real-valued

functions and h is the small parameter. The analogous asymptotic solution in the nonlinear case due to the plane waves correspondence to the cnoidal ones has the form of a "distorted" cnoidal wave; 1 U = y (S (x,t)/h)

(1.6)

where y'(~) is the same elliptic function as in (1.1). First in case of nonlinear ordinary differential equations G.E. Kuzmak (1959) suggested to find asymptotical solutions in the form (1.4) and proposed the method of their construction, then G.B. Whitham (1968) made the same for equations with partial derivatives. In a great number of papers the construction method of solutions in the form (1.6)was further developed and improved, and its application to investigation of some real physical problems was considered. According to 1171 we shall call this method a nonlinear one. Rather full bibliography is available in 16, 8, 19, 20, 23, 24, 91. The solutions of plane waves or cnoidal waves form (1.31, (1.1) are self-similar: i.e. at any time T they can be represented as a single sinusoidal or cnoidal wave,

3

F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS

if at any time t < T they were represented in this form. The analogical statement does not hold for asymptotic WKB-solutions of the form (1.3) even in the linear case: as a rule at some critical time T and in some point (called focal) the cr the phase S(x,t) ceases to be a single-valued function. Then at times t > T cr asymptotic solution is constructed by means of a canonical operator r l 3 l and, generally speaking, has the form distinct from (1.3). In particular, outside the vicinity of branching points of function S(x,t) (focal points) at times t

>

T

cr

the corresponding asymptotical solution is represented as a sum: u = A (x,t) cos

1

s1 (x,t) h

~

...

+

sQ (x,t) + A a (x,t) cos ___ , e r 1

thus it is a multiphase asymptotic solution. Here the phases S

j

(1.7)

and the amplitudes

A j are again smooth functions. Note that the solutions of such type arise not only due to focal points, but they necessary arise in boundary value and mixed problems. Analogical multiphase solutions appear as well in nonlinear equations. The necessaty of their investigation was pointed out already in 1967 by Lighthill, Whitham and others. Whitham e.g. writes in c221: "An extended theory with possibility of more than one principal mode would be necessary. It is possible that such a theory could be developed and the interaction treatment for nearly linear modes would help in this connexion" ( p . 1 8 ) . Each summand in (1.7) is an asymptotic solution, thus (1.7) is the superposition

of "distorted" plane waves. Such a representation as a sum of separate solutions is possible only in the linear case, so the nonlinear generalization of the solution has the form 11,231:

where y

a(

...,T a )

so.(x,t)

s1 (x,t)

~

u = y

-(

,

.

.

.

I

h

)

'

(1.8)

is the same function as in (1.2), which holds for exact almost

T ~ ,

periodic solutions. The construction scheme of multiphase asymptotical solutions of the form (1.8) is practically the same (in any case for the principal term) as in case of the single phase solution (1.6). The lack of methods allowing to deter-

a

mine the function y ( T l , . . . , ~) prevented it from realization till the mentioned

a

above paper by S.P. Novikov was published, Nowadays the methods of constructing exact almost periodic solutions (1.2) are essentially developed, see e.g. the review and bibliography in [lo, 16, 241. The authors of this paper used the finite gap almost periodic solutions in the problem of reflection from the boundary for the generalized Sine-Gordon equation: [4,

5, 61:

S.Yu. DOBROKHOTOV and V.P. MASLOV

4

and in wave train existance problem for Korteveg-de Vries equation which was set by Lighthill and Whitham 1221. Independently, H. Flaschka,

G.

Forest and

D.W. Mc Laughlin C8l investigated the "distorted" almost periodic finite gap s o l u tions of equation (1.9). This paper deals with the results obtained in the wave train superposition problem for Korteveg-de Vries equation. First let us mention the following. So far we dealt only with asymptotical solutions with real-valued phases S.(x,t), but S.(x,t) can also be complex-valued 1

1

functions. In linear case the asymptotical solutions with single complex phase S have the form:

and boundedness of these solutions as h

-f

+O implies the inequality

ImStO

(1.11)

The asymptotic solution (1.10) is both fast oscillating (due to the factor exp iRe S/h) and fast diminishing as h equation

r

= {

the vicinity

(x,t)

r,

:

+

+O outside the set

r

defined by the

In S(x,t) = 01: Thus the function (1.10) is concentrated in

i.e. for each fixed x,t

E

r

m

U(x,t,h) = O(h

).

The solutions of

linear 'equations of form (1.10) concentrated in the vicinity of curves being Hamilton systems trajectories are well known and can be obtained by the model equations method 121 or by the complex canonical operator (complex germ) method 1141: the latter concerns the solutions concentrated in the vicinity of some kdimensional surfaces, k > 1 , or more complicated manifolds (see 113,141. In the nonlinear case solutions concentrated in the vicinily of curves in Rn were obtained in the following papers [3,141. Such asymptotical solutions analogical to (1.10) have the form:

1 u = y (S(x,t)/h)

(1.12)

where the complex phase S is a smooth function satisfying (1.11) and the elliptic 1

function y (T) (see (1.1)) is supposed to be bounded in the whole complex halfplane Im

T

2

0. It turns out the last condition defines among solutions (1.1) the

soliton (degenerate) solutions. Thus the asymptotical solutions of form (1.12) of nonlinear equations with complex phase are expressed in terms of soliton functions. Note that solutions (1.10) and (1.12) as well (1.4) are complex. To obtain the real-valued solutions of initial equation, one should set v = Re U = (of course, if the function

4 (U +

-

iS,/h

U) = 5 $ e

~

Li

p

.-is/h

(1.13)

a, complex conjugate to the solution U, is a solution).

In case of a real-valued phase S the function v again can be represented as a

5

FINITE GAP ALMOST PERIODIC SOLUTIONS single phase function

/$I

cos(S/h + Arg

$)

,

in case of an arbitrary complex

function satisfying (1.11) there does not exist such a representation, and (1.13) is a twe phase solution with phases S

1

= S

iSl/h

v = % ~ $ e

and

s

2

=

-

-S:

iS20h + + T e

(1.14)

Hence in linear case the real-valued asymptotical solution with the complex phase S(x,t) is always a multiphase (at least two phase) solution. The same is true for nonlinear equations: the real-valued asyrnptotical solutions are always multiphase, being expressed in terms of exact multisoliton (i.e. degenerate finite gap) solutions of the form (1.2). The multiphase solution of such form, both real and complex, were obtained for the Sine-Gordon equation by the authors of this paper, and for the equation describing a proper nondegenerate semiconductor as well: (1.15) In particular for the last equation the asyrnptotical solutions of Steklov-type problem, concentrated in the vicinity of stable closed geodesics on 252 are obtained in a convex compact domain R with the smooth boundary. Analogical solutions are obtained in a half-finite straight cylinder, and for equation (1.15) with more general nonlinear terms. In the latter case the soliton functions y

I!

(T,

...,7 9.)

can-

not be expressed in terms of elementary functions and are expressed as Dirichlet series. At last note that an exception among asymptotical solutions with the complex phase

S is the case Re S = 0, where the single phase asyrnptotical solutions can be realvalued. Such asymptotical solutions turning into shock wave or infinitely narrow solitons as h

-f

0 were obtained in 1151.

Another domain of finite gap solutions applications is based on the fact that these solutions form Lagrangian tori (generally speaking, in infinitely dimensional phase space), which can be quantized in quasi-classical approximation by the methods of 113,141. The phase space of periodic Toda lattice is finite dimensional and the quasi-classical quantization can be grounded strictly (see 1 6 1 ) .

2.

WAVE TRAIN INTERACTION PROBLEM IN LINEAR CA SE

Let us consider Korteveg-de Vries equation in the following form: Ut - 6UUx

+

2

h Uxxx = 0

,

x E R

,

(2.1)

were h > 0 is a small parameter depending on a number of physical constants. At first we consider the wave train interaction problem for the linearized Korteveg-

S.YU

6

. DOBROKHOTOV

and V . P. MASLOV

de Vries equation: V

2

t + h Vxxx

,

= 0

(2.2)

f o r which w e s e t t h e Cauchy problem w i t h f i n i t e f a s t o s c i l l a t i n g i n i t i a l v a l u e s : V(t,O

= $ ( x ) cos

kx

h

,

(2.3)

where k > 0 i s t h e wave number and $(x) i s a smooth f u n c t i o n w i t h t h e s u p p o r t belonging t o an i n t e r v a l i n R. The a s y m p t o t i c s o l u t i o n of problem (2.21, (2.3) i s e a s i l y o b t a i n e d by means of WKB-method, i t s p r i n c i p a l term h a s t h e form; V = $(x

+

2

3k t ) cos

3

-(

kx+k t h ) '

(2.4)

and a p p e a r s t o b e a p l a n e wave t r a i n . The sum of any two s o l u t i o n s i n form (2.4) c o r r e s p o n d i n g t o d i f f e r e n t wave numbers k l and k2 i s a l s o a s o l u t i o n of e q u a t i o n

(2.2).

L e t us c o n s i d e r s u c h a s o l u t i o n :

v

=

4

1

(X

+ 3k

2

1

3

t)

k x+klt

2 t $ ( x t 3k t ) cos(-2

)

COS(-

3 k2x+k t 2 h

I f

(2.5)

> k2 > 0 and t h e s u p p o r t s of smooth f u n c t i o n s $ ( x ) , $2 (x) belong t o d i s 1 1 j o i n t i n t e r v a l s i a O , b O ] and [ c d 1 r e s p e c t i v e l y , and d < a o . The s o l u t i o n ( 2 . 5 )

where k

0' 0

0

s a t i s r l e s the i n i t i a l conditions: VltzO

=

(XI

cos

k x

1

~

h

+

$ (x) cos

2

k x 2 h

-.

(2.6)

I t c o n s i s t s o f two d i s t a n t running t r a i n s of p l a n e waves till a c e r t a i n i n s t a n t of

t i m e . A s k l > k2 t h e t r a i n w i t h t h e impulse k l r u n s f a s t e r , t h a n t h a t w i t h t h e impulse k2

,

and c a t c h e s i t . They superposed f o r some t i m e . Then t h e t r a i n w i t h

t h e impulse k l r u n s ahead and t h e s o l u t i o n a g a i n c o n s i s t s of two d i s t a n t t r a i n s . Note t h a t t h e f u n c t i o n (2.6) can be r e p r e s e n t e d i n t h e form of an o s c i l l a t i n g function with a s i n g l e phase:

where $

0

,

S o a r e smooth f u n c t i o n s

0

(one s h o u l d s e t :

$ (x) =

tJ1(X)

+ $,(x)

,

(2.8)

and demand t h a t So = k x f o r x E [ a O , b O 1 and So = k x f o r x E [ c o , d o l ) . Consider 1 2 t h e Caushy problem f o r e q u a t i o n (2.2) i n case, when S o i s an i n c r e a s i n g f u n c t i o n e q u a l t o k x f o r x > a. and t o k x f o r x 4 d o . The p r i n c i p a l t e r m of t h i s problem 1 2 a s y m p t o t i c a l s o l u t i o n i s e x p r e s s e d i n terms of s o l u t i o n s of t h e f o l l o w i n g Hamilton i a n system:

FINITE GAP ALMOST PERIODIC SOLUTIONS

7

The set of these solutions has the form: 0 p = p (a)

,

0 2 q = -3(p (a))t

and for each t defines in the phase space the curve:

1

At

=

0 0 2 (p = p (a) , g = -3(p 1 t + a)

.

= min Before the critical time t the I curve A1 can be projected one-tocr t a 6P Pa one on the axis q and the asymptotical solution of problem (2.2), (2.7) has only 1 one phase. After the time tCr the focal points appear on the curve At, which is

not already projected one-to-one on the axis q , and outside the focal points vicinity the asymptotical solution can be represented in the form: 0

vo

=

z

j

@ ( a .(x,t))

1 1 cosl0 h 1-6p0pa(", (x,t))tl

JI

(S

0

( a .(x,t) 3

1

where C . is the index of the curve point l 1 3 1 with the coordinate u.(x,t) defined 1

3

by the equation q(a,t) = x. In case the function Q0 has the form (2.8) this sum consists only of two summands and (2.9) coincided with (2.5). Thus the wave train interaction problem can be regarded as a simplified variant of asymptotical solution construction problem for time greater than critical.

3.

WAVE TRAIN INTERACTION PROBLEM IN NONLINEAR CASE

Traditionally cnoidal wave (real-valued) solutions of Korteveg-de Vries equation (2.1) are defined in terms of elliptic Jacobi function dn(z,k) and have the form: u = b -

u

=

2 dn2(=,

b

4 2 -L

12 K(k) '

v=

t co,

k),

T =

ux + Vt ___ h '

J . 2 2(6b + i e ( k 2 - 2)) , 12 K(k) 2 3

where a, b, co, k are real-valued parameters, a > 0, 1 > k

2

0, K(k) is the com-

plete elliptic integral of the first kind. The function is completely defined by 1 - E(k)/K(k), the parameters a, b, c0 and k. It means value is equal to C = b - 6 where E(k) is the complete elliptic integral of the second kind. Thus u is a fast oscillating function (with the frequency

*

lih as h

+

0) with the mean value C.

S . Yu. DOBROKHOTOV and V . P .

8

MRSLOV

I t i s more c o n v e n i e n t t o e x p r e s s u i n t e r m s of t h e second d e r i v a t i v e of 0, where

0 i s a Riemann f u n c t i o n 116,241. Namely, l e t H , C , C0 and U b e r e a l p a r a m e t e r s , / H I < 1. Cnoidal wave s o l u t i o n s of Korteveg-de V r i e s e q u a t i o n can b e e x p r e s s e d i n

& f u n c t i o n as f o l l o w i n g :

u = y

1

ux + V t ( 7 B) ,

(3.1)

where B = -i I n H , 0 - f u n c t i o n i s d e f i n e d by means of t h e s e r i e s :

k=-m

k=-m

and V i s a f u n c t i o n of H ( o r B ) , U , C. The c o r r e s p o n d i n g formula f o r V ( H , U , C ) be g i v e n below.

will

(We do n o t g i v e t h e r e l a t i o n s between H , U , C and a , b , k because

we need o n l y ( 3 . 1 ) ) . Note t h a t t h e mean v a l u e o f t h e f u n c t i o n u i s e q u a l t o C and that u

-t

C for H

+

0.

The " d i s t o r t e d " f a s t o s c i l l a t i n g c n o i d a l wave can be r e p r e s e n t e d by formula ( 3 . 1 1 , where U, H , C, C

0

and V are f u n c t i o n s smoothly depending on x , t and, p r o b a b l y , on

the parameter h , h e [ O , l l .

This function i s a sel f - si m i l ar

asymptotical s o l u t i o n

o f Korteveg-de V r i e s e q u a t i o n on small time i n t e r v a l s , i . e . f o r s u f f i c i e n t l y small time t t h e a s y m p t o t i c a l s o l u t i o n of Caushy problem w i t h i n i t i a l c o n d i t i o n i n t h e form of a " d i s t o r t e d " c n o i d a l wave (and s a t i s f y i n g some c o n d i t i o n s on U , H and o t h e r s a c c u r a t e t o O ( h ) ) c a n be r e p r e s e n t e d i n t h e same form, of c o u r s e , U , H and o t h e r s depend on x ( s e e C9,231) i n t h e d i f f e r e n t way. The f u n c t i o n d e f i n e d by e q u a l i t y ( 3 . 1 ) , where

T =

s ( x ) and h

C ( x ) , S ( x ) , H(x) = exp

iB are 2

smooth f u n c t i o n s

w i t h supp H e [ a , b l , we s h a l l c a l l a t r a i n c o n c e n t r a t e d on t h e i n t e r v a l Ca,bl of " d i s t o r t e d " c n o i d a l waves o s c i l l a t i n g on a smooth background. N a t u r a l l y , two d i s t a n t c n o i d a l wave t r a i n s on a smooth background a r e d e f i n e d a n a l o g i c a l l y , w i t h t h e s u p p o r t H(x) b e l o n g i n g t o two d i s j o i n t i n t e r v a l s . The e x a c t s t a t e m e n t of t h e wave t r a i n i n t e r a c t i o n problem i s t h e f o l l o w i n g . One s h o u l d o b t a i n t h e e q u a t i o n ( 2 . 1 ) s o l u t i o n , s a t i s f y i n g f o r t = 0 t h e f o l l o w i n g condition:

where B = - 2 i I n H ( x , h ) and C ( x , t ) , S ( x , t ) , H a r e smooth f u n c t i o n s i n x h

E

c O , l l , w i t h H ( x , h ) b e i n g such t h a t I H ( x , h )

I

E

R and

< 1 'dx E R and s u p p H b e l o n g i n g t o

two d i s j o i n t i n t e r v a l s [ a , b l and [ c , d l . On a s u f f i c i e n t l y s m a l l t i m e i n t e r v a l t h e a s y m p t o t i c a l s o l u t i o n of problem ( 2 . 1 ) , ( 3 . 2 ) i s d e f i n e d ( s e e C231) a c c u r a t e t o O(h) by t h e same formula ( 3 . 2 ) , b u t h w i t h o t h e r (depending on t i m e ) f u n c t i o n s S , H and C , t h e f u n c t i o n s u p p o r t H b e l o n g i n g t o two d i s j o i n t i n t e r v a l s [ a t , b t l

and

9

F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS

depending on t i m e . The a s y m p t o t i c a l s o l u t i o n w i l l c o n s i s t of two d i s t a n t

[ct,dtl

" d i s t o r t e d " c n o i d a l wave t r a i n s on a smooth background, d e f i n e d by t h e f u n c t i o n C ( x , t ) . I f t h e i n i t i a l f u n c t i o n s S,

H and C are s u c h t h a t a s y m p t o t i c a l s o l u t i o n

a t time t t h e d i s t a n c e becr' cr becomes e q u a l t o z e r o , t h e n one of t h e

p r e s e r v e s t h e mentioned above form f o r time t < t

tween t h e i n t e r v a l s [ a t , b t l and [ c d 1 t' t wave t r a i n s c a t c h e s t h e o t h e r and t h e y s u p e r p o s e a s i t i s i n t h e l i n e a r c a s e . I n t h i s case f o r t i m e t > t

t h e asymptotical s o l u t i o n r e p r e s e n t a t i o n ( 3 . 1 ) does not cr h o l d . Namely, w e s h a l l c o n s i d e r i n t h i s s i t u a t i o n t h e problem ( 2 . 1 ) , ( 3 . 2 ) and show, t h a t f o r t > t t h e a s y m p t o t i c a l s o l u t i o n c a n b e o b t a i n e d by means of " d i s cr t o r t e d " f i n i t e gap s o l u t i o n s . The e x a c t c o n d i t i o n s on t h e i n i t i a l f u n c t i o n s S, H and C which d e f i n e t h e problem d i s c u s s e d a r e g i v e n below.

4.

MULTIPHASE ASYMPTOTICAL SOLUTIONS

According t o t h e i n t r o d u c t i o n w e s e a r c h f o r t h e a s y m p t o t i c a l wave t r a i n i n t e r a c t i o n problem s o l u t i o n i n t h e form o f t h e series: S

v =

yk(h, x

k=O

,

t)hk

(4.1)

where y ( ~ , x , t ) ,T = (T ,..., T ~ ) ,S = ( S ( x , t ) ,...,S L ( x , t ) ) a r e new unknown k 1 1 smooth f u n c t i o n s , x E R, t E R , T E Rn; yk b e i n g 2 ~ - p e r i o d i c f u n c t i o n s i n each argument T l,...,~k. (We c o n s i d e r t h e method of c o n s t r u c t i o n t h e r n u l t i p h a s e solut i o n s ( 4 . 1 ) w i t h a n a r b i t r a r y number of p h a s e s , though i n t h e wave t r a i n i n t e r a c t i o n problem i t i s s u f f i c i e n t t o c o n s i d e r t h e case o f two p h a s e s : L: = 2 ) . I n s e r t i n g t h e f u n c t i o n ( 4 . 1 ) i n t o e q u a t i o n ( 2 . 1 ) , d i f f e r e n t i a t i n g and e q u a t i n g t o z e r o t h e c o e f f i c i e n t s of e q u a l powers o f h , w e o b t a i n t h e f o l l o w i n g system of relations: (4.21 (4.3) where A and

Fk

= S

a

+ aT 1

- * *

Skt

+

a 5

t

-

B = s

lx

a

aT1

+

...

+

sa.x

a

%-a. '

a r e p o l y n o m i a l s i n f u n c t i o n s ~ ~ , . . . , y ~S j-t ,~ ,S j x and t h e i r d e r i v a t i v e s .

I f the functions Sl , . . . , S k

and yk s a t i s f y t h e r e l a t i o n s ( 3 . 2 ) and ( 3 . 1 ) up t o t h e N

o r d e r N i n c l u s i v e l y , t h e f u n c t i o n uN = k$o y k ( S ( x , t ) / h , x , t ) h k s a t i s f i e s t h e equaN

t i o n ( 1 . 1 ) a c c u r a t e to O(h ) tions $(x,t,h), (x,t)

I

E

R2

X,t'

.

(By O(hN) we d e n o t e , as u s u a l l y , t h e smooth funch t (O,ll,

f o r which t h e f o l l o w i n g estimate h o l d s :

) . The v a r i a b l e s x , t s e r v e as p a r a xtt ( 4 . 3 ) and t h e e s s e n t i a l i n d e p e n d e n t v a r i a b l e s i n t h e s e r e l a t i o n s

l $ J ( x l t l h ) < c o n s t ( l ( ) h on each compact l( c R meters i n ( 4 . 2 ) ,

10

S.YU. DOBROKHOTOV and V . P . MASLOV

are Tl,...,~p..

All the relations ( 4 . 3 ) are linear uniform equations with partial

derivatives in variables ' r l , vatives in

,

T l,...,~Q

..., T~ ,

(4.2)

is also an equation with partial deri-

but a nonlinear one with constant coefficients of

(The number of unknown functions in the system of first relations ( 4 . 2 ) ,

Tl r * *

tTQ.

(4.3)

is

more than N, hence it can not be solved without any additional conditions. Thus the function yo should be 2n-periodic in should be bounded for each fixed x,t). Consider equation ( 4 . 2 ) .

T

~

..., , T~

and the functions y lf...'YN+l

Though the number of independent variables

T

in this

equation is equal to S, in fact there are only two independent variables

n

and 5

defined by the equalities:

Thus by changing the variables in ( 4 . 2 ) T .

1

= T . ( ~ , Q )= S .

1

n +

It

:

w(6,rl) = y (T ( 5 , r l ) ,..., r p . ( S , r i ) ) 0 1

S . 5 , j = 1,. ..,f,: IX

(4.4)

we again obtain the Korteveg-de Vries equation, but without the small parameter h:

w

n

- ~ W W

5

+ W

155

= O

Due to 2n-periodicity of yo in argument

(4.4')

the solution w is a conditionally pel riodic function for each fixed (x.t). Thus the construction problem of multiphase T.

asymptotical solution of Korteveg-de Vries equation is reduced in the principal term to the already solved problem of obtaining the conditionally periodic solution of the same equation. We give the explicit formulas in 0-functions for these solutions, obtained by A.P. Its and V.B. Matveev (see r 1 6 , 2 4 1 ) , which generalize the solutions of the form (3.1), ( 3 . 1 ' ) .

Consider the parameters which define them

to be functions in x and t. Let E (x,t),...,E2p.(x,t)be real-valued functions such that for any fixed (x,t) 0

holds max(Ezj+z,E2j+l) < min(E2j,E2j-1). Consider a set of hyperelliptic Riemann surfaces

r2&

w 2 = '2Q+1'

of the kind S given by the algebraic curves 2p. = Il (2-E.),which present for each (x,t) two complex planes 'ZP+l j=o 1

[E2p.-2rE2S-3]t*.-r[E2' E 1 1 1 glued on cuts, which are the intervals iE2p.~E2Q-llr and glued on the ray [EO,ml in the usual way. We choose from the set r on each

...,

b such that i) the intersecr Xlt the basis of cycles al,..., aS, bl' a. tion matrix of this basis has the form a 0 a . = hi 0 b . = 0, a. 0 b = Aij: ii) i i 1 1 1 and it surrounds each cycle is situated on the upper sheet of the surface r x,t only one cut which connects the points Ezj, E2j-1; iii) each cycle b . is situated surface

on both sheets of

r

it crosses only the cuts [E

1

,E2j-ll and [EO,ml, transi-

21 x,t' tion from the lower sheet to the upper and backward happens at the intersection-

points of b

1

with the cuts [E

1 and [Eo,m) respectively when moving along

211~21-1

11

FINITE GAP ALMOST PERIODIC SOLUTIONS

the cycle b . . One moves along the cycle a . clockwise, if E 3

3

clockwise otherwise; along the cycles b . clockwise. On each surface dW

j

=

E 2j-l, and counter




0, in case under

consideration we have Re B . = 0 for k # J and Re B . = ~i (J-m), if m inequali3k 11 ties E < E and (j-m) inequalities E > E ,i-j hold. 2i 2i-1 2i 2i-1 By means of the matrix of B-periods we construct the Riemann 0-function of dimension 2 , which is the generalization of (3.1 ' )

i O ( T ~ B )= C exp(2 where

T =

t

( C ~ , . . . , T ~k) ,=

t

:

+

Bk,k >


1. Analogically to (4.17) we have for small H . , = exp(i B . ,/2), j = 1 ,..., e : I1 I1

,..., T P )

yO(T1 e

= C

COS(Tl+C01)+...+4U~HPP COS(Tk+CoP) + O(H 2 +... + 4U1Hll 2 11

+ H P2e )

and hence, as it was in one-dimensional case, the functions H . . j = 1 , ...,P Ill have the sense of "nonlinear amplitudes". Nevertheless in this case there exists such a set of values iB . H . = e 1k/2, P 1 k > j 2 1, that define the "interaction' 0 s the components L Ik with different phases of the solution Y ~ ( T ~ ~ . . . , T ~For ) . P = 2 there exist four relations (4.12) by solving which with respect to V following relations:

G =

9 (UIH,C) , H . 0 lk

=

1'

G, H12 one can obtain the

(4.20)

$jk(U,H,C)r. ]=1,2, k=2 '

where f i = (Hll,H12),U = (U ,U ) and $ and 0 are functions. 1 2 jk The relations (4.19) are analogous to (4.15) and it is naturally to call them the multidimensional dispersion relations. Distinct from one-dimensional case ( V . = 1) the relations in functions $ , can be obtained only in the form of series in terms 1 of nonlinear amplitude powers H I under the condition U . # 0, IU1 # IU2 These 1 series coefficients expressions as well of the series, which express H12 and G by

I

1.

U, f / and C, can be obtained by a standart methods using the equations (4.12) in

the following form:

(equation (4.131, n

=

(%,O), n = (O,$))

,

15

FINITE GAP ALMOST PERIODIC SOLUTIONS

-

(U1 H12 =

U2)

(U1 + U 2 )

U4 1

G =

2

R

2 +

2

12

+ U2)

U1U2(U1

4

+ H2 U 2 ) =

(4.221

2

(4.23)

Ro

(equation ( 4 . 1 3 1 , n = ( 0 , o ) Here R . , j = 0 , 1 , 2 , R 1 2 3

-

are polynomials in U , C , V .

I'

1

with coefficients analitically

< 1 are such that the expansion of depending on H . and H . , 0 < l H j k l < m, l H j j ik 33' R . in a series in terms of the powers of ff begins from the third degree, and of

3 R12

-

from the second degree. The corresponding formulas for these series coeffi-

cients and their convergence proof we de not need further, and we do not give them here. Not only, that it follows from ( 4 . 2 1 ) , in particular, that the disper-

1 1 .3 )

sion relations accurate to O ( ff

do not contain the phases interaction: every

dispersion relation coincides accurate to O( I If I

3

)

with the one-dimentional disper-

sion relation, has the form ( 4 . 1 5 ) and contains only the values with the single index j . From this fact it follows immediately a weeker statement that the phases S.(x,t) satisfy the same Hamilton-Jacobi equation ( 4 . 1 0 ) in the (linear) limit 1

H

-f

0. For Q > 2 the investigation of the relation system ( 4 . 1 3 ) becomes essen-

tially complicated, it is overdeterminated in the sense the relations ( 4 . 1 9 ) and ( 4 . 2 0 ) , which express V j , B j k ,

j # k and G by U. and B . . can already be obtained 1 31'

from any ( P . + l ) Q / 2 + 1 relations of ( 4 . 1 3 ) , while the total number of relations ( 4 . 1 3 ) is equal to 2'.

The point which teh relations from ( 4 . 1 3 ) one should take in order

to obtain the equalities ( 4 . 1 9 ) satisfying the whole system ( 4 . 1 3 ) (and existing according to lemma 1 ) is yet open. Nevertheless, when we take the small values of iB,. f o r these relations, it is easily seen from lemma 1 that one can H . ,= e I3

'"'

take the relations with the indexes n = (n

,..., nQ )

such that 0 5 nl

+...+ n2

5

1.

The latter can be expressed in the form ( 4 . 1 9 ) , ( 4 . 2 0 ) by means of series in terms of the powers of H l l , . . . , H L Q

5.

analogically to the case P. = 2 .

Higher approximations

iB = e k 1 / 2 ) and Col ,...,Coe, which (or H kj define accurate to O(h) the asymptotical solution ( 4 . 1 ) , cannot be obtained from

The functions S l ,

...,S Q

the system ( 4 . 9 )

(or ( 4 . 1 3 ) ) , because the number of unknowns is greater than the

and E

,...,E 2 Q

number of relations. One should regard the equations for higher approximations in order to get the full system of relations. As we already know, all of them are linear non-homogeneous equation with 2n-periodic with respect to

..., 'T

T ~ ,

16

S.Yu.

DOBROKHOTOV and V . P . MASLOV

coefficients. The necessary existence condition of 2n-periodic with respect to variables

...,T~

T ~ ,

solution of each of these equations if the orthogonality con-

dition of the kernel right-hand parts F joint to the operator the operators L and

L

of the operator L

(see (4.3)). In the variables

-* L , evidently, have

a a a' L=--~w-+-,L

^*

1

','an

=

A

and

a

-

6y B + B3, ad-

/aE,

'

(see (4.4))

the form

a a' 6w - + -

= - _a the leading role in the investigaa53 an dE at3 tions of these operators properties has the function xK(n, 0 a r b i t r a r y s m a l l , independefit of h , such t h a t

F I N I T E GAP ALMOST P E R I O D I C SOLUTIONS

23

Under the assumptions 1) - 3) for any N 2 0 when t E [O,t 1: i) There 0 N N exist smooth functions S S : , CN, G and HN = exp iB. j,m = 1,2 satisfying 1' im Im'N+l the problem (4.9), (4.13), ( 6 . 6 ) , (7.1) accurate to O(h 1 ; ii) The function

THEOREM.

where yk are the solutions of equations (4.2), satisfies the problem (2.11, (3.2) N+ 1 accurate to O(h 1. Consider the behaviour of the solution (7.2). Assume that in addition to 1) - 3) the condition holds: 4) on the time interval t


dt, which t' t does not intersect with [c , d 1. Outside these intervals we have, Uo = CN (x,t,h), t t and in each of these intervals the function uo presents a cnoidal wave train on

-

the smooth background. At an instant of time tl the point at reaches the interval [ct,dtl and at the points x from [a ,b 1 n cc ,d 1 in the solution two phases t t t t arise, i.e. the wave train interaction occurs. Then after some time t > tl the

- -

train with the phase SN leaves behind the train with the phase S';', and the inter2 vals [ct,dtl, [at,btl became disjoint, [at,b 1 is situated on the left of [c d 1. t t' t Again is the solution describing two distant wave trains on the smooth background. Conditions l), 2) of THEOREM will hold e.g. in case when

Remark.

S =

k x, for 1

[ao,bol, S = k x, x e [co,dol (i.e. when the wave trains almost coincide with 2 plane cnoidal waves). Assumption 4) then holds automatically, if k l > k2. x

E

The following statement implies the proof of the Theorem: LEMMA 4.

Let conditions l), 2) hold and the function H(x,E) smoothly depend on

E > 0 ( i < E ) such that (HI .: c const. Then for t 6 r0,t 1 and x c R 0 (n) (n) (n) (n), c(n) , n =O O,l,. that there exist such smooth functions S m " m I m I m' -M S(n) for any integer M > 0 the functions S (Xtt)En, n=O

parameter

-M 1 3 =~

M

c

n=1

..

-M (x,t)En, m=1,2, til2 =

M

c

n=O

fz)

n -M (x,t't , G

M =

c ) ;9

n=2

(x,t)En,

satisfy the problem (4.9), (4.13), .P = 2 ( 6 . 6 ) , j = 1,2,3, (7.1) accurate to

. . DOBROKHOTOV

24

S YU

and V . P

M O(E ) . If the expansion of such a solution exists

MASLOV

n terms of the powers of E ,

this expansion is unique. In order to prove this Lemma we insert the expansions of functions S etc. into m the equations mentioned and using a standard method we obtain from (4.21)-(4.23), (6.9)-(6.10) the recurrent system of equations for the expansions coefficients. The solvability of this system which coincides with (4.18), (6.11) in the zero approximation follows from conditions 1 1 , 21. To prove the Theorem we use the results of items 4-G setting

CM, M

BN -M CN = exp i-lk = H , 2 Ik' coincides with (3.2).

=

"/A

1

t

-P.l SN = S . ,

I

1

1 and note that for t = 0 the function (7.2)

In conclusion note that we did not actually regard the sufficient conditions of

equation (4.3) solvability in periodic functions, which demands additional complicated investigations and is based on considering of eigen functions of the ^* -1 operator L which have the form ( x ( v , S , < ) ) exp(i!X(v,5,5)dnl .

REFERENCES

111

Ablowitz M.A., Benny D.Y., The evolution of multiphase modes for nonlinear

121

Babich V.M., Buldyrev V.S., Asymptotic methods in short waves problems

131

Dobrokhotov S.Yu., Maslov V.P., Spectral boundary problem asymptotics for

dispersive waves, Stud. Appl. Math. 49 N3 (19701. ("Nauka", Moscow, 1972) (in Russian). non-linear equation of semiconductor, Dokl. &ad.

Nauk SSSR 243, N4 (1978)

897-900 (in Russian). 141

Dobrokhotov S.Yu., Maslov V.P., Solution asymptotics of mixed problem for 2 non-linear wave equation h O u + ashU = 0 Uspekhi Math. Nauk 34, N3 (1979)

151

Dobrokhotov S.Yu., Maslov V.P., Boundary reflection problem for equation 2 h Ou + ashu = 0 and finite gap conditionally periodic solutions, Funkz.

225-226 (in Russian).

Analis i Primenen. 13, N3 (1973) (in Russian).

C6l

Dabrokhotov S.Yu., Maslov V.P., Finite gap almost periodic solutions in asymptotic axpansions, Modern Probl. of Math. 15 (VINITI, Moscow, 1980) (in Russian).

171

Dubrovin B.A., S.P. Novikov, hypothesis in 0-functions theory and non-linear equations of KdV and KP type, Dokl. &ad.

Nauk SSSR 251, N3 (1980) (in

Russian). C8

I

Flaschka H., Forest G., McLaughlin D.W., Multiphase averaging and the inverse spectral solution of KdV, Preprint LA-UR 79-365, Los Alamos Scintific Laboratory.

F I N I T E GAP ALMOST PERIODIC SOLUTIONS

25

191

Karpman V.I., Non-linear waves in dispersive media, ("Nauka",Moscow, 1973).

1 101

Krichever I.M., Algebraic geometry methods in non-linear equations theory,

Clll

Kuzmak G.E., Asymptotical solutions of nonlinear differential equations of

Uspekhi E4ath. Nauk 32, N6 (1977) 183-208 (in Russian). second order with variable coefficients, Priklad. Math. i Mech. 23, N3 (1959) 515-526 (in Russian). 1121

Luke J.C., A perturbation method for non-linear dispersive wave problems, Proc. ROY. SOC. A292, N 1430 (1966) 403-412.

C131

Maslov V.P., Operational methods ("PTIR", Moscow, 1976).

1141 Maslov V . P . ,

Complex WKB-method in non-linear equations ("Nauka", Moscow,

1977) (in Russian).

El51

Maslov V . P . ,

Tcupin V.A., &-like generalized according to Sobolev solutions

of quasi-linear equations, Uspekhi Math. Nauk 34, N1 (1979) 235-236 (in Russian).

C161

Matveev V.B., Abelian functions and solutions, Inst. Theor. Phys., Univ. Wroclaw, Preprint N 373, 1976.

C171

Miura R.M., Kruskal M.D., Application of non-linear WKB-method to KdV equation, SIAM Appl. Math. 26, N2 (1974) 376-395.

C181

Novikov S.P., Periodical problem for KdV-equation, Funkz. Analis i Premenen. 8, N3 (1974) 54-66 (in Russian).

1191 Scott A.C., Active and non-linear wave propagation in Electronics (Wiley Interscience, New York, 1970). r201

Scott A.C., Chu F.Y., McLaughlin D.W., The soliton: a new concept in applied

C211

Whitham G.B., Non-linear dispersive waves, Proc. Roy. SOC., A283, N 1393

science, Proc. IEEE 1 (1973) 1443. (1965) 283-291. c221

Whitham G.B., Variational methods and applications to water waves, Proc.

C231

Whitham G.B., Linear and non-linear waves (Wiley Interscience, New Yourk,

1241

Zakharov E., Manakov S . V . ,

Roy. SOC., A299, 6 (1967). 1974). ("Nauka",Moscow, 1978).

Novikov S.P., Pitaevsky L.P., Theory of solitons

This Page Intentionally Left Blank

A N A L Y T I C A L AND NLI'IZRTCAL APPHOAClfC.? TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S. Axelsson, L.S. F r a n k , A . v a n d c r S l u i s ( e d s . ) @ N o r t h - H o l l a n d P u b l i s h r n y Company, 1981

PERIODIC SOLUTIONS NEAR EQUILIBRIUM POINTS OF HAi'lILTONIAN SYSTEMS

Hans Duistermaat Mathematisch Instituut Rijksuniversiteit Utrecht Utrecht, The Netherlands This is a report on work in progress in collaboration with Richard Cushman. Consider a Hamiltonian system with n degrees of freedom:

m

where F is a smooth ( = C

or real analytic, everything which follows

holds in both categories) real-valued function of the 2n coordinates z = (q,p). Also F may depend smoothly on a finite number of parameters. The vectorfields (in the z-space) on the right hand side of (1) will also be denoted by v(z), it will be assumed throughout that the origin is an equilibrium point of the system ( I ) , that is

-

v(0) = 0, corresponding to dF(0) = 0. The substitution of variables z = E . Z leads to the equation (2)

dz dt

-

E-l

5

.V(EZ) = A(z)

+

higher order terms in

E,

which is the "scaled version" of the system (1). Here A = Dv(0) is the linear part of v at the equilibrium point, the correspondinq linearized system ((2) for E=O) is eaual to the Hamiltonian system defined by the quadratic part of the Taylor expansion of F at the origin. Its solutions are given by =

etZ(z 0 ) )

r

periodic with period w if and only if E N

=

def

ker (ewA-I) = (kEz

E2Tiik,w'IR.

Here EX denotes the eigenspace of A for the eigenvalue X and the suffix IR means that one has to take the real part of the corresponding complex vector space. Note that the dimension of N is always 37

28

H . DUISTERMAAT

even, because the eigenvalues of A occur in complex conjugate pairs. A classical theorem of Lyapounov [ 5 1 states that if for w = w 0 , v = vo the matrix A = Ao has only one complex conjugate pair of eigenvalues in Z -2.rri/w0(each having algebraic multiplicity equal to 1) then the non-linear Hamiltonian system ( 1 ) with v near vo has a family of periodic solutions with period w (depending on the solution) close to w o and filling up a smooth 2-dimensional manifold P through the origin. At the origin the tangent space to P is equal to N and the orbits in P are concentric around the equilibrium point. Also the family depends smoothly on the parameters in the Hamiltonian function F , reflecting the fact that the theorem can be proved by a direct application of the implicit function theorem (see for instance [31).

However, if some eigenvalues of A = A o on the positive imaginary axis are rationally related, called the case of resonance, then for a suitable choice of w = u o the dimension of N is higher than 2. Nevertheless the periodic solutions of period o close to w o of ( 1 ) in the generic case will fill up a set which looks like a 2-dimensional algebraic variety in a (dim N)-dimensional space, having some complicated singularity at (resp. near) the origin for v = vo (resp. for v near v 0 ) . (The genericity assumption is essential here, because in the linear case the set of periodic solutions even has a higher dimension.). It is our aim to make this statement as precise as possible, but let us first mention some results obtained before. For two degrees of freedom and purely imaginary eigenvalues with a ratio 2:l it had been shown in [ 3 1 that generically one gets 3 families of periodic solutions, one filling up a smooth 2-dimensional manifold through the origin, tangent to the plane of shortestperiodic solutions of the linear system, and the other two having a cone-like singularity at the origin, the tangent cone at the origin not being contained in a 2-dimensional plane. The result itself indicates that in this case the implicit function theorem cannot be applied directly but only after a suitable blowing-up procedure, leading to the dividing out of a factor from the periodicity equation which caused the degeneracy at the origin. A description of the situation for other ratio's k : Q ( k , Q E Z) is given in Henrard [41. The analysis in [ 3 1 , 1 4 1 was incomplete because it did not treat the perturbation to eigenvalue ratio's close to, but different from k:Q. The Lyapounov theorem then predicts only two families of periodic solutions (of nearby period), filling up smooth

29

HAMILTONIAN SYSTEMS

2-dimensional manifolds through the origin with complementary tangent spaces at the origin, and the interesting question is how the bifurcation to the singularity at exactly k:R takes place. This has been analyzed by Schmidt [ 9 1 . His essential tool is the use of the Birkhoff normal form which allows to recognize a factor in the equations which vanishes of order k+R-2 at the origin (also after using a blowing up procedure). After dividing out this factor the implicit function theorem can be used to obtain existence of certain families of periodic solutions which appear to be attached to the Lyapounov families outside the origin if the eigenvalue ratio is different from k:R and drawn into the origin (forming singularities there) if the eigenvalues ratio passes through k:R. In [ 9 1 the qeometry of the attaching of the additional families to the Lyapounov ones has not been described in detail and a still more complicated question is how all the families bifurcate at the origin if the eigenvalue ratio passes through k:R. It is our purpose to answer these questions in the following very precise way. To begin with, we allow an arbitrary number n of degrees of freedom, of which only 2 resonate in the sense that

(5) dim N o

=

4

if

0 0

No = Ker(ew A -I)

and Ao NO has 2 eiqenvalues on the positive imaginary axis, each of (6)

I

algebraic multiplicity equal to 1 and with ratio k:L, k # Q, k,R being integers without a common factor.

Then we have the following Theorem. (I) There is a 4-dimensional smooth manifold N through the origin (depending smoothly on the parameters LI in F) with tangent space at 0 equal to N, such that every periodic solution of (1) close to the origin and with period close to w o is contained in N.

(Note that N is not assumed to be invariant under the flow!).

(11) Under a genericity assumption for the coefficients of the Taylor expansion of F up to the order k+R there is (IIa) A standard Hamiltonian function Gu in 2 degrees of freedom 4 which is a polynomial in w EIR of degree k+L depending linearly on at most 4 real parameters u. (IIb) A smooth mapping p a local diffeomorphism

H

0

0

(defined near p = p , ~ ( )p = 0) and from W4 to N ( u ) depending smoothly on p , u(p)

H. DUISTERMAAT

30

such that for 1-1 close to '1the set of periodic orbits of ( 1 ) near the origin and with period near w o is equal to the Q'-image of the set of periodic orbits of the Hamiltonian system is given a s the set of w E IR4 where:

Gul

u

= U ( U ) ~which

0

(7) grad Gu (w) is a multiple of grad G2(w). Here

G:

denotes the quadratic part of the Taylor expansion of Go at

the origin.

(111) The standard Hamiltonian function Gu can be taken as follows (see (11) I (13), (16) below). By a linear canonical change of coordinates and a suitable time scaling (including time reversal) the quadratic part of F o on N can be brouqht into the form (8)

0

G 2 = ilpl

+

kp2

, k,R

Zwithout common factors, 0

E

il




I

0, resp. k

.

3 = 1,2.

0 , write

( 1 0 ) p 3 = Re(ql+ipl)k-(q2-iP2)Qlresp. p 3 = Re(ql+iP1)- k - (q2+ipi)Q

For Ikl + il ( 1 1 ) G'

>

5 we can take u E IR3 and

pl+kp2+apl+2 2 (b+v2)P ~ P ~ +2 C P ~2 + ~ ~ P ~ P ~ + P ~ .

= (il+u,)

Here a,b,c are fixed real number which can be chosen arbitrarily within the components of the set determined by the inequalities: (12) a

ka

=

def

-

Rb # 0, y

Lc

=

def

-

kb # 0, k a

-

Ry

0 , ka

+

Ry # 0 .

under the conditions (14) y

0, 3a

( 1 5 ) 15a2

-

For k = '2, ( 1 6 ) G"

6ay

-

y # 0 , 3a

-

y2

$2

+

y

#

0 and

0 if k = + 3 , no extra condition if k = - 3 . 2

R = 1 we can take u E IR

= (R+ul)p

I

+kp2+ap1+2(b+v2)P1P2+CP2+P3 2 2

31

HAMILTONIAN SYSTEMS

under the conditions (17) y # 0 , 2a

-

y # 0 , 2n

+

y # 0.

The point is that using polar coordinates in the (q ,p.)-planes, j i the set where (7) holds can easily be analyzed and the theorem says that for the original Hamiltonian system the set of periodic orbits is equal to the image of this set for v = v(’) under a smooth embedding @’

depending smoothly on the parameters p , w(p) also

depending smoothly on 11. The proof of the theorem consists of a combination of the following techniques i)

The Weinstein-Moser method [71 for the reduction of the search

for periodic solutions to the problem of finding the pointswheredH 0 is a multiple of dH2. Here Hw is a smooth function on dim N variables, H; is the quadratic part of the Taylor expansion of H 0

W

.

One also gets that HV is invariant under the periodic flow of the 0 0 Hamiltonian system defined by H2, that is HV and H2 Poisson-commute. If dim N = 4 this means that the Hamiltonian system defined by HV is completely integrable. ii)

The Birkhoff normal form theorem which asserts that by a

canonical change of coordinates (depending smoothly on the parame0 ters) F can be made to Poisson-commute with F 2 up to any order one likes. As a consequence, the function Hw in i) can be identified with FV up to any desired order. IN

iii) By the theory of stability of functions of Mather in the equivariant version of Wasserman [lo] one can, by a diffeomorphism (depending smoothly on the parameters, being e uivariant with respect 0 G to a to the flow of the Hamiltonian system G 2 = H$ and mapping :

8

function of G 02 ) , combined with replacing H’ by a suitable function of H’ and G 02 , bring H’ into the standard form G V ( ’ ) . Here it is essential that the condition that the change of coordinates is canonical is dropped, because the Birkhoff normal form contains infinitely many invariants, whereas here we end up with at most 4

parameters in the normal form.

A detailed proof, which needs much more space, will be published elsewhere. Instead, let me close with some additional remarks.

32

H

. DUISTERMAAT

Remark 1. In the analytic category, Brjuno [ 2 1 has the statement that the Birkhoff normal form converges on a set of periodic orbits determined by an equation of the form (7). This can be considered as a paraphrase of the Weinstein-Moser method. However, [2] deals also with quasi-periodic solutions, suggesting that the Weinstein-Moser method should have an extension to quasi-periodic solutions as well. Remark 2 . The techniques i) - ii) can also be used to obtain strong asymptotic results (for long time intervals) about all solutions near the equilibrium point (not only the periodic ones). This is closely connected with the averaging method, see Sanders [ 8 1 . Remark 3 . The techniques i) - iii) also work for more degenerate Hamiltonians, and if more than 2 degrees of freedom are in resonance. However, the algebra of standard forms becomes rapidly more complicated. For a description of the 2:l:l resonance (with positive definite F 2 ) see v.d. Aa and Sanders [l]. Remark 4 . The case of an eigenvalue on the positive imaginary axis with multiplicity 2, but with dim N = 2, is very interesting because it describes the transition from an elliptic to a hyperbolic equilibrium point. In this case ' A has a non-zero nilpotent part on the corresponding 4-dimensional generalized eigenspace, so the theorem of Moser [ 7 1 does not apply directly. However, we are convinced that a suitable variant also works in this case. An example is given by the Lagrange equilibria in the restricted 3-body problem for a special value of the mass-ratio. Van der Meer [ 6 1 has computed the relevant part of its Taylor expansion and checked that it is nondegenerate. Remark 5. If one prescribes the period w of the sought periodic solution then it is easy to show the existence of w-dependent diffeomorphisms which bring the periodic solutions into normal form. Oneof our basic problems was that we wanted an w-independent diffeomorphism on a full neighborhood of the origin which brings the whole family of periodic solutions into normal form at one stroke.

33

HAMILTONIAN SYSTEMS

REFERENCES [ 11 Aa, E. van der and Sanders, J., The 1:2:1 resonance, its

periodic orbits and integrals, in 'Asymptotic Analysis, from Theory to Application', ed. F . Verhulst, Lecture Notes in Plath. 711, Sprinqer-Verlag, Heidelberg 1979.

[ 21 Brjuno, A.D., Integral analytic sets, Dokl. Akad. Nauk SSSR

220

(1975), 1255-1258 - Soviet Math. Dokl. 16 (1975), 224-258. See also his Preprints 97, 98 of the Inst. Appl. Math. Acad. Sci. USSR, Ploscow, 1974.

[ 31 Duistermaat, J . J . , On periodic solutions near equilibrium points

of conservative systems, Arch. Rat. Mech. Anal. 160.

5

(19721, 143-

[ 41 Henrard, J . , Lyapounov's center theorem for resonant equi-

librium, J. Diff. Eq.

14

(1973), 431-441.

[ 51 Liapounoff, A.A., ProblZme gi?nSral de la stabiliti? due mouvement,

Ann.of Math.Studies

11,Princeton

University, 1947. (Original 1892)

[ 61 Pleer, J.C. van der, On the computation of 4th order coefficients

of a normalized Hamiltonian at 1:l resonance, Preprint, Utrecht 1980.

[ 71 Moser, J . , Periodic orbits near an equilibrium and a theorem by

Alan Weinstein, Corn. Pure Appl. Math.

2

(1976), 727-747.

[ 81 Sanders, J . A . ,

On the theory of nonlinear resonance, Thesis, Utrecht 1978. See also: Are higher order resonances really interesting?, Celestial Mechanics 16 (1978), 421-440.

[ 91 Schmidt, D . S . ,

Periodic solutions near a resonant equilibrium of a Hamiltonian system, Celestial Mechanics 2 (1974), 81-103.

[lo] Wasserman, G . , Classification of singularities with compact abelian symmetry, Regensburger Math. Schriften I, 1977.

This Page Intentionally Left Blank

A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS 1 N A N A L Y S I S S . A x e l s s o n , L . S . Frank, A . van d e r Sluis [ r r i s . ) @ N o r t h - H o l l a n d Pub1 i s h i n q C o m p a n y , 1981

ASYMPTOTICS OF ELEMENTARY SPHERICAL FUNCTIONS by Hans Duistermaat ldathematisch Instituut Rijksuniversiteit Utrecht Utrecht, The Netherlands

This is a report on work in progress together with J.A.C. Kolk and V.S. Varadarajan, in continuation of [ l ] . 1. Symmetric spaces of negative curvature.

The most fruitful approach to the analysis of symmetric spaces is via group theory. S o for us a negatively curved symmetric space S is a space on which a noncompact connected semisimple Lie group G acts transitively and such that the stabilizer of a point of S is a maximal compact subgroup K of G. That is, S may be identified with K\G and the space of functions (resp. distributions) on S with the space of left K-invariant functions (resp. distributions) on G. A continuous linear operator 4 : Cm(S) D' ( S ) which commutes with the (right-)action of G on S is always of the form +

c"(K\G) 3 f

(1.1)

;a *

f E c"(K\G)

where is a uniquely determined compactly supported distribution on G which is both left and right K-invariant. A left and right Kinvariant distribution on G is called spherical and the space of compactly supported spherical distributions on G will be denoted by E'(K\G/K). Note that 4 actually is continuous linear: Cm(S) Cm(S), C:(S) C:(S) and extends to continuous linear mappings: D' ( S ) D ' ( S ) , E' ( S ) E ' ( S ) . Also that E ' (K\G/K) is an algebra with respect to convolution, the convolution corresponding to the composition of the corresponding operators on S. +

+

+

If

+

= TeG, resp.%

OJ

=

TeK denote the Lie algebra's of G, resp. K,

then let 4 be the orthogonal complement of

k

in

bd

with respect to

the Killing form (1.2)

:

K

Because on which

(X,Y)

+

Tr(ad X

0

ad Y ) .

is non-degenerate and k is a maximal linear subspace of

K K

is negative definite,g = 35

k 86 and

K

is positive o n h .

,

36

H. DUISTERMAAT

Furthermore the map (x,X) x.exp X is a diffeomorphism: K x /3 G and w : x.exp XI-+ x.exp -X ( x E K, X E n ) defines an automorphism -1 for any of G called the Cartan involution. Writing x' = v(x) x E G, it follows that +

(x,y)

+

xyx'

:

G

x

expb

+

exp4

defines an action of G on e x p h allowing to identify exp.3 with the symmetric space S, having K as a Riemannian metric for which G acts by isometries. L e t a b e a maximal linear subspace o f 4 which at the same time is a commutative sub Lie algebra of 9. Correspondingly, A = exp M is a maximal flat subspace of S through e, r = dim A is called the rank of the symmetric space S. Because the ad X, X E 4 form a commuting set of symmetric (with respect to K @ - KI ) linear operators-in g, one has a common I6 I* decomposition of into eigenspaces:

7

(1.3)

y

=

Z' aEA

, 9,

91,

non-zero linear subspace of

43

such that (1.4) (ad X) (Y) = [X,Y1 = a(Y).Y

for Y E (aa.

Of course the eigenvalues a(X) are real and depend linearly on X E bt, so the a E A actually are elements o f m * , called the roots. if we write m = k n Also N C @do, more precisely q o = m @3

vo.

The null spaces in otof the a E A,a f 0 are called the root hyperplanes. Thier common complement has connected components which are convex open cones with piecewise flat boundaries, called the Weyl chambers. If k E K normalizes tn, that is Ad k ( W c M, then (because Ad k is a Lie algebra homomorphism) Ad k permutes the root hyperplanes and therefore also the Weyl chambers. Furthermore each orthogonal reflection in a root hyperplane arises in this way. In fact the Weyl group W, defined as the group generated by the orthogonal reflections in the root hyperplanes, is equal to the group (1.5) {Ad k l a ; k

z

E

MI

-

,M

=

normalizer of &in

K. In other words,

W = M/M if (1.6) M = {k E K; Ad k(X) = X for each X E

-

denotes the centralizer of &in K. The Lie algebras of and M are both equal t o m , so M is open in M. On the other hand M is compact 5

31

ELEMENTARY S P H E R I C A L FUNCTIONS

as a closed subgroup of K and therefore W is finite (a fact which puts very severe restrictions on the system of root hyperplanes). It is easy to see that W acts transitively on the set of Weyl chambers, it turns out that this action is simple as well, so in fact#(W) is equal to the number of Weyl chambers. Now choose a Weyl chamber, which will be called the positive + Weyl chamber OL from now on. (Because of the above it does not matter which one we choose.) The non-zero roots do not change sign + , write on (1.7)

A+ = {a E A ;

a(X)

>

0 for some (all) X E

m+)

for the corresponding set of positive roots. Then A = ( - A + ) U ( 0 1 u A + and writing (1.8) Z = {a E A+ ; not a = B+y for some 0,y

E

A+]

for the set of simple roots, it turns out that is a basis of m* and that each positive root is a linear combination of simple roots with non-negative integral coefficients. As,a consequence the walls of &+ are contained in the null spaces of the simple roots. Because [ ga~QdBl C (1.9)

Z

?t =

aEA+

qa+Bl the

sum

%a

is a nilpotent sub Lie algebra of OJ and N sub Lie group of G. The map (k,X,Y)

+

=

exp N is a nilpotent

k.exp X.exp Y

is a diffeomorphism: K x O'L x U + G and as a consequence each x E G can be written as x = k.a.n with uniquely determined k E K , a E A, n E N, depending smoothly on x (Iwasawa decomposition). For any 41

E C:(K\G/K)

(1.10) ( A $ ) (X) = e-p'x)

here (1.11)

p =

5

Z

write

I N

@(exp X.n)ds, X

E

n,

dim g,.a.

U€A+

A is a continuous linear map: C I ( K \ G / K )

,

extending to a E' (a).More importantly A is a continuous linear map: E' ( K \ G / K ) homomorphism with respect to convolution and it maps to W-invariant distributions on oi. The deepest result however, on the proof of which we shall comment later,is that A actually is an isomorphism: +

+

Cz(W)

30

H. DUISTERMAAT

resp. (.8(K\G/K), * ) ( d ( M ) W , * ) . For Schwartz functions the result is due to Harish-Chandra and for compactly supported functions to Helgason ( in some special cases) and Gangolli 1 2 1 . In particular the algebra of G-invariant linear operators on S ( € I (K\G/K), * ) is commutative. For example, the G-invariant differential operators D on S are, since they are local operators on K\G, given by convolution with @ E E'(K\G/K) satisfying supp @ C K. But then supp A @ c {O}, that is A @ = Dgt6 for some uniquely determined differential operator D D&is an isomorphism from with constant coefficients on&, and D the algebra Diff (S)G of G-invariant differential operators on S onto the algebra Dif f (a) "61 of W-invariant differential operators with constant coefficients o n m . NOW, if u E D ' ( S ) is a common eigendistribution of all G-invariant differentialoperators D on S, that is (1.12) Du = A ( D ) . u for all D E Diff(S)G , +

+

A(D) is a homomorphism : Diff(S)G then D identification with Diff (a)"& is given by +

+

C, which via the

for some X EdC;f: which is uniquely determined modulo the action of the Weyl group. The space of common eigendistributions for a given X consists of analytic functions. It contains a spherical one, unique up to a factor, which is given by

Here H(y) is the element of ot defined by (1.15) y E K.exp H(y).N.

In view of the Iwasawa decomposition, H is a smooth mapping: G called the Iwasawa projection. In fact @ A is equal to the image of the exponential function X e(xrx' under the transpose: C"(0t) Cm(K\G/K) of the continuous linear map A : E' (K\G/K) E l (OL) , the integration over K is needed in order to make the functionright K-invariant as well. It may also be remarked that @ A is a common eigenfunction for all G-invariant C m ( S ) and that Fourier continuous linear operators : Cm(S) +

+

+

+

+

ELEMENTARY SPHERICAL FUNCTIONS

39

decomposition in ot shows that every element of E ' (K\G/K) can be expanded in @ p E a*.So the name elementary spherical function ip for the $ A is fully deserved. 2. Harish-Chandra's assptotics. The map kl.exp X.k2

-+

X (kl,k2 E K,X E m+) defines a fibration a

+

from an open dense subset U of G to OL I if f E C"(ot+) then a*f is a spherical function on U. For any differential operator D on G I

+

D(a*f)I + is a differential operator onot , called the radial M part Drad of D. On the other hand one has the integration formula f

-+

(2.1) J f(x)dx = J A(X). I f(kl.exp X.k2)dkldk2 dX G KxK R+ where dim q, (2.2) A(X) = const. II (f(ea (X)-e-a (X)) ) c i a +

and following Gangolli [2] it turns out to be somewhat more convenient to work with

Then D

+

6

is a homomorphism from Diff (S)G to the algebra of

+

differential operators on m with smooth coefficients, the conjugation with the factor A' makes that also D* = ( 6 ) * so in particular symmetric operators get mapped to symmetric operators. = A f $ / + are common eigenfunctions of all 6, Obviously the hot

D E Diff(S)G. If the 6 would have constant coefficients then it would follow that the $ A are sums of W(W) many exponential functions which are easy to analyse. This is not true,but asymptotically when + (away from the walls) then D approaches a constant X m in ot coefficient operator in an exponential fashion. In order to describe this, write -+

Then (2.5)

+

01.

=

[z EIR'

: 0


1 the system of partial differential equations actually leads to a system of ordinary differential equations with regular singularities along +

+,

41

ELEMENTARY SPHERICAL FUNCTIONS

each curve in the z-space, to which then the same asymptotic theory can be applied.

as p

3. Asymptotics of I$~"(X)

+

m

in in*.

In [ 1 1 we studied the asymptotic behaviour of the common spectrum of the G-invariant operators on S, pushed down to a compact quotient S/T of S in order to make the spectrum discrete. It is not surprising that for more detailed information we need the asymptotic

1x1

(keeping Re A bounded) which is complementary to the asymptotics of Harish-Chandra.

behaviour of I$,(X) as

+

m

Here the approach is to incorporate Re X in p in (1.14) and read (1.14) as an oscillatory integral

i( p,H(xk)) g (H(xk)) dk e K with phase function f : k (p,H(xk)) and amplitude k g(H(xk)) . UfX + In view of the decomposition G = K . e x p 4 .K we may take x = exp X, + X E OL and then H(xkm) = H(xk) for m ' E M = dentralizer of M. in K (see (1.6)) , so (3.1) can actually be seen as an integral over the flag variety K/M rather than over K. (The name is because for the classical groups, M is the stabilizer group for a natural action of K on a set of flags.) The first step is to observe that the asymptotic expansion of (3.1) for p = w.v, w E I R , w m is determined by the behaviour of the amplitude near the stationary point of the phase function fv,X. (3.1)

I

+

+

+

3.1. Theorem. The set of stationary points of f

VfX

(3.2)

Cv,x =

is equal to

K'GK',

which is a smooth submanifold of K, and the rank of the Hessian of is equal to the codimension of C in K. f v f X at each point of C vtx V,X (Clean stationary point set in the sense of Bott.) Obviously f is constant on the connected components of C /PI v,x v,x and because W acts transitively on the set of these connected components, an application of the method of stationary phase immediately leeds to the asymptotic expansion

as p = o.v, w

+

m.

42

H . DUISTERMAAT

An explicit calculation of the Hessian of f at c gave us an v,x v,x explicit formula for the leading coefficient cw,ot if both v and X are regular (not in a root hyperplane), then

Although the proof is quite different, the result is very similar to the asymptotics of Harish Chandra. In fact I believe that in the same way as Harish Chandra's asymptotics implied that tA*°F-l is a unitary embedding: L 2 ( O t * / W , B ) L 2 ( G ) , our asymptotics implies that the adjoint F o A is a unitary embedding:-L2 ( K \ G / K ) --* L2( */W,B). As a consequence A is injective, or equivalently A*OF-' has a dense range, which was the missing bit of information in the proof that A is an isomorphism. With a lot more work we are able also to get uniform asymptotic estimates when 1 ~ --*1 m , allowing u to pass through the root hyperplanes. (There the behaviour is quite' singular because the dimension of C then suddenly increases, leading to a jump in the order of v,x the asymptotic expansion ( 3 . 3 ) . This is an example of the Stokes phenomenon.) Here X is kept in compact subsets: we are also working on the still harder problem to understand what happens if both p and X run to infinity, that is to unify our asymptotics with the asymptotics of Harish-Chandra. I would like to close with some comments on the remarkable properties of the Iwasawa projection +

(3.4)

k

H(xk)

:

K/M

+

OL

which are a consequence of the description of the set of stationary points of fv,x. (fv,x is nothing else as testing the image with the linear function v.) The remarkable fact is that the set of stationary points in K/M does not move continuously with v : K v only depends on the set of roots orthogonal to v and this varies within a finite set, the dimension only going up when v enters the intersection of more root hyperplanes. Assuming for convenience that X is regular + ( = not in a wall of O t ) then if v is regular as well, /M = W all v,x the time, so for instance the maximum value of fv,.j( is equal to ( v, (w-l(X)) for some w E W, all the time. It immediately follows

ELEMENTARY SPHERICAL FUNCTIONS

43

that the image of K/M under the Iwasawa projection is contained in the convex hull of the finite set {w-l(X) ; w E W) and expanding this argument a little further one finds back the 3.2.

Theorem (Kostant [ 6 1 ) . The image of K/M under the Iwasawa projection ( 3 . 4 ) is equal to the convex hull of the Weyl group orbit of X in 4. This result is very remarkable indeed because K/M is a beautiful smooth compact manifold without boundary, the Iwasawa projection is analytic, nevertheless the image is a polyeder, full of faces, edges and corners. Of course for us the surprise came already at an earlier stage, namely that the asymptotics of ( 3 . 1 ) could always be obtained by just applying the methods of stationary phase on a set of stationary points depending in a discrete fashion on the parameter V.

For more convexity results, generalizing Kostant's theorem, see the thesis of Heckman [ 4 ] , defended yesterday in Leiden.

REFERENCES Duistermaat, J.J., Kolk, J.A.C. and Varadarajan, V.S., Spectra of compact locally symmetric manifolds of negative curvature, Inv. Math. 2 ( 1 9 7 9 ) , 2 7 - 9 3 . Gangolli, R . , On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Liegroups, Ann. of Math.

93

( 1 9 7 1 ) , 150-165.

Harish-Chandra, Spherical functions on a semi-simple Liegroup I,II, Amer. J. Math. 80 ( 1 9 5 8 ) , 2 4 1 - 3 1 0 , 5 5 3 - 6 1 3 . Heckman, G.J., Projections of orbits and asymptotic behaviour of multiplicities for compact Lie groups, Thesis, Leiden, 1 9 8 0 . Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T. and Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Annals of Math., 107 ( 1 9 7 8 ) , 1-39.

Kostant, B . , On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ec. Norm. Sup. 6 ( 1 9 7 3 ) , 4 1 3 - 4 5 5 .

This Page Intentionally Left Blank

A N A L Y T I C A L AND NUMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS IN ANAL.I'SIS S . A x f l s s o n , L . S . F r a n k , A . vdn d c r S l u i s @ North-Ho1lCina Publishing C o r n p i n y , 1 9 8 1

(rds.)

ON THE QUESTION OF THE EXISTENCE AND NATURE OF HOMOGENEOUS-CENTER TARGET PATTERNS IN THE BELOUSOV-ZHABOTINSKII REAGENT Paul C. Fife' Mathematics Department University of Arizona Tucson, Arizona USA

This continues previous work by Tyson and the author on the application of multiple-scaling techniques to the modeling and analysis of the expanding concentric rings of chemical activity seen in the Belousov-Zhabotinskii reagent. The model is based on "Oregonator" kinetics. Previous work with these kinetics concentrated on heterogeneous-center patterns, induced by an external particle or other stimulus; this explores the possibility of homogeneous center structures, more controversial but reportedly observed. 1.

INTRODUCTION

In [ l ] , Tyson and the author presented an analysis, based on multiple-scaling techniques, of the phenomenon of expanding concentric rings of chemical activity As model we took Tyson's seen in the Belousov-Zhabotinskii reagent [ 2 - 5 1 . scaled version of the differential equations of the Oregonator [6], reduced to two equations by a pseudo-steady-state argument, and supplemented by a diffusion term for one of the reacting components (the lack of such a diffusion term for the other component was inessential for our argument). These equations contain a "stoichiometric parameter", which we symbolized by "b". We supposed that a catalyst particle or other externally imposed heterogeneity effected a spatial variation in b in a neighborhood of one point, and we found how such a variation could induce the formation of an expanding circular pattern centered at the point mentioned. We invisaged an initial state in which the reacting chemicals were uniformly distributed in space. Away from the heterogeneity, the value of b was such that a stable rest state was possible, and the initial uniform distribution was taken everywhere to equal this state. Near the center, however, the values of b were such as to make the reaction kinetics oscillatory. It was shown how the tendency to oscillate near the origin induces the periodic formation of abrupt (steep-profiled) expanding chemical wave fronts and backs. These generated and propagating fronts and backs eventually result in a periodic target structure. The existence of such heterogeneous-center patterns has been well corroborated by experiment; but there have also been sparse reports [7-91 suggesting the existence of homogeneous-center ones (target patterns without a catalyst particle or other externally-imposed heterogeneity at the center). The purpose of the present paper is to examine the question of their existence, again within the context of the Oregonator model. The supposition here is that the value of b is related to the concentration of some third chemical, besides the two entering into the basic equations, and that the production rate of this third chemical is affected slightly by the others. 45

46

P.C. FIFE

Specifically, we suppose the fast species ( u = HBr02 in the notation of ( 1 1 ) weakly catalyzes the production of the third, and also that the latter slowly decays into an inert (for our purposes) state. No attempt is made here to interpret these effects in terms of the known ingredients of the reaction. As in [l], we work only with one-dimensional patterns; the spatial coordinate should be interpreted as distance from the center.

(XI

The process we describe begins with an assumed initial nonuniform distribution of b (or third chemical), possibly due to uneven mixing, such that a target pattern will be produced according to the process described in 111. Then as a result of the weak interaction between b and the other components, the distribution of b, hence the characteristics of the patterned structure, will slowly evolve in time. Under certain reasonable assumptions, it is shown that a special target pattern is approached in the limit. This special pattern has a unique period; and is such that b is uniformly distributed (independently of x), the waves propagate at constant speed, and successive fronts are equally spaced. It is not known, at least by the author, whether homogeneous center patterns with these characteristics have been observed. Of course heterogeneous-center patterns with diverse periods are in evidence. other hand, it should be noted that the uniqueness in period of the here constructed is seen only after sufficient time has elapsed; in stages, no set period is predicted.

target On the patterns the first

Another analysis of model homogeneous-center patterns was given by the author in [lo]; that analysis differs from the present one in that both the wave fronts and wave backs are of "trigger" type (in Winfree's terminology), whereas we here follow the more reasonable approach in [l], taking the backs to be phase waves and the fronts to be trigger waves. Another difficulty is that the chemistry supposed in [lo] (at least as regards the moderate speed component) was not necessarily linked to the Oregonator model. In [ll] a model for homogeneous center patterns was also proposed, similar in concept to that in [ l o ] , but a minimal amount of analysis was supplied. Approaches to chemical wave propagation by multiple-scaling techniques with sharp wave fronts have appeared elsewhere in the literature; see, for example, [12-191. Other analytical approaches for a class of models have been given in [20-221, and a numerical study was made in [23]. In Section 2, the model described above, but without diffusion or spatial variation, is formulated as a system of ordinary differential equations with three separate time scales. It is shown that such a system may have both a stable rest state and a stable periodic solution, and the domain of attraction of the latter is estimated. In Section 3, small diffusion is added, and the resulting spatial pattern and its slow evolution are investigated. Conditions are given under which the pattern will evolve to a final state. The period of the final state will be unique, and b will be distributed uniformly. Some discussion remarks are given in the final section. I am grateful to J. Tyson for helpful comments regarding this paper.

BELOUSOV-ZHABOTINSKII REAGENT

2.

47

THE KINETICS

As in [ l l , we begin with a system of kinetic equations derived from the Oregonator model:

d * u = dt d A dt

where Ce+4,

=

u and v can be interpreted respectively, E

0

.

Condition ( 1 ) is feasible. In fact, by assumption F(bo) = 0 (see ( 7 ) ) ,and so (F(bl)) is small for bo - bl small, and the finite upward jump makes Condition ( 2 ) is also clearly feasible, from the form of the F(bl - 0 ) < 0. dependence of F on B. The main result in this section is:

-

Under the above assumptions, there exists, besides the stable rest state ( 7 ) a unique (up to translation in time) stable periodic solution of ( l ) , (21, ( 3 ) If I for which b lies in the interval (b2,bl) and is approximately constant. initially, b(0) lies in this same interval, then the solution of ( 1 ) - ( 3 ) evolves into this periodic solution as t +

-.

P.C. FIFE

I I

1

I

Figure 2 Schematic representation of

U(b)

and

F(b).

To show this conclusion, we simply note that our assumptions imply that F(b) decreases through 0 at some unique point b* in the interval (b2,bl), as desired. *This value produces a stable *limit cycle. In fact, F(b) > 0 for b e (b2,b 1 and F(b) < 0 for b f (b ,bl), so that if b(0) E (b2,bl), b ( s ) + b* as s + m . In this sense, the oscillatory solution evolves into the unique periodic solution. For future reference, we denote the latter's period by T*. All that remains is to show that a trajectory on L,, for b = bl, reaches (ul,v) in finite time (under the approximation that the trajectory tracks L 1 exactly). Near the bottom point on L,, the curve L 1 is approximated by the function 2

k (v Hence Prom (2) with

-

b = bl, v

= (u

-

- ul)2 ,

small,

k > 0

.

51

BELOUSOV-ZHABOTINSKII REAGENT

Any solution of this with 3.

u

1

-

-k(v

b v 1-

-

k(v

- v) 'I2.

- 1)'I2

v(0) > 1 reaches the value

in finite time.

THE PATTERNS

As basic model, we take ( l ) , ( 2 1 , ( 3 ) , ( 5 ) , with the diffusion term adjoined to (l), and (3) rewritten in terms of b:

u

t

= EU

xx

1 + f(u,v) ,

-

b + y)

xx (11)

E

vt=bu-v, bt = 6(-Bu

EU

(12)

.

(13)

We retain all the assumptions of Section 2 , so that the main conclusion there still holds. As before, b evolves on a slow time scale: b = b(x,s), averaging over a t-cycle, we approximate ( 1 3 ) by

s =

6t, and by

-

bs=-Bu-b+y.

(14)

As mentioned in the introduction, we assume an initial nonuniform distribution b(x,O). We suppose it attains a minimum at x = 0 , such that

b(O.0) < bl;

lim b(x,O) = b 0 1x1-

'

We assume, for simplicity, that the initial distributions of u and v are constant and equal to the rest state corresponding to b = bo: u(x,O) z u 0' v(x,O) :v 0'

On the moderate time scale of order 1, b remains essentially constant in t We assume and a target pattern develops, as detailed by Tyson and Fife in [ll. 6 is so small that this pattern has time to become established as an approximately periodic structure before b changes appreciably. Its period To will be that of the relaxation oscillator with fixed b = b(0,O). x > 0, thz u-v traiectory of the oscillation will be such that the range of v is v (x) < v < v, where in general v+(x) > 1. This results in a typical period loop as shown below: On the other hand, it was shown in [ l ] that for each fixed value of

52

P.C. FIFE

V

1

Figure 3 The approximate time it takes to traverse this loop can be calculated from a knowledge of b and v+, using ( 1 2 ) and the fact that most of the time is spent on L 1 and L3, where u and v are related by f(u,v) = 0. Thus for some function g, we have T = g(b,v+)

.

(15)

This relation, together with the conditions T = To, b = b(x,O), yields v+ a function of x (in fact, see ( 1 7 ) below). The trajectories of the wave fronts can then be found from the differential equation

as

c is the characteristic speed of a front at the value v = v+ (see [ l I The trajectories of-the wave backs are determined from the conditions that they occur where v = V.

where

[lo]).

,

But our main concern is with the subsequent (slow) evolution of this pattern. The basic evolution equation on this slow time scale is ( 1 4 ) . We are assuming it is slow enough that at each value of s, the pattern is an established structure of the type described above and in [ l l , periodic in the faster time variable t. Thus, we are looking for slow modulations of periodic solutions. Therefore (15) holds for each (x,s). In addition, as previously indicated, T will be a function of b(O,s),_ namely the period of a relaxation oscillator with that value of b. Setting b(s) = b(O,s), we have T = h(g) for some function h.

(16)

53

BELOUSOV-ZHABOTINSKII REAGENT

It is pretty easy to see that the function g variable, so (15) may be solved for v+: v+ = V(b,T) Finally, the average

in (15) is monotone in its second

.

(17)

over a cycle will depend on

b

and :'v

This function is related to the function of a single variable in ( 8 ) by U(b) = U(b,v). From (14), (18), and (171, we have

_ ab as

Specializing this to we find

-BU(b,V(b,T))

x = 0

-

b + y E G(b,T)

and using the fact that

. 1 at

V+ =

(191 Y,

= 0

always,

Our problem has now been reduced to

(i)

solving ( 2 0 1 , with the given initial value

(ii) finding T(s)

u(b,v+) > uo

b(0) = b(O,O);

from (16); and

(iii) solving (19) with T = T(s), Our main result is the following:

If

. d

for all

using the prescribed initial data

b(x,O).

(b,v+) admitting oscillations,

b < b(0,O) = g(0) < b l , b(x,O) = bo for large 1x1, and 8 is small 2 enough, then the solution b(x,s) & (19), (20) is such that for all x, lim b(x,s) = b*,

lim T("bs)) = T*

S+m

S+m

.

(21)

Before showing this result, a comment is in order regarding the assumption. For b > b l or €or b < b2, there is a rest point €or ( 1 1 , (2) on branch L1 or Lg in Fig. 1. This means a periodic solution of the type shown in Fig. 3 is impossible unless v+ is larger than the value of v at that rest point. This is the meaning of (b,v+) "admitting oscillations", and of course U(b,v+) is only defined for such (b,v+). If b2 < b < b l , the problem does not arise. For a solution of thfs type, time is spent on both L 1 and Lj, so the condition u = U(b,v ) > uo is apparently not very restrictive. The demonstration of (21) proceeds on the basis of the following lemma, whose proof is straightforward and will not be given: Lemma:Let where F (i) F I

y(s)

satisfy

has the following properties for some interval is differentiable in x

y

and continuous in

I = (y',~"):

(y,s), uniformly in

54

P.C. FIFE

(ii) lim F(y,s) = F(ylm) exists for all S+-

(iii) F

Y


0 for b = b + W. In fact, ( 7 ) shows y to be near bo than, so G(h,T) will be near bo2- b > 0. The only remaining condition to be verified is that G(bo,T) < 0 €or all S . But G(bO,T) = -BU(bo,V(bo,T)) = -BU(bO)

- bo + Y

- bo + Y

+ 8[-U(bo,V(bo,T)) + U(boll

= B[-U(bo,V(bo,T)) + U(bo)l =

B[-U(bO,V(bO,T)l + u,]


0

0

A p p l y i n g a g a i n ( 2 . 1 6 ) we f i n d c h a t

E

(2.19)

che i n i t i a l v a l u e problem

Rn

L E L R n ] = O ( s n + l ) , u n i f o r m l y i n a n y bounded domain w i c h

Rn(x,O) = R:(x,O)

Setcing

+ X"(X,t)

i=(j

IRn( +

S u b s t i t u t i o n of

vi(x,;)

cions y i e l d consecucively t h e funccions u ( x , t ) :=

t 2

(X,C) + EVc(X,:)

+

O(E)

,

u n i f o r m l y i n a n y bounded domain i n t h e r e g i o n F o r d e r a i l s s e e l i c 1 4 1 , p p . 64-71,

o r lit-151.

c 1 0

.

64

DE JAGER and R . GEEL

E.M.

3

THE QUASI LINEAR PROBLEM

3.1

The Formal Approximation

We consider the initial value problem

with au u(x,O) = (x,o) = at

o ,

where the coefficients they are of class (x,t) tive

wich

Cm

0

t 2

< x
la/ for all values of

u , A formal approximation of the solution of the

and all values of

t

,

initial value problem is given by

-u(x,c)

= w(x,c) +

w(x,c)

E

"(X,T)

.

(3.2)

denotes again the solution of the reduced problem

aw aw a(x,t,w) - + b(x,c,w) - + d(x,t,w) ax at w(x,O) E

=

0 ,

v(x,L)

-m

< x
0

aT E

av (x,O) = av (x,O) = at

aT

aw - -ac

(x,O)

,

lim

V(X,T)

= 0

T+=

and hence

(3.4)

Under the assumpcion thac w Y

u

is sufficiently regular

sacisfies the initial value problem (3.1) upto order

(w

E

3

C )

L)(E)

,

,

the funccion

SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE

65

In fact we have

-

, u(x,O)

L E G = I)(€)

=

v(x,O)

E

D with

uniformly in compact domains PutLing

-

-

u ( x , t ) = u(x,t)

-

, u (x,O) t 2

= 0

0

,

(3.5)

, where

w

E

C

3

.

E

+

= ;(X,t)

(3.6)

we obtain according t o ( 3 . 5 ) and ( 3 . 6 ) for the remainder term

R

the initial

value problem

2) a 2R + a(;

+ R) aR + b(;

+ {a(a + R)

-

a(u)]

+ {d(ii + R)

-

d(;)}

+ R) aR

at

ax

ax

a;

+ {b(u + R) - b(u)}

au at

(3.7)

, uniformly in 'D ,

= I)(€)

with the initial conditions R(x,O) = R (x,O)

0

=

I n order to simplify the notation we have omitted in the coefficients

d

the arguments x

and

I:

.

a,b and

I n the same way as in the linear case one may construct a better formal approxiu

mation of

by putting

"

U(X,t) =

i

1

E

n

W.(X,t) +

i=O -n

:= u

i=O

(x,t) + Rn(x,t)

where the terms

E

i

wi(x,t)

E

i+ I

t

v.(x,-) 1

E

-

E

n+l

vn(x,o) + Rn(x,t) (3.9)

, and

E

i+ I

v.(x,l) 1

E

are determined consecutively

by differential equations obtained from ( 3 . 1 ) by means of substitution of ( 3 . 9 ) into ( 3 . 1 ) , expansion into powers of of

E

E

and equating terms with equal powers

; for details see licC41, pp. 85-91.

In this way we obtain for the remainder term Rn

the initial value problem

66

E.M.

DE JAGER and R . GEEL

uniformly valid in any compact domain 0 , where

wo

... n)

and so w.(i=1,2,

are sufficiently regular,and with the initial conditions Rn(x,O)

=

Rn(x,O)

(3.8*)

= 0

-

u

In order to prove or to disprove whether the formal approximation

(or more

generally ;")is really an approximation of the unkwown solution u similarly as in the linear case, to estimate R generally Rn However

R

-

from (3.7*)

(or

from (3.7)

-

(3.8)

we have, (or more

(3.8*)).

Rn) satisfies a non linear equation and we need now a more

sophisticated reasoning than in the linear case. 3.2

A Fixed Toini; l'heorem

Van Harten 181 has considered elliptic singular perturbations of quasi-linear operators of the first order and in order to prove the justification of a formal approximation he applied a fixed point theorem for estimating the remainder rerm. I n this seccion we introduce a modification of this cheorem which will serve our

purpose, namely the estimation of the remainder term

.

R

The theorem reads as follows: Fixed Point Theorem. Let and

B

N

be a normed linear space with norm

11 11

a Banach space with norm

N

mapping

+

B with

, z

. Let

B

E

I .I

,

y

E

N

b e a non linear

F

F ( 0 ) = 0 and with (3.10)

where

L is the linearization of F at y

The following conditions are imposed on i)

The mapping

L

from

N

to

B

L

=

0

. I

and

is bijective and

L

-1

is continuous, i.e. (3.11)

R

where ii)

Let

(jN

is some number independent of

There exists IIy ( Y , )

where

Iy c N , IyI

( p ) :=

-

o

y(y2)11

m(p)

5

z

.

PI

such that 5 m(p)

I y,

-

is decreasing for

y2(

p

, v yi

6

nN(p)

decreasing with

, (i

= 1,2),

lim m(p) P-fO

o

= 0

5 p

.

5

P ,

(3. 12)

SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE

67

Finally define (3.13)

Assereion: If

f

B

E

chen there eXiSKS

g

11 f 11

wich E

N

5

? LP o ,

(3.14)

2

wich

Proof: The relation F(y)

=

L(y) + Y(y)

=

f

is equivalent wich che

relacion z =

T(Z) := f z

where

=

L(y)

o L-l(z) ,

Y

.

Consider now the ball

RB(p)

Whenever

and

11 f 11

< fkp

Iz

=

0

,

B

E

5 p 5 po

flB(Lp)

and is strictly contractive in

So the exiscence of a unique fixed poinc

11

.

5 .pi

then

.

z

=

T maps z*

QB(lp)

into

RB(P.p)

is quaranteed and hence

chere exists a solution of che equacion F(y) = f , namely y := g = L- 1 (z* ) , -1 lying in L [R,(P.ep)l c blN(p) Taking P = 2 Q-' 11 we obtain the result

1/f

(3.15). For details of the proof see 1 4 1 , pp. 2 0 - 2 4 , i8l

.

i 6 1 , pp. 30-32 , or

3.3 The Approximation of the SoZution I n order to estimate the remainder term (1 c

Rn

of secEion 3.1 in a trapezium region

D , we cake

q - Qco(n)

N = ( y ; y c C ( R1 ) ,

2

2

ar:

ax

€

, y(x,O)

wich

I I y

=

3 at

(x.0)

=

0)

(3.16) =

max

Iy1

;z

E

11

+

E3'4

max R

I

y,

/

+

=

max

E3'4

max R

lytl

and B = Iz

Co(R)l

wich

/Iz 1 1

12

1z I

(3.17)

68

E.M.

The map

F

L ( R ~ =)

E(7

from

N

i s given by ( 3 . 7 )

B

to

2 n

a2Rn

a R2 ) + - -

at

DE JAGER and R . G E E L

ax

a R" + K

a(?)

or

(3.7*)

and hence

a R" a t

b(?)

(3.18) and Y(Rn) =

If

a,b

{a(;"

+

Rn) - a(?)}

+ {a(;"

+

R")

+ {b(?

+ Rn)

a R" + ax

-

a(;")

-

-

b(?)

-

+ {d(? + Rn) - d(;")

-

and

d

che o p e r a t o r s

are

L

and

Cm Y

in

{b(;

aa -n au (u ) ab

a Rn K

+ Rn) - b(?)I

n R I

n (in) R }

a;" ax

a;" (3.19)

( i n ) Rn} (x,t)

for a l l

u

s a t i s f y the conditions

and

c2

i)

and

in

u

ii)

p o i n t theorem of che p r e c e d i n g s u b s e c t i o n . For t h e q u a n t i t y

9.

-1

for all

(x,t)

of t h e f i x e d we o b t a i n

according (2.16) k

-1

=

C(n)

E

-114

(3.20)

It may be remarked h e r e t h a t t h e norm (3.16) i n t h e s p a c e

N

has been chosen

i n accordance w i t h t h e r e s u l t (2.16) The non l i n e a r p a r t (3.19) of t h e o p e r a t o r calculation

the factor

C(O)

The parameter

po

E -

21C(n)12

E

-

,u

n

yields after a l i t t l e

i s a g e n e r i c c o n s c a n t , dependent on

satisfies

and hence Po =

F

and

il

but independent of

E

69

SINGULAR PERTURBATIONS OF HYPERBOLIC TYPE

Therefore in order c o escimace che remainder cerm we need according to (3.15)

,

a formal approximacion of ac lease order 2

n

i.e.

=

, (compare

I

(3.7*)).

Application of che fixed poinc cheorem yields finally IR

1

I=

max I R

1

I

n

= O(2e-I

+ s 3 I 4 max

J R ~ +/

n

E2)

=

O(E

I

I R ~ I

max

E3'4

n

714)

uniformly in any crapezium region

lying in

D , where wo is sufficiencly

regular. Finally ic follows from (3.9)

I

-

u - w O - E W I - E V O

I

V]

E2

= O(E7I4)

or /u

au

1- ax au

-

wo(

= O(E)

aw

0

-(=

(3.21)

O(E)

ax

- _ ac

-Iae

E

=

O(E)

.

uniformly in che crapezium n

We summarize chis result in the following theorem Theorem Suppose the coefficients a, b

and

of che differential equation ( 3 . 1 )

d

sacis-

fy che following condicions:

i)

a, b

ii)

b > la( for

d

and

C2(u)

are of class Cm(x,c) -m

+

< x
0

,

-m

< x < +

cU

in some compacc domain D , chen che initial value problem

2 2) ax =

,

aw

0 ar: +

b(x,c,w o )

is of class c"(x,c) 2

m

and of class

of che initial value problem

wo(x,o) = 0

E(%a t

(x,t) wich

for all values of

u

for all values of

=

0

au ax + b(x,t,u) ,

-a

< x
0

,

-

< x
> E , why do we not just calculate xo(h) and y o ( h ) from (5.1.40)? We give several reasons. Remark:

a) The exploitation of the methods of singular perturbation theory for the development of numerical techniques usually proceeds with the numerical determination of values of one or more terms in the asymptotic expansion supplied by that theory [3]. The extrapolation formulas (5.3.6) break through this limitation of approach.

76

F .C.

HOPPENSTEADT and W . L . MIRANKER

b ) Equation ( 1 ) with E = E' is not stiff and may be easily and reliably solved by simple explicit numerical methods. While (2) is also not stiff, employing it for the determination of x o ( h ) and y o ( h ) requires the solution of the nonlinear system g = 0 at each mesh point. This is usually a costly computation. c) Unlike the extrapolatory method, the conventional approach requires that the initial value problem be placed in the form ( I ) , and this may require considerable work.

I .2 COMPUTATIONAL EXPERIMENTS The following two computational experiments compare the extrapolation method (7) and the asymptotic expression itself.

i)

A linear sysrern

We consider first the following linear example

d x- - y - x , dt

x(0) = 5,

d y = - Y + l , y(O)=TJ. dt

E

The exact solution is given by the formulas x ( r ) =e-'[

y(r)= &

+ ( l - e - ' ) E - ( ~1)- (& T J - & ) ( e - " F - e - ' ) ,

+ e-"yTJ-E),

and the leading terms of the matched asymptotic expansion solutions are X(t)=e-'5+€[(1)-1)e-'+ y(r)= E

I ] + ... ,

+ ... .

For any value of the leading terms of the matched asymptotic expansion ( x o ( h ) , y o ( h ) )and the exact solution ( x ( h , E ) , y ( h , E ) ) agree to about four figures. Thus while in Table 1-1, E = lo-' is employed in the column labeled E ' / E , the results in that table are otherwise valid for any ~ < 1 0 - ' . The results using the extrapolation method and evaluation of the matched asymptotic expansion ( t o leading order) are presented in Table 1-1. Since K = 1 + I T J I and 8 = 1 in this case, we take T = - e n [ h P + ' / ( l + 17 I ) ] . In spite of the involved form of this formula for T , the values of the latter should be taken only approximately. The calculated values of E' = h / T are E' = .0082 and E' = ,00304corresponding to h = .1 and .01, respectively. Since these values of E' a r e to be taken only as approximate, calculations for nearby values of E ' , are also presented in the table. Note that p = 4 in the computation displayed in Table 1-1 as well as in Table 1-2 to follow. The relaxed equations are in fact solved on the interval (0,h ) by means of a fourth order Runge-Kutta method for which we employ a submesh with submesh increment k = he'. This choice of submesh increment assures high accuracy and stability of the Runge-Kutta computation.

COMPUTATION BY EXTRAPOLATION

.$=q=l

p=4 h=.l

EXTRAPOLATION M E T H O D

MATCHED SOLUTION

x(h) ,8632 .R648 ,8657 ,8707

-

,9049

E’

h = .01

MATCHED SOLUTION

E’

.005 ,0082 .0 1 .02

EXTRAPOLATION M E T H O D

77

E’

,003 ,00304 .004

x(h) ,9866 .9866 ,9871

-

,9901

E’

= ,0082

y(h) ,0025 ,004 ,005 .o 10

E’IE

500 819 1000 2000

0.0 = ,00304

y(h) ,00155 ,00157 ,00255

E‘/E

3 00 304 400

0.0

Table 1 - 1 Notice that the extrapolation method gives a 4 % answer for h = . I , but it gives better than a 1% answer h = .01.

We compare this extrapolation method with the frequently used package of C. W. Gear for for the second integrating stiff differential equations [4]. We use this package with E’ = case in Table 1 - 1 with the following result. The package reaches h = 0.01 by employing a variable submesh of points which it determines adaptively. To produce a 1Yo answer, the package requires an average submesh size of 1 . 9 6 ~ The extrapolation method with E‘ = .00304 produces its 1% answer with a fixed stepaize of 9 . 7 5 ~ nearly an order of magnitude difference. Of course as E decreases the latter remains invariant, but the average stepsize employed by the package will decrease even further. ii) A

model enzyme reaction

A simple enzyme reaction involves an enzyme E , substrate S , complex C and product P. Schematically, the reaction is

E

+ S+C,

C e E

+ P.

After some preliminary scaling, this reaction can be described by a system of differential equations for the substrate concentration (x) and the complex concentration b) as

dt

= --x

+ ( x + k)y,

x(0) = 1,

78

F.C. HOPPENSTEADT and W.L. MIRANKER

where E measures a typical ratio of enzyme to substrate (O(IO-')), and k and k ' ( k denote ratios of rate constants suitably normalized (O(1)).

< k')

(although as The Table 1-2 summarizes the result of these numerical calculations for E = h = .I noted in (i) above, except for the column E ' / E , the results are valid for any and 0.1, k = 1 , k' = 2. In this case, K = 1 , 6 = k', so we take T = -(&)en h. The calculated values for E' = h/T are E' = .04 and E' = .0009, respectively. Calcu?ations are also presented for some nearby values of E ' . x(0) = 1 y(0) = 0 p = 4 h = .I

EXTRAPOLATION M E T H O D

= .04

.05 .I .I5 .2

x(h) .9530 .9596 .9617 ,9726 .9882 ,9937

y(h) ,3229 .3247 ,3253 ,3285 .3350 .3406

-

.9888

.3308

E'

.01 .04

MATCHED SOLUTION

E'

h = .01 E

I

EXTRAPOLATION M E T H O D ,0004 .0008 .0009 .001 ,0016 MATCHED SOLUTION -

E'

E'/E

1000 4000 5000 10000 15000 20000

= .0009

x(h) ,995 1 .9952 .9952 ,9952 ,9954

y(h) ,3322 .3323 ,3323 .3323 .3323

.99 17

.3315

E'/E

40 80 90 100 160

Table 1-2 The extrapolation method gives a 3% answer for h = . l , but it gives better than a 1% answer for h = .01. As in the case of Table 1-1, a comparison here produces an average step size of 2 . 4 ~ for a 1 % answer for Gear's package as opposed to a fixed stepsize of 2.7 x for a 1 % answer for the extrapolation method with E' = .0009.

2. AVERAGING PROCEDURES Problems to which the Bogoliubov averaging method and various multitime schemes are applied frequently reduce to problems of the following form.

where x, f, SeR" and where f(7, e ) is an almost periodic function of

7.

Multi-time perturbation

COMPUTATION BY EXTRAPOLATION

where x o is determined from the initial value problem dx,

(10) Here

7 is the average o f f ,

-

= f ( x 0 ) . xo(0) =

6,

defined by

The coefficient x1 is determined from the formula (11)

u

~ ~ ( I/&) 1 , = xl(r)

+

i"'[fC~.

xo)-f(xo)]dr.

(For details see [ S ] . ) Thus,

This approximation suggests several numerical schemes for determining x ( h , E ) . In Section 2.1, we consider the computation of the average f; first by the customary method and then by a second difference method which accelerates the computation of 7 in some cases. Then in Section 2.2, we describe an extrapolation method for approximating x ( h , E ) . As in the extrapolation method introduced i n Section 1 , certain larger values E' of E are introduced, and (2.1) is used with this larger value of E' to furnish approximations to x ( h , E ) itself. In Section 2.3, the results of computational experiments which compare the methods are presented. Finally a discussion of these various computational procedures is given .in Section 2.4. 2.1 ACCELERATED COMPUTATION OF 7 We propose two methods for calculating 7. i) Direct evaluation of lim

T-m

SO fdr.

-!-

T

A convergence criterion is first set, and then the integral L T f d r is calculated for increasing values of T until the criterion is met: Given a tolerance 6,txere is a value T ( 6 , x) such that

and

79

80

F .C.

HOPPENSTEADT and W . L . MIRANKER

for all T , , T , 2 T ( 8 , x). Thus, we can write

and proceed t o solve (10). Unfortunately, there is n o certain way of finding T ( 8 , x).

In order to find a candidate for T ( 6 , x), we calculate Y ( T , x) =

L T f ( T , X)dT

for 0 5 T 5 2 P , and keep increasing P until the condition sup

0 m'

>0

R

. We

+

9,

, of

et the

( c f . [81). denote by

C

and

( - l ) I p l DP(cpq Dq)

II (-1) Ipl,Iql 6 m' where t h e c o e f f i c i e n t s

Wr(R) t h e Sobolev space o f t h e P I q l 4 r , equipped w i t h t h e

for

Dp( bpq Dq)

c and b a r e smooth enough i n P9 Pq t h e s e s q u i l i n e a r forms a s s o c i a t e d w i t h C and B :

R

, and by c and b

D. HUET

88

The f o r m c [ r e s p . We s e t B, = CC t B

b l i s well defined i f and b, = E C t b

.

u,v E Hm(Q)

[resp.

.

u,v E H " ' ( n ) I

I now g i v e t h r e e s i m p l e examples o f e l l i p t i c s i n g u l a r l y p e r t u r b e d p r o b l e m s . F o r t h e sake o f s i m p l i c i t y , I r e s t r i c t m y s e l f t o t h e D i r i c h l e t b o u n d a r y c o n d i t i o n s . Example 1.1. L e t C and B be t h e o p e r a t o r s d e f i n e d by (1.1). We suppose t h a t B, and B a r e c o e r c i v e i n t h e f o l l o w i n g sense : t h e r e e x i s t p o s i t i v e c o n s t a n t s a, B , y such t h a t I b E ( u , u ) I 3 a, I l u II

(1.3)

H'"(Q)

f o r any

I b ( u , u ) I 3 Y I I u II 2 Hrn'(Q) L e t h be g i v e n i n t h e d u a l space problems :

H-"'(Q)

uE E i m ( Q )

,

BEuE = h

(1.5)

u E l?'(Q)

,

Bu = h

.

that

u,

I t i s known ( s e e [23 and [61)

h E L2(Q) 1

IIU,

-

u II

Hm' (Q)

=

0

(E

0

u E H"'

.

of

l?(Q)

in

Hm'(R)

1

(n)

@(Q)

u E

.

We c o n s i d e r t h e D i r i c h l e t

.

(1.4)

Moreover, i f , f o r i n s t a n c e ,

f o r any

+BIlUll2 Hm' ( Q )

+

,

u

m' P 0

, we

,E

~ "

u,

+

0

in

Hm(n)

have t h e e r r o r e s t i m a t e

&) *

Example 1.2. k!e c o n s i d e r t h e same p r o b l e m as i n example 1.1, b u t we suppose t h a t t h e o p e r a t o r s C and B a r e G i r d i n g e l l i p t i c i . e . t h e r e e x i s t c o n s t a n t s B1 , B2 , a l > 0 , a2 > 0 such t h a t

2

Re b(u,u) t B2 I I u II L2(n)

5 a2 IIU II

Hrn'(Q)

f o r any

0

u E H~

1

(n)

.

F. Stummel i n [lo1 o b t a i n e d t h e same r e s u l t s as i n example 1.1 ( s e e a l s o [51)

.

.

We suppose t h a t t h e o p e r a t o r C ( n o t n e c e s s a r i l y o f Example 1.3. L e t 1 c p c m t h e f o r m ( 1 . 1 ) ) i s e l l i p t i c i n t h e u s u a l sense, o f o r d e r 2m , and t h a t i t s p r i n c i p a l symbol c ( x , c ) s a t i s f i e s t h e Agmon c o n d i t i o n

.

89

A NORMED SPACE AND S I N G U L A R PERTURBATIONS

5 # 0 and any x

f o r any r e a l v e c t o r

u,

and l e t

E

(see [ l l and [ l l l ) . L e t

fl

h E L

be t h e s o l u t i o n o f t h e O i r i c h l e t problem

P

(n)

(1.8) I t i s known ( [ 3 1 , [41) t h a t

U,

+

h

in

L (0) and P

u,

E

--t

0

in

WZm(K?) P

.

The r e s u l t s o b t a i n e d i n t h e above t h r e e examples a r e i d e n t i c a l . B u t t h e p r o o f s , g i v e n i n t h e r e f e r e n c e s , a r e q u i t e d i f f e r e n t . The purpose o f t h e f o l l o w i n g s e c t i o n s i s a c o n s t r u c t i o n o f a s u i t a b l e a p p r o x i m a t i o n o f a Banach space and t h e statement o f p r o p e r convergence theorems which c o n t a i n t h e above r e s u l t s as s p e c i a l cases. 2. PROPER APPROXIMATION OF A NORMEO SPACE Let

nE,

be a f a m i l y o f normed spaces. We

t h e C a r t e s i a n p r o d u c t o f t h e spaces

nEE ( i . e . uE E E E f o r a l l

of

(EE), >

be a normed space and l e t

E

denote by

D e f i n i t i o n 2.1. L e t

and

(u,)

-

l e n t i f and o n l y i f l l u E

(u,)

an element

> 0).

E

(v,)

vE II

and by

E,

--t

0

EE

.

nEE

be i n

.

They a r e s a i d t o be e q u i v a -

D e f i n i t i o n 2.2. L e t R : u !-+ R(u) be a l i n e a r map o f E i n t o t h e s e t o f t h e We say t h a t e q u i v a l e n c e c l a s s e s , such t h a t R(u) # 0 f o r a l l u E E &(E, nEE , R) i s a p r o p e r a p p r o x i m a t i o n o f E i f and o n l y i f we have

.

l l U & II

--t

IIU IIE

EE

for all

u E E

and f o r a l l

(REu)

We denote by D e f i n i t i o n 2.3.

Let

elements

,

(E,

+

u E E

E ; R)

(2.1)

{

Remark 2.1.

E R(u)

(u,)

.

an element o f t h e e q u i v a l e n c e . c l a s s

nE, , R)

d(E, uE E

E ,

E

.

R(u)

be a p r o p e r a p p r o x i m a t i o n o f

> 0 , t h e p r o p e r convergence

u,

.

E

For

L, u

i s d e f i n e d by 4

Ut

i.e.

uE

u (EE u,

+

+

E ; R)

E R(u)

u (E,

+

lluE

i f and o n l y i f

E

E ; R)

l l u E IIE + I I u l l E

implies

be a normed space. F o r

I I - [ I E = 1 1 . I I E , and f o r

REu II

6

E

> 0

, we

+

0

EE

.

F o r more d e t a i l s on t h e above d e f i n i t i o n s see [91. Example 2.1. L e t

-

.

set

E,

-

u IIE

=

E ,

u E E

&

T ( u ) = {(u,)

(2.2) Then

&(E,

nEc , T )

E nEE ; IIu,

-

u II

= EE

i s a proper approximation o f

llu,

+

0

1 .

E called the t r i v i a l

-

90

approximation, and usual sense.

uc

D. HUET

u

(EE

-t

, T)

E

i f and o n l y i f

uE

u

-+

in

i n the

E

Example 2.2. (see [ l o ] and 151). L e t D and E be two H i l b e r t spaces such t h a t D cC We suppose t h a t D i s dense i n E and t h a t t h e i d e n t i t y mapping o f D into i s continuous. We s e t E c = D , f o r E > 0 , and

i

= E ( u , v ) ~t ( u , v ) ~

(u,v)E

(2.3)

for

E D

U,V

&

.

equipped w i t h t h e s c a l a r p r o d u c t (2.3) i s a H i l b e r t space. We now EE d e f i n e l i n e a r o p e r a t o r s RE E %(E, EE) by

Then

(v,U)E = ( v

(2.4 R

Let

u E E

be d e f i n e d , f o r

u E E

for all

I

,

.

v E D

and a l l

by

(2.5 We have

d(E, l l E E , R)

Theorem 2.1.

1)

2) I f cp E D

, one

3)

uE

4u

(EE

E llEE

, R)

E

+

Let

R&U

u E E

-t

and l e t lluE

-

u

, lemme

.

(u,)

@PI E&

uE

+

01

u

in

E

in

D

-,

. and

uE -, 0

cl/'

in

D.

1.1, t h a t

E ,

in

-

; IIu&

i f and o n l y i f

P r o o f 1 ) I t was shown i n [51 (2.7)

E

has

R(cp) = {(u,)

(2.6)

i s a proper approximation o f

R & U -,

o

.

u E E

f o r any

E R(u) ; (2.3) and ( 2 . 5 ) g i v e :

REu llE

+

0

and

c1j2

JluE

-

REu llD

+

.

0

Therefore l l u e 11'

E&

=

1 1 ~ ~( u' C ~- REu + REu) 1l0 2 + llu,

2) I t f o l l o w s from (2.3) and (2.7) which proves ( 2 . 6 ) .

t h a t , f o r any

(2.8)

Ilu

-

wIlE

E

s6

nEc ;

cp

IIE 2

E D , IIcp

-

+

IIu llE 2

REcpllE

+

0

.

I t f o l l o w s from Stummel's r e s u l t s (see

R(u) = {(u,)

R&u + R,u

&

3) i s an obvious consequence o f ( 2 . 3 ) and ( 2 . 7 ) Remark 2.2. uE:t

-

V 6 > 0

and I l u E

-

,

[91, ( 4 . 1 . 6 ) )

3 cp E D

REcpII d 6 E&

c0 > 0

and

for

t h a t , f o r any

E

6 co1

such t h a t

.

91

A NORMED SPACE AND S I N G U L A R PERTURBATIONS

Exam l e 2 . 3 . I now want t o e x t e n d t o Banach spaces t h e c o n s t r u c t i o n o f example h and E be two Banach spaces such t h a t D c E . We suppose t h a t D i s dense i n E and t h a t t h e i d e n t i t y mapping o f D i n t o E i s c o n t i n u o u s . The f a m i l y o f Banach spaces

EE = D

(2.9)

i s d e f i n e d by

(EE)

IIu l l E

where

. We

a > 0

DE

denote by

llE

1111

t h e space

(2.10)

, TIEc , s )

&(DE

Then,

seep

we can t a k e Lemma 2.1.

vIlE

+

,

01

&

.

D

E

U,

i s a p r o p e r a p p r o x i m a t i o n o f t h e normed space f o r any

E

,

s)

(EE

cp

'pE+

.

E

= cp

-

; lIwE

~ ( c p )= { ( W E ) €FEE

by

s(@)

H

+

DE

0

z

DE

.

i f and o n l y i f

cacpE

D

in

0

-+

.

1 1 . IIE

D equipped w i t h t h e norm

I d e f i n e t h e map cp

L o o k i n g a t p a r t 2) o f theorem 2.1,

in

IIu I I D

= ca 6

and

and

.+ 'p

p ',

T h i s lemma f o l l o w s f r o m (2.9) and (2.10). u E E

We now d e f i n e f o r

i

(2.11)

S ( u ) = {(u,)

E

Ilu - v l l E

6

,3

v 6 > 0

;

, and lluE

'p

E D

and

- scvllE 4 6

co > 0

for

such t h a t

E Q to)

E

Then, i t i s known, s i n c e

E

a p p r o x i m a t i o n of 4.1.9). Remark 2.3.

nEE

i s dense i n

which s a t i s f i e s

D and

If

0

E

, that

E

S('p)

nEc , S )

&(E,

f o r any

= s(q)

a r e H i l b e r t spaces, and

'p

i s a proper

.

E D

21 , t h e

a =

. ,

(See [91

R

mappings

and

S d e f i n e d by ( 2 . 5 ) and (2.11) r e s p e c t i v e l y s a t i s f y R(u) = S ( u ) f o r any u € E Therefore, i n t h i s case, d ( E , n E E , R) and d ( E , nEE , S ) a r e i d e n t i c a l . d ( E , nEE , S )

Theorem 2.2. L e t

(2.9), in

D

Proof

exists any

E

(2.10),

(2.11). Then

and

-t

uE

u

in

1) Suppose t h a t cp E 0

4 c

0

and

. Hence

HUE

- u

E

.

uE*

111.1

IIu II

E

=

a'

uE4u

u

IIE 6 lluc

- u IIE

+

0

(EE

(EE

+

E

, S)

26

.

-

-,E , S) . l e t - QIIQ

SC'pllE

t

IISE"

&

t

E

i f and o n l y i f

6 > 0

-

.

d e f i n e d by a E u E -, 0

By ( 2 . 1 1 ) , t h e r e 6

and l l u E - sE'pII

such t h a t IIu

c0 > 0

6

Therefore

be t h e p r o p e r a p p r o x i m a t i o n o f

.

6

5

for

+

0

i n 0.

EE

+ II'p - u l l E

"HE &

IIsEcp - v l l E

for

E Q

c0

E

.

Now, i t f o l l o w s f r o m remark 2.1 t h a t

l l u c IID t l l u c l l E

-t

IIu l l E

.

Hence

cauE -, 0

in

u

and

D

.

FE

2) L e t

u E E , (u,)

E nEE . We suppose t h a t

uE

-t

in

E

cauC

D. HUET

92

We want t o show t h a t (u,) 6 IIcp UII 6 4 . Therefore

lIuc if

-

'PllE

6 c

a

Example 2.4. L e t We denote by G,, family

a

(u,)

-

lI(P11D + lluc

E S(u)

u l l E + IIu

IIu l l H

.

f o r any --f

E

IIu llH

> 0

.

u E G

f o r any

E G we d e f i n e E nHE ; 1P'l-;

q(Q) = {(cp,)

QllH

+

.

01

&

, nHE , q)

d(GH

q(cp)

(cpp,) E

i s a proper approximation o f

Q(u)

(2.14)

HE = E

E

.

cp E G

nHc

-

11

q,cp

Q

6

;

v

,3

6 > 0

f o r any

cp

E Q E

&

, Q)

and

E G

> 0

such t h a t

0

.

1

i s a proper approximation o f

E~

H

.

Example 2.3 i s a p a r t i c u l a r case o f example 2.4 w i t h

Remark 2.4.

,

since, f o r

u E H , Q ( u ) by

= {(u&) E

llut

d ( H , r[H&

GH

:

We now d e f i n e , f o r any

Then

.

&

(2.13)

for

'PIIE d 6

.

{

Then

-

such t h a t

H be a norrned space and l e t G be a dense subspace o f H t h e space G equipped w i t h t h e norm o f H We c o n s i d e r a

HE = G

(2.12)

cp

E

cp E D

exists

o f normed spaces such t h a t :

(HE)

F o r any

110 +

llUE

. There

6 > 0

6

i s small enough. Hence

c

. Let

E S(u)

-

Indeed IIu l l E = c a l l u l l D + I I u I I E

+

IIu l l E f o r any

H = E

u E D

E

.

,G

= D

,

The purpose o f t h e n e x t s e c t i o n i s t h e statement o f t h e main theorems o f t h i s paper. 3. PROPER APPROXIMATION THEOREMS

Theorem 3.1.

Let

F

, F, , E

proper a p p r o x i m a t i o n o f example 2.4. i)

A

ii)

AE

(3.1)

F AE E %(HE

Let

i s surjective, and

A

Ac

.

, be normed spaces and l e t d(F,TTFE , P) G , H , HE , d ( H , r [ H E , Q) be t h e same

> 0

Let

, FE) , A

E .%(H, F )

i s bijective,

t

and suppose t h a t

> 0 ;

s a t i s f y the consistency condition :

AEq

L+

Acp

(FE

+

F

, P)

f o r any

cp E

G

,

be a as i n

A NORMED SPACE AND SINGULAR PERTURBATIONS

At

i i i ) the operators

K > 0

a constant

A

Then,

uc-

4 l l A t u IIF

Ht

1) L e t

u E G

+ l l A u IIF

ACuE-

.

, Q)

l l A E u II

there exists

.

f o r any

u E G

Au ( F t

F

and a n y

E

.

> 0

E

i s b i j e c t i v e , and

u ( H E -, H

___ Proof

satisfy the inverse s t a b i l i t y condition i.e.

such t h a t

K IIu II

(3.2)

93

.

From ( 3 . 1 ) ,

,

implies

P)

.

Au ( F t -, F , P )

Q

, I I u llH

From ( 2 . 1 2 )

FE Therefore, t a k i n g t h e l i m i t

Atu

-,

+

.

IIu IIH

Thus

E

o f (3.2) y i e l d s

K I I u IIH < IlAu l l F f o r any

.

u E G

Since

u E H

holds f o r any

.

6 > 0

. We

IIIAIII = IIAIIx(H,F))

.

Au ( F E -, F , P )

There e i x s t s

A E %(H,

and

H

, this

We w a n t t o show t h a t

-

IIcp

such t h a t

E G

cp

u llH 4

have

< llU& -

l l u c - qtcpllH

F)

inequality

i s injective.

A

ACut+

2) We suppose t h a t Let

i s dense i n

G

and

&

Il(p

It f o l l o w s f r o m ( 2 . 1 3 ) t h a t

-

Q I I H + Ilcp t

-

qE(pllH

311IAIIl

.

.

q,cpII

.

0

+

ut E Q ( u ) (where

6K

Ht From ( 3 . 2 ) and ( 3 . 1 )

&

-

llue

Thus,

qtcpllH 6 6

C o r o l l a r y 3.1.

E

Let

if

, be Banach spaces and l e t d ( F , n F t , P ) b e D , E , Ec , & ( E , nEE , S ) be t h e same as EX(EE, F t ) , A E X(E, F ) and suppose t h a t

F , FE ,

a proper approximation o f i n example 2.3. L e t i) ii)

At

f o r any

and cp

A

E D

a copstant

A

t

.

> 0

Let

i s bijective,

t >

0

.

s a t i s f y the consistency condition i.e.

AEcp

r-)

(FE

Acp

-,

F

.

iii)The o p e r a t o r s Then

At

F

i s s u r j e c t i v e , At

A

.

uc E Q ( u )

i s s m a l l enough and

c

satisfy the inverse s t a b i l i t y condition i.e. there exists

At

K > 0

, P)

K IIu I I E S IIA,u

such t h a t

i s b i j e c t i v e and

Acue

E

r-t

Au ( F E

+

I1 F

F&

f o r any

, P)

implies

u E uE-

D

.

u(EE

+

E

, S)

.

T h i s c o r o l l a r y f o l l o w s f r o m t h e o r e m 3 . 1 and r e m a r k 2.4. Theorem 3.2. L e t

F , FE

proper approximation o f example 2.3.

Let

operators o f

EE

of

t

,

E

> 0

b e Banach spaces and l e t

F

.

Let

D , E , EE ,

A EIf(E, F) into

and dense i n

E

Fc

.

and l e t

, whose

A&

domain

&(E,

d.(F , n F t , P ) be a be t h e same as i n

nEE , S )

b e a f a m i l y o f l i n e a r unbounded G

i s a subspace o f

D ,

independant

94

D. HUET

We suppose t h a t i)

A

ii)

A&

A

and

(3.3)

i

A&

there exists any

, P)

F

+

f o r any

u E G

K > 0

> 0

.

A C u E G Au ( F c

+

and f o r

t

&

,

F

Thus,

A& E

,

%(HE

FE)

K I I u I1

A E

and

+ 1)

6 (K

A

i s i n j e c t i v e since

We now suppose t h a t

. Let

(u,) E S(u) 6 > 0

if

E

llF

.

IIA,u

.

Since

vIIE

cp

-

Let

IIF

for &

uc+ u ( E c

-.

E

,

S)

.

u E G ; ( 3 . 4 ) and (3.5) y i e l d

. We

.

IlAu llF

(FE

+

F

.

H

i s dense i n

G

.

, P)

I t f o l l o w s from theorem 3 . 1 t h a t

s t i l l have t o prove t h a t

. Then,

E Gc0

uc E Q ( u )

6 IIu

llA,u

I t f o l l o w s from ( 2 . 9 ) and ( 3 . 3 ) t h a t

AEuC 4 Au

, Q)

u E o u (Hc .+ H

> 0

&

K I l u IIH < ( K t 1 )

-

.

HE

We now t a k e t h e l i m i t o f ( 3 . 6 ) .

IIu

g(H,F )


, Y I I u l l w

d

We d e n o t e b y v

a(u, v ) u E D(&),dLu

d

where

= cdb t

N

t h e unbounded o p e r a t o r i n

I

. For

any

i s a constant. Therefore, there e x i s t s

&(F

G = D ( J ~ )= ( D =

,nF,

, P)

=

v)

IIU

I I ~ 6) II

= {U

E V ;

,

K > D such t h a t

u llW

dEcp +

c ( E = w)

d(W,nW,

D(&)

domain i s

.

Furthermore, i t i s obvious t h a t Let

, whose

, one has ( s e e e.g. [61 , p. 135)

u E D(&)

~ ( c l ” I I 1 1~” t

(4.9)

W

on V e q u i p p e d w i t h t h e norm II. l l w 1 . F o r a n y b ( v , & u ) = a(v,u) f o r any v E V Let

i s continuous i s d e f i n e d by

-t

.

.

u E W

f o r any

u E V

f o r any

a =

in

cp

,

f o r any

W

F = W

,

u E D(A)

f o r any

cp

.

E D(4)

.

,

, T ) ( t h e t r i v i a l a p p r o x i m a t i o n o f W ) . Then, we c a n

a p p l y t h e o r e m 3.2 and t h e o r e m 3.3, t o

AE = d c

,A

= I

and we o b t a i n t h e m a i n

97

A NORMED SPACE AND S I N G U L A R PERTURBATIONS

theorem o f [ 2 1 Now l e t

.

0

1

V = @(Q) , W = Hm ( Q )

.

Let

c

and

b

be d e f i n e d by ( 1 . 2 ) .

Suppose t h a t ( 1 . 3 ) h o l d s . Thus, problems ( 1 . 4 ) and ( 1 . 5 ) a r e e q u i v a l e n t t o (4.10)

u&

J1,

E&

UE

=

and t h e r e s u l t s o b t a i n e d i n example 1.1 f o l l o w from theorem 2 . 1 and remark 3.2.

.

L e t 1 < p < + m L e t C be a l i n e a r d i f f e r e n t i a l o p e r a t o r . Assumpa r e t h e same as i n example 1.3. L e t ( 6 . ) , j = 1 ,..., m be a J normal system o f l i n e a r d i f f e r e n t i a l o p e r a t o r s w i t h c o e f f i c i e n t s d e f i n e d on t h e Example 4.3. t i o n s on C

W2m(Q; ( B . ) ) t h e subspace o f WZm(Q) o f a l l f u n c t i o n s u P J P B.u = 0 , j = 1, 2, ..., m , on t h e boundary aR o f C . ( I n example J a j-1 1.3 , E. = , where v denote t h e normal t o a a ) ( s e e [ l ] ) . I t was proved J F i n [l] t h a t , f o r any E small enough, f o r any u E W2m(Q; ( 6 . ) ) P J

boundary. We denote by satisfying

I t i s obvious t h a t , f o r any

D = WZm(Q; ( B . ) ) , P J

Let

= L

P

P

(Q) ,

7

a = 1

, -

t Ccp

+

cp

-, cp

in

,nFc

P ) = Jb(L

=

.

L

.

-,E

Lp(R)

(a) and l e t P be t h e t r i v i a l a p p r o x i m a t i o n o f ,F

(a) , Lp(R) , T) P Then, we can a p p l y c o r o l l a r i e s 3.1 and 3.2, t o AE = - E C + I , A = I Therefore u c h 1.3, t h e problem ( 1 . 8 ) i s e q u i v a l e n t t o Acu& = h &(F

,

E

cp E Wzm(0; ( B j ) )

.

L

(a).

P I n example

(a) , S ) and t h e r e s u l t s f o l l o w f r o m theorem 2.2. P Remark 4.1. The r e s u l t s o b t a i n e d i n s e c t i o n s 2 t h r o u g h 4 were announced i n [71 (Ec

= L

L - b e h a v i o r ( p # 2 ) o f t h e s o l u t i o n uE P boundary v a l u e problem a s s o c i a t e d w i t h t h e e q u a t i o n c C u E + B ut = h Remark 4.2. The s t u d y o f t h e

C

and

B

are e l l i p t i c l i n e a r operators o f order

2m

and

2m'

with

.

o f some

, where # D ,

m'

i s , a t my knowledge, an open problem. I n a f u t u r e paper I hope t o a p p l y p r o p e r convergence theorems t o t h a t problem. BIBLIOGRAPHY

[11 Agmon, S., On t h e e i g e n f u n c t i o n s and on t h e e i g e n v a l u e s o f g e n e r a l e l l i p t i c boundary v a l u e problems, Comm. Pure. App. Math 15 (1962) 119-147.

[21 Greenlee, W.M., Rate o f convergence i n s i n g u l a r p e r t u r b a t i o n s , Ann. I n s t . F o u r i e r 18 (1969) 135-192. 131 Huet, D., Sur quelques problemes de p e r t u r b a t i o n s s i n g u l i P r e s dans l e s espaces Lp , Rev. Faculdade Cienc. L i s b o a 11 (1965) 137-164. [ 4 1 Huet, D.,

Remarque s u r un theoreme d'Agmon, B o l l . U.M.I.

2 1 (1966) 219-227.

[51 Huet, D., P e r t u r b a t i o n s s i n g u l i P r e s de problemes e l l i p t i q u e s , L e c t u r e notes i n Math. S p r i n g e r Verlag, 594 (1977) 288-300.

D. HUET

98

[61 Huet, O . , Decomposition s p e c t r a l e e t o p e r a t e u r s , Presses U n i v e r s i t a i r e s de France, P a r i s , 1977. [ 7 ] Huet, D., C. R. Acad. Sc. P a r i s 289 (1979) 69-70

and

595-596.

[8] L i o n s , J . L . , and Magenes, E., Problemi a i l i m i t i non omogenei, Scuola Norm. Sup. d i P i s a , 15 (1961) 39-101.

[91 Stummel, F., D i s k r e t e Konvergenz l i n e a r e r Operatoren, Math. Ann. 190 (1970) 45-92. [ l o ] Stummel, F., S i n g u l a r p e r t u r b a t i o n s o f e l l i p t i c s e s q u i l i n e a r forms, L e c t u r e Notes i n Math. S p r i n g e r Verlag, 280 (1972) 155-180.

1113 V i s i k , M.I. and L y u s t e r n i k , L.A.,

E l l i p t i c problems w i t h a parameter and p a r a b o l i c problems o f general t y p e , Uspehi Mat. Nauk 1 9 (1964) 53-155 and Russian Math. Surveys 19 (1964) 53-159.

A N A L Y T I C A L AND NlJMERICAL APPROACHES TO ASYMPTOTIC PROBLEMS IN A N A L I ' S I S 5'. A x e l s s o n , L . S . P r a n k , A . v a n d e r S l u i s O N o r t h - H o l l a n d P u b l i s h i n g C o m p a n y , 1981

leas.)

AN ANALYSIS OF SOME FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION PROBLEMS Claes Johnson and Uno Navert Department of Computer Science Chalmers University of Technology 412 96 Goteborg, Sweden

We analyze two ways of improving the performance of the usual Galerkin finite element method when applied to a stationary advection-diffusion problem with small diffusion having a non-smooth solution. The first method consists in postprocessing the usual Galerkin solution by smoothing which increases the accuracy in regions where the exact solution is smooth. The second method is the result of applying the usual Galerkin method to a modified advection-diffusion problem obtained by adding extra diffusion in the streamline direction and compensating by modifying the right hand side. We.prove error estimates and give the results of some numerical experiments. INTRODUCTION The usual Galerkin finite element method when applied to an advection-diffusion problem may give an oscillating approximate solution which is not close to the exact solution in e.g. the L2-norm in case the exact solution is non-smooth and h > E/a where h is the mesh size, a is the velocity of the advection and E the diffusion coefficient. In this note we shall analyze two different ways of improving the performance of the Galerkin finite element method for such problems. Roughly speaking, the first method consists in post-processing the (oscillating) Galerkin finite element solution by a smoothing process, thus obtaining a non-oscillating approximate solution and reducing the L2-error. The success of the smoothing procedure relies on an error estimate in a Sobolev norm with negative index (cf.16)). The second idea, first proposed by Hughes and Brooks 1 5 1 , is to apply the Galerkin finite element method to a modified advection-diffusion problem obtained essentially by adding a suitably chosen diffusion term to the original problem. This "artificial viscosity" involves diffusion only in the direction of the streamlines and no "crosswind" diffusion. The resulting finite element schemes have an "upwind" character (cf.[2]) and produce non-oscillating solution but smear sharp fronts relatively little in the cross-wind direction. This is in contrast to the performance of more classical upwind or artificial viscosity 99

100

C.

JOHNSON and U. NAVERT

schemes obtained by adding diffusion in all directions which in general smear too much. Another important feature of the streamline diffusion method is that it is possible to compensate for the added diffusion by modifying the right hand side of the equation thus obtaining a higher-order accurate method. Such a modification cannot be done in practice if diffusion is added in all directions and classical upwind or artificial viscosity methods are only first order accurate. To briefly indicate some main features of the two methods to be considered below let us consider an advection problem with vanishing (or very small) viscosity. In such a problem information is propaga-

ted downstream in the direction of the characteristics (streamlines). If we apply the usual Galerkin method to such a problem we obtain a discrete model where information may be propagated also in other directions with relatively little damping. This is manifested by a discrete solution which oscillates in case the exact solution is nonsmooth and which is very sensitive to the outflow boundary conditions specified. However, by adding diffusion in the streamline direction one obtains a discrete model where the propagation is damped in directions transverse and opposite to the streamlines. In this way, we thus obtain a discrete analogue which better models the essential features of the continuous problem. In this note we shall consider a stationary constant-coefficient model problem with non-smooth solution. For simplicity we consider in the analysis only the case of vanishing viscosity. However, the problem considered is a representative model for linear advectiondominated flows with the main difficulties still present. For extensions of the results presented here to time-dependant and variable coefficient problems with non-vanishing viscosity, see [ 8 1 . An outline of the note is as follows: In Section 1 we introduce the model problem and in Section 2 we consider the usual Galerkin method together with smoothing. We prove in the case of continuous piecewise linear trial functions on a general triangulation that the global L 2 -error is at least of order O(h 'I4) and that with smoothing this can be improved to almost (3(h5'4) in regions where the exact solution is smooth. In Section 3 we analyze the streamline diffusion method and prove in the same case as in Section 2 a global L -estimate of order (3(h'I4) and a local L2-estimate of order (3(h3$) to-

ADVECTION-DIFFUSION

101

PROBLEMS

gether with a local L2-estimate of the derivative of the error in the streamline direction of order O(h). The results of Sections 2 and 3 can be extended to the case of higher order polynomial approximation with the corresponding increase of powers of h in the local estimates. Finally, in Section 4 we present some numerical results which indicate rates of convergence which are the same as or slightly better than the theoretical minimum rates just mentioned. Below C will denote a positive constant, possibly different at different occurences, which does not depend on the mesh parameter h. Further, for s a positive integer and R a bounded region we let H S ( R ) denote the usual Sobolev space with norm closure of C;(n) in the norm 1 1 - l l s , R .

11

s,n

and HE ( R )

is the

1. A MODEL PROBLEM As a model problem we shall consider the following stationary advec-

tion-diffusion problem with constant coefficients:

(l.lb)

-E-

auE + u n = gnx an X au' an

(1.lc) where

E-=

o

on on

Find

uE such that

r-,

r+u r0

is a small positive parameter, R is a bounded domain in the au Further, the boundary r is (x,y)-plane with boundary r and ux = -ax' divided into an inflow part r - l an outflow part r', and a characteE

ristic part Y o defined by (cf. Fig. 1.1)

~r

:nx(x,y) < 01 , E r :nx(x,y) > 0 1 , E r :nx(x,y) = 0 ) , where n = (nxln ) is the outward unit normal to r . We note that the Y boundary condition ( .lb) corresponds to a given total inflow and that the outflow condition ( 1 . 1 ~ )is of Neumann type. Below we shall a l s o briefly comment on the following Dirichlet boundary conditions as alternatives to the natural boundary conditions (l.lblc): (1.2) (1.3)

uE= g uE= o

on on

r-, r+ u ro.

C.

102

JOHNSON and U. NAVERT

ro Fig. 1.1 The r e d u c e d problem o b t a i n e d by s e t t i n g s = 0 i n ( 1 . 1 ) r e a d s a s f o l - lows : (1.4)

lx u

+ u = f

in

R,

u = g

on

r .

-

W e n o t e t h a t i n t h e r e d u c e d problem t h e boundary c o n d i t i o n s a r e g i v e n

o n l y on t h e i n f l o w p a r t

r - . The c h a r a c t e r i s t i c s of t h i s problem a r e

o r i e n t e d i n t h e d i r e c t i o n of t h e p o s i t i v e x - a x i s . I n g e n e r a l ( c f . [ 4 1 ) t h e s o l u t i o n of

( 1 . l a ) t o g e t h e r w i t h a combina-

t i o n of t h e boundary c o n d i t i o n s g i v e n by ( 1 . 1 b , c ) , w i l l have a boundary l a y e r a t t h e boundary I'+ U Y o .

l a y e r i s i n general of order O ( E ) a t

r+

( 1 . 2 ) o r (1.3)

The w i d t h of t h e

f ) along and C ( E

ro. Further,

i f t h e i n f l o w d a t a g h a s a jump d i s c o n t i n u i t y t h e n t h e s o l u t i o n w i l l

O(Ef ) i n t h e i n t e r i o r o f R ( i n t e r n a l l a y e r ) p a r a l l e l t o t h e s t r e a m l i n e s . I n t h i s case t h e s o l u t i o n u of t h e rehave a l a y e r of w i d t h

duced problem ( 1 . 4 ) w i l l have a jump d i s c o n t i n u i t y i n fi a c r o s s a characteristic. L e t u s now r e t u r n t o t h e model problem

( 1 . 1 ) . The f i n i t e e l e m e n t

methods f o r t h i s problem t o be i n t r o d u c e d below w i l l be b a s e d on t h e 1 f o l l o w i n g v a r i a t i o n a l f o r m u l a t i o n o f ( 1 . 1 ) : Find us E H (R) s u c h t h a t (1.5)

Ea(uE,v) + ( u i , v ) + (us,v)

-

< u E I v > -= ( f , v ) - < g , v > v v E H1(Q),

where a ( v , w ) = 1 Qv-vw d x d y , R ( v , w ) = I vw dxdy, R -= 1 vw n X d s .

r-

W e n o t i c e t h a t t h e boundary c o n d i t i o n s a r e imposed weakly i n ( 1 . 5 ) .

103

ADVECTION-DIFFUSION PROBLEMS

Remark. If the inflow data g and r- are smooth then it is possible auE in to compensate up to a term of order O ( C2 ) for the extra term can (l.lb) as compared to (1.2) by modifying the right hand side of (1.lb). In this case we may thus, neglecting terms of order O ( E2 ) , consider also the Dirichlet inflow condition (1.2) to be included in the formulation (1.5). However, if we replace the Neumann outflow condition ( 1 . 1 ~ )by the Dirichlet outflow condition (1.3) we will have to impose this condition strongly in the corresponding variational formulation.,

2. THE USUAL GALERKIN METHOD WITH SMOOTHING For simplicity, let us suppose that R is convex with polygonal boundary. Let Th = { K } be a family of triangulations of Q. indexed by the positive parameter h E (0,l) representing the maximum of the diameters of the triangles of Th. We shall assume that the usual minimum angle condition ( [ 3 1 ) is satisfied, i.e., the angles of the triangles K E Th are bounded below by a positive constant independent of h. Let us introduce the finite element space

vh

=

tv E H'(R): v is linear on K,

K E T ~ I ,

and let us then formulate the following discrete analogue of (1.5): Find u i E Vh such that (2.1)

(u;,v)-Ea (u;1,v)+ (u;lrx,v)+

Taking here v = u; (2.2)

lIv~;ll

c

r,

(f,v)--

VVEVh.

we easily obtain the following stability estimate: +

1IU;lI

+

Iuil 5 C(IIfl/ + 141

with C an absolute constant. Here

r'

=

(1.11

r

- ) r

is the L2(R) norm and for

1vlr, = ( /v2 Inxlds)+.

r' For simplicity we write 1v/ instead of Ivlr. Below, we assume that f E L2(R) and that 141 - < From the estimate (2.2) we obtain

-.

r uniqueness and thus also existence of a solution of (2.1).

We want to analyze the method (2.1) in the case E < h. In particular E may be very small. For simplicity we shall in the rest of this section consider only this case and we may then effectively put E

= 0 in (2.1). Thus

, we

shall analyze the following finite element

104

C. JOHNSON and U. NiVERT

method for the reduced problem ( 1 .4): (2.3)

Find

u h E Vh such that

= u(f,v)-~Uh,x'~~+~uh'v~-~ hlv~~

V V E Vh'

If the solution u of the reduced problem ( 1 .4) is smooth we easily find (cf. (2.4) below) that lIu-uhll 5 Ch. Here we want to analyze the case when u is discontinuous having a jump discontinuity across a characteristic corresponding to an internal layer in the full problem (1.1).

It is easy to guarantee immediate success of the finite element method (2.3) also in this case simply by choosing the finite element mesh to follow the streamlines where u is discontinuous. In practice, however, one does not in general have this freedom of adapting the mesh and thus we do not here assume any special properties of the triangulation Th except the minimum angle condition. Let us now prove some error estimates for the method (2.3) for the reduced problem (1.4) in the case the solution u of (1.4) is discontinuous. We have with e = u-uh: Theorem 2.1. There is a constant C independent of u such that (2.4)

IIelI

+

5 C(IIu-v/I + Iu-vI

lei

+

vv E Vh.

l~ux-v~l)

If u is peicewise smooth with a discontinuity across a characteristic

then

inf (11u-ql + Iu-vI + IIux-vx/I) 5 Ch 1 /4 . vEh' Proof. Taking v = uh-w with w E Vh in (2.3) and in the corresponding (2.5)

equation for the exact solution u we easily obtain (2.4) using the stability estimate (2.2) with E = 0. To indicate a proof of (2.5) suppose that u has a jump discontinuity across the x-axis, or more precisely, let us assume that

I

u(x,y) =

1

for

y > 0,

0

for

y < 0.

Let us now choose v E Vh to interpolate the function (XIY) E 1R

U(X,Y) = e(y),

2

,

at the nodes of Th, where 8 E C ' (R)is a transition function to be defined below satisfying 8 (y) E [ 0,1] for y E P I e(y) =

I

1

for

y > d,

0

for

y < 0,

and d > 0 is the width of the transition region D = I (x,y)E n:O

5 y 5 dl.

105

A D V E C T I O N - D I F F U S I O N PROBLEMS

S i n c e v a g r e e s w i t h u o u t s i d e t h e r e g i o n D h = i (x,y) E 0:-h

5

y < d+hl

o f w i d t h ( d + 2 h ) and v t a k e s v a l u e s b e t w e e n 0 and 1 , i t i s c l e a r t h a t (2.6)

< C(d+2h), llu-q12 + Iu-vl2 -

t h a t ux : 0

.

N e x t , t o e s t i m a t e I [ u -v 1 1 w e r e c a l l x x and t h a t vx = 0 o u t s i d e D h . By a T a y l o r ' s e x p a n s i o n

where C depends o n l y on

I?

u s i n g t h e f a c t t h a t fix = 0 it e a s i l y f o l l o w s t h a t , a s s u m i n g 0 " t o be piecewise continuous, P

where C d e p e n d s o n l y on t h e minimum a n g l e o f K . t h u s h a v e w i t h Ti = { K E T h : K

+ Ch2 I: J [ max Kt Ti K (x,yE K I f w e now c h o o s e d = h' e(y) = y

2

1

D

Recalling (2.6) we

# 81,

l e " ( y ) 11 2dx d y .

and

(3d-2y)h

we easily find, since

n

E K,

-3/2

18" ( y )

for 0

I 5

6h-l,

5

y

5

d,

that for h

5

1,

which proves t h e d e s i r e d r e s u l t . Remark 2 . 1 .

Theorem 2.1 c a n e a s i l y b e g e n e r a l i z e d t o t h e f u l l p r o b l e m

( 1 . 1 ) w i t h d i s c r e t e a n a l o g u e ( 2 . 1 ) . I n t h i s c a s e t h e Neumann o u t f l o w 1

boundary l a y e r c o n t r i b u t e s t o t h e e r r o r w i t h a t e r m of o r d e r O ( E ' ) and a n i n t e r n a l l a y e r w i t h a t e r m of o r d e r ( 3 ( ~ ' / ~ ) .

Remark

2.2.

I f w e r e p l a c e t h e Neumann o u t f l o w c o n d i t i o n

t h e Dirichlet outflow condition

( 1 . 1 ~ )by

( 1 . 3 ) and i n t h e c o r r e s p o n d i n g

d i s c r e t e analogue use a f i n i t e element space of functions vanishing

r + w e c a n n o t o u t o f a n a b s t r a c t e s t i m a t e o f t h e form ( 2 . 4 ) o b t a i n a p o s i t i v e rate of convergence. This i s because i n a D i r i c h l e t o u t -

on

f l o w l a y e r a t r + d e r i v a t i v e s o f t h e e x a c t s o l u t i o n o f order k b e h a v e -k l i k e O ( E ) ( c f . [ 41 1 . However, away f r o m t h e o u t f l o w boundary the exact

s o l u t i o n i s a f f e c t e d by d i f f e r e n t o u t f l o w b o u n d a r y c o n d i t i o n s o n l y up t o a t e r m o f o r d e r W E ) . M o r e o v e r , s i n c e t h e w i d t h o f t h e o u t f l o w boundary l a y e r a t

i s o f o r d e r O ( E ) and h >

E,

such a l a y e r cannot

b e r e s o l v e d by t h e c h o s e n d i s c r e t i z a t i o n and t h u s w e c a n n o t r e a l l y

106

C. JOHNSON and U. NAVERT

hope t o model d i f f e r e n t o u t f l o w c o n d i t i o n s o n

rt

in the discrete

model. Hence, o n e may as w e l l c h o o s e t h e o u t f l o w boundary c o n d i t i o n t h a t i s b e s t from c o m p u t a t i o n a l p o i n t o f v i e w , i . e . , t h e Neumann au = 0 on rt. A l a y e r a t Y o h a s t h e w i d t h O ( Ef ) outflow condition an and t h u s s u c h a l a y e r may b e r e s o l v e d t o some e x t e n t i f h < E'. The problem c o r r e s p o n d i n g t o a D i r i c h l e t c o n d i t i o n on Y o i s s i m i l a r t o

a problem w i t h a n i n t e r n a l l a y e r and c a n b e t r e a t e d a c c o r d i n g l y . From Theorem 2 . 1

it follows t h a t t h e g l o b a l L2(R)-error f o r t h e

G a l e r k i n method ( 2 . 3 ) i s a t l e a s t of o r d e r O(h1'4)

(the error i n the

( 0 ) - p r o j e c t i o n o n t o Vh i s of t h e o r d e r (!)(hi)). One c o u l d hope t o 2 g e t a h i g h e r r a t e of c o n v e r g e n c e away from t h e jumps i n u . However,

L

numerical experiments

( c f . S e c t i o n 4 ) show t h a t t h e a p p r o x i m a t e

s o l u t i o n o b t a i n e d by t h e u s u a l f i n i t e e l e m e n t method ( 2 . 3 ) i s o s c i l l a t i n g g l o b a l l y and t h a t t h e r a t e of c o n v e r g e n c e i n t h e L2-norm o v e r r e g i o n s where u i s smooth i s n o t b e t t e r t h a n t h e r a t e o f c o n v e r g e n c e i n t h e g l o b a l L2-norm. L e t u s now p r o v e an e r r o r estimate i n a S o b o l e v norm w i t h n e g a t i v e

index u s i n g a d u a l i t y argument. For s

L

0 w e i n t r o d u c e t h e norms

where 0' i s a bounded r e g i o n . B e l o w w e s h a l l w r i t e 0 ' c c 0 t o mean 0 and t h a t t h e d i s t a n c e from t h e c l o s u r e o f R ' t o t h e com-

t h a t R' c

plement o f n i s p o s i t i v e . Theorem 2 . 2 .

If

0' c c 0 t h e n t h e r e i s a c o n s t a n t C i n d e p e n d e n t o f

e such t h a t

2

P r o o f . Given X E H o ( n ' ) l e t cp b e t h e s o l u t i o n o f t h e p r o b l e m

I t i s e a s y t o see t h a t t h e r e i s a c o n s t a n t C i n d e p e n d e n t of X s u c h

that

F u r t h e r , by (2.7), cph

=

o

on

r+,

( 2 . 3 ) and (1.4 ) w e have f o r any cp

E Vh w i t h

ADVECTION-DIFFUSION PROBLEMS

By c h o o s i n g find that

I

vh E

(erx)

107

Vh t o be t h e i n t e r p o l a n t of cp and u s i n g

1 5 chllell I I X I I ~ , ~ ~

V

(2.8) we

2

X E HO(Q'),

which p r o v e s t h e d e s i r e d e s t i m a t e . By a smoothing p r o c e s s it i s p o s s i b l e t o o b t a i n an L 2 ( R ' ) - e s t i m a t e s i m i l a r t o ( 2 . 6 ) . To d e s c r i b e t h i s i n somewhat more d e t a i l , l e t K ( x , y ) be a smooth f u n c t i o n w i t h compact s u p p o r t s u c h t h a t

(cf.[6])

I K ( x , y ) d x dy = 1 , n,m

/ K ( x , y ) x n y m dx dy = 0 ,

2

0 , n+m

< r,

where n,m and r a r e i n t e g e r s , and d e f i n e

K6

=

1 7 K(5,:)

>

6

8

0.

I n t r o d u c i n g t h e smoothing o p e r a t o r S 6 S 6 v ( X , y ) = K,*v(X,y) =

d e f i n e d by

i K6 (x-x,y-y)v(x,y)dx dy,

we have i f R' c c n, f o r 6 s u f f i c i e n t l y s m a l l , (2.9)

II'-S6V11L2(R')


0: Find u: A

E

E V h--s u c h t h a t

E

(uh,v) = L(v)

v v E Vh/

where A€(W,V)

with

= E(VW,VV)

-

+ 6(wX+w,vx) + (wX,v) + (Wrv)

-

<w,v>-r

6 = max(0,h-E).

The problem ( 3 . 6 ) i s t o b e c o n s i d e r e d t o b e a d i s c r e t e a n a l o g u e o f t h e p r o b l e m ( 1 . 5 ) . L e t u s n o t e t h a t t h e s o l u t i o n uE o f ( 1 . 5 ) s a t i s f ies (3.7)

A€ (uE,v) = L ( V )

+

ET(AU€

vv E H

,vX)

1

I n p a r t i c u l a r w e n o t i c e t h a t t h e r i g h t hand s i d e o f from t h a t o f smooth.

( 3 . 6 ) w i t h a term o f o r d e r

For a n a n a l y s i s o f

( Q ) .

(3.7) d i f f e r s

(3(~8) i n r e g i o n s where u i s

( 3 . 6 ) see [ 81.

110

C.

Remark 3 . 2 . (l.la),

JOHNSON and U. NAVERT

The c o r r e s p o n d i n g d i s c r e t e a n a l o g u e o f t h e problem

( l . l b ) together with t h e D i r i c h l e t outflow condition

(1.3) reads:

Find

WE E

Wh

such t h a t

A E (wL,v) = L ( v )

(3.9)

v v E Wh,

where

w h = I V E vh:

v

= o

L e t u s now a n a l y z e t h e method

on

r+ u ro1.

(3.5) c o n s i d e r e d as a d i s c r e t e

( 3 . 1 ) and l e t u s t h e n from now on c h o o s e 6 = h .

analogue of

We have

which p r o v e s e x i s t e n c e o f a u n i q u e f u n c t i o n uh E Vh s a t i s f y i n g ( 3 . 5 ) i f 6 = h < 1 . By a r g u i n g a s i n t h e p r o o f o f Theorem 2 . 1 w e o b t a i n t h e f o l l o w i n g g l o b a l e s t i m a t e o f t h e e r r o r e = u-uh, where u s a t i s f i e s ( 3 . 1 ) : There i s a p o s i t i v e c o n s t a n t C independent of u such

Theorem 3.1.

< 4

that for h

I n p a r t i c u l a r , i f u i s p i e c e w i s e smooth w i t h a jump a c r o s s a c h a r a c t e r i s t i c , then t h e r e i s a constant C such t h a t

L e t u s now p r o v e a n estimate o f t h e e r r o r away from a jump o f u

across a c h a r a c t e r i s t i c i n R and away from t h e o u t f l o w boundary r + u Y o . T o b e s p e c i f i c l e t u s assume t h a t u h a s a jump d i s c o n t i n u i t y across t h e x - a x i s

n’

regions

= {(x,y) E

L e t us define

nl

c

n2

c

n

,

and t h a t u i s smooth i n t h e non-empty

>

n:y < 0)

e x c e p t p o s s i b l y close t o

(see F i g . 3 . 1 ) t h e subdomains n l and n 2 ,

with boundaries

r+u r 0 .

ADVECTION-DIFFUSION

r.

=

rI

= { (x,y) E

where 0 < a 2

r ia r e

PROBLEMS

I (x,ai) E n i u I (x,bi) E G I u

r

- :ai

< al < b l < b2

< y < bil

111

-

ri u

+

Ti'

i = 1,2,

and t h e d i s t a n c e s between

r +l I r 2+

and

positive.

Y

Fig. 3.1 L e t u s assume t h a t t h e e x a c t s o l u t i o n u i s smooth i n 0 2 . F u r t h e r ,

l e t u s assume t h a t t h e t r i a n g u l a t i o n Th i s q u a s i - u n i f o r m so t h a t t h e f o l l o w i n g i n v e r s e e s t i m a t e h o l d s f o r K E Th and v E Vh: IIvIIH1 ( K ) '

(3.9)

5

Ch

-1

IIv/IL2 ( K )

'

W e c a n now s t a t e and p r o v e t h e main r e s u l t o f t h i s s e c t i o n . Here

11 -11

d e n o t e s t h e L2 (ni)-norm.

Theorem 3 . 2 .

Under t h e a b o v e a s s u m p t i o n s w e have f o r h

Ile.Jn

1

< 4,

5 Ch.

P r o o f . By ( 3 . 4 ) and ( 3 . 5 ) w e have t h e f o l l o w i n g e r r o r e q u a t i o n (3.10) L e t now JI

I

ah:H

A(e,v) = 0 JI

vv E

be a smooth " c u t - o f f

vh' f u n c t i o n " d e f i n e d on D such t h a t

1 in R $= 0 i n R \ R 2 and J1, 5 0 i n R . F u r t h e r , l e t 1' 1 (il) + Vh b e t h e u s u a l i n t e r p o l a t i o n o p e r a t o r and d e f i n e

112

I - = u -

- uh so t h a t e = I- + 8 h ( c f . [31 ) w e h a v e

e

and

nhu

= II u

s t a n d a r d estimates

I1$nl1+

(3.11b)

2 1$17I-zCh

/uI

r

II+e -

l.IS,O

2 < Ch 2 Ch Iu12,1)2 -

hi1 ( $ n ) A I 5

(3.11a)

where

JOHNSON and U. NAVERT

C.

< Ch 2,r- + hi1

nh(+e)lI

Iqe12,K

(3.12b)

/$e

By

,

, 2

2

'

( $ e l x - ~ ~ ( ~ 1 e 5) ~Chl I(Cl$e12,K)', K

d e n o t e s t h e seminorm o f d e r i v a t i v e s o f o r d e r s i n t h e

e i s l i n e a r on e a c h

S o b o l e v s p a c e Hs(o). S i n c e

so t h a t ( c f .

2

and 8 E Vh.

D])

5

K w e have

Cilelll,K

using a l s o t h e inverse estimate

-nh($e)

lr-

5

(3.9),

.

Ch/81r-

I t i s e a s y t o check t h a t

2 h Ile.J*

E

= A(e,$e)

-

1-h

+

2 lel

2

2

IIelI*

+

h(ex,$,e)

+

1 -h ~ ( e , $ .e ) ,

where

Thus s i n c e $x

5

0 and h

and t h e f a c t t h a t e = E

5

A(e,$n)

< 1 w e have u s i n g also (3.10) w i t h v

I-

+ 0

+

A(e,$e-nh($e))

+

hile.Jln IIeII 2

R2

= ~ ~ ( $ 8 )

.

R e c a l l i n g (3.11) and ( 3 . 1 2 ) w e t h e r e f o r e conclude t h a t E

5 Ch 2 lleAIJI

+

c4le I I I I d n2

R e p l a c i n g h e r e 8 by e-n fact that

+ Ch311e

R2

+

11

+ Ch211elI$ + Ch 2 l e ( $ + N l e A l n

n2

2

cNelln I I e l I 2

R2

+

Ch e l

-

r2

+

1IelI

h311ell

R2

R2

and u s i n g a g a i n (3.1 ) t o g e t h e r w i t h t h e

.

A D V E C T I O N - D I F F U S I O N PROBLEMS

113

L

+ C h / e l2r 2

=

T h u s , s i n c e J,

+

Ch 3 l e l r 2

1 i n R,

Finally, since

a1ex/

w e have

41eJ

I t e r a t i n g t h i s i n e q u a l i t y s i x t i m e s on a s e q u e n c e o f d o m a i n s r e l a t e d

as a l and

R2

a n d u s i n g t h e s t a b i l i t y e s t i m a t e g i v e n by

(3.8) t o

f i n a l l y bound norms o v e r R w e o b t a i n t h e d e s i r e d r e s u l t . Remark.

Theorem 3.2 c a n e a s i l y b e g e n e r a l i z e d t o t h e case 0




01

,v

=

{xE

s"

:-1/2
O Combining (3.6) w i t h ( 3 . 1 ) , w e o b t a i n M i s n o t compact t h e n ( 3 . 6 j i s s t i l l v a l i d f o r p , q i n

123

PARABOLIC EQUATIONS

a compact subset of M - say, if p,q E A , where A = I z E M :d(x,z) Ia,d(z,y) I a] and a is a sufficiently large real number. On the other hand, the hyperbolic theory leading to (3.6) yields also the global L2 estimate

valid for every E > O if q ranges in a compact subset of M , and by considering A* instead of A one obtains

We now decompose the second integral in the right hand side of (3.5) into an integral over (MLV) fl A , and an integral over (M\V)\A For the integral over (MLV) n A we utilize (3.61, and for integration over (M\V)\A we use (3.7) and (3.8) along with the Cauchy-Schwarz inequality, thus absorbing those terms in the right hand side of (3.3).

.

If we have more specific information on the relationship between 2 and 7 , we can obtain more explicit information from (3.3). As an illustration, consider the case of a cut point which is not a conjugate point. (This happens for example on the circle S1 and on the torus.) In this case 7 is a regular value of exp-

.

Hence the set

(expZ)-l(y)

(finitely many) N

C

Tx(M)

unit vectors

T1

is discrete, and in particular there are

,...,TN E

T-(M)

with

7

1 6 i 1 N , (that N > 1 follows from the fact that the exponential map

exp-

V

C

ti

M

is a cut point of

is locally invertible near containing ?

exist neighborhoods U 1 , U 2

d(Z,y)Ti

-t

x,y E U1xU2

T(M)

.

, 1 1 i s N , such that Si(x,y) E Tx(M)

It follows that each pair

desic curves g(t,i,x,y) ( g(t,i,x,y) geodesics

=

x,y E U1xU2

expxtSi(x,y)/ lSi(x,y)

I),

7 ,

ii ) and

Thus there

smooth maps and

expxSi(x,y) = y

can be joined by

N

for

geo-

1 6 i C N , the functions

being smooth functions of their arguments g(t,i,x,?)

.

=

7 , respectively, and an open set

and

such that (3.1) and (3.2) are satisfied, and N

: U1xU2

exp.-(d(x,y)Ti)

t , x and y

, and the N

are all the minimizing geodesics joining X

7

with

.

Without loss of generality we may assume that for all x,y E U1xU2 , the geodesic minimizing the distance between x Set di(x,y) di(Z,y)

=

=

d(Z,y)

unit circle and

, 16 i E N

ISi(x,y)I and

d(x,y)

?=n/2

=

and y

.

is one of

We find that

g(t,i,x,y)

,

1IiIN

di(x,y) E Cm(UlxU2)

.

,

.

min di(x,y) in U1xU2 (Thus, if M is the 16iSN take dl(x,y) =y-x , d2(x,y) =a-(y-x) .)

, 7 =3n/2 , we

We want to express u(x,y,t) asymptotically (in U1XU2)by means of the functions di(x,y) For this purpose we assume (by decreasing U1 and U2 if necessary) that V contains not only the midpoints of minimizing geodesics joining x and y for x,y E U1xU2 , but all points of the form g (di(x,y)/Z,i,x,y), 1 5 i I N , i.e., all midpoints zi(x,y) of each of the N geodesics joining x and y constructed above. Under these circumstances, the only local minima of the exponent d2(x,z) +d2(z,y) in z E V are zi(x,y) , 1 6 i C N , and the value of All those minima are nonthe exponent at the i-th minimum is dZ(x,y)/2 degenerate. Applying Laplace asymptotic integral to the integral in (3.3) and noting that a multiplicative factor of the order ctnl2 is introduced, we see that there exist N families of functions v~,~(x,Y), v ~ 6 ,C (U1xU2) ~ , 1 s i 1 N , k=0,1, , such that

.

.

...

124

Y . KANNAI

as

t +O+

~-n/2

-a;(x,>

=

O(t

,

uniformly in U1xU2

i=l

Note that the functions di(x,y)

. are local solutions of the ciconal equation

Thus, the functions di(x,y) "branch" at x = z , y = j , and the formula (3.9) exhibits a "Stokes" phenomenon of the leading term passing from one di(x,y) to another one. The integral in (3.3) can be analyzed i n other cases as well, see [14]. The ndimensional integration over V can always be replaced by an (n-1)-dimensional integration (compare also [61); in many cases lower dimensional integrations suffice (thus, if 7 is a cut point which is not a conjugate point, the zero-dimensional integration (3.9) suffices), but there are cases (like Sn ) where we need all n-1 dimensions (in Sn the integral over V reduces asymptotically to an integral over the equator).

We pass now to the case (ii) of a Riemannian manifold with a non-empty boundary M If Z , y E int M and all minimizing geodesics joining x near X to y near 7 stay away from the boundary M , then the former analysis applies. New phenomena occur if either (i) X or 7 are near (or at) the boundary, or if (ii) X and 7 cannot be joined by a minimizing geodesics. Here we will discuss in detail only case (i) (for case (ii) see [14]).

.

The simplest case (analogous somewhat to the case of a cut point which is not a conjugate point) occurs if % E aM , 7 E iiit M and M is locally geodesically convex in the sense that there exist neighborhoods U1 ,U2 of f , 7 respectively, such that every two points x,y E U1,U2 can be joined by a (smooth) minimizing geodesic contained (except for possibly one end point) in int M Assume also that 7 is not a cut point of Z , and that the minimizing geodesic joining % and 7 makes a non-zero angle with aM at X Let u(x,y,t) be a fundamental solution of (l.l), where A is the zero Dirichlet data realization. Then u(x,y,t) = 0 for x E a M and all t > O , I n this case we set (as our new "branch" of a solution for the ciconal equation (3.10))

.

.

.

inf [d(x,z) +d(z,y)] E aM Then e(x,y) is a well defined smooth function in U1xU2 , e(x,y) = d(x,y) for x E aM n U1 , and e(x,y) , regarded as a function of x , is a solution of (3.10) (differing from d(x,y) i n U1 int M ) . (The function e(x,y) yields the "reflected" distance between x and y .) The characteristic curves of the nonlinear equation (3.10) with respect to the solution e(x,y) are the geodesics joining z to x , where z is the point in aM n U1 , where the infimum in (3.11) is attained. Hence transport equations can also be set with respect to e(x,y) It follows that in U1xU2 the fundamental solution for the mixed problem for the wave equation (with zero Dirichlet data) is given by (3.11)

e(x,y)

=

z

.

(3.12) where

R(x,~,s) = R~(x,Y,S) Rl(x,y,s)

-

R2(X,Y,S)

is given asymptotically by (2.6)

rewritten as

PARABOLIC EQUATIONS

125

where the functions gZ,k(x,y) are determined by means of the transport equations (corresponding to e(x,y) ) and the initial conditions g2,k(X,Y) = gl,k(X,Y) Applying the transmutation formula (1.7) to (a/as)R(x,y,s) for x E aM n U and using (3.13f and (3.14), we obtain (using a suitable finite propagation of supports argument) the asymptotic formula

.

Similar results can be obtained for the Neumann realization or the Robin realization (see [14]). If we let 7 tend to aM (in particular, if we want 7 to be near 5 ) then diffraction phenomena become important. Using a useful observation due to Seeley [191, we can say something in this case too. I n fact, the formula (3.15) still holds if (3.16) d(x,y) < C[dist(y,aM)] 112 for a certain constant C > 0 , as no reflected ray from x to y would hit the boundary once more (between x and y ) if (3.16) holds. .Hence the eiconal equation and the transport equations (relative to e(x,y) ) make sense in this case too. The recent advances in the theory of propagation of singularities (close to the boundary) for solutions of hyperbolic mixed initial boundary value problems (due to K . G . Anderson, J . Chazarin, G.I. Eskin, R. Melrose, M.E. Taylor, and others) have not led, s o far, to explicit formulas for w(x,y,s) (even in the second order case) - only micro local formulas are known. We can therefore argue only implicitly and obtain a !general formula" similar to the one obtained in Theorem 4.6 in [14]. Thus let R(x,y,s) -be a parametrix for a mixed initial boundary value problem in the sense that R(x,y,s) contains all the singularities of the fundamental solution w(x,y,s) where w(x,y,O) = 6(x,y) , (a/as)w(x,y,O) = 0 and w(x,y,s) satisfies (as a function of x , s) the boundary conditions. We assume also that the speed of light for the mixed problem is equal to one. Then $(x,y,s) = w(x,y,s)-R(x,y,s) is a C" function of all its arguments, and $(x,y,s) = -R(x,y,s) if I s 1 O

(5.2)

/e(x,y;Xm)dU(")(X)

=

/e(x,y;X)du(n)(X1/m)

=

O(e-En)

[O,") and satisfying such that

as

n+-

.

noted i n Section 4, formulas such as (5.2) (or the other statements (iii) and (iv) of Theorem 4.1) imply that, as a function of X , e(x,y;h) is either very small for large X , or else that e(x,y;A) is highly oscillatory. For example, if M = R ~and A = (l/i)(d/dx)m , then

As

Note that we have used here only one direction of Theorem 4.1 (that (i) implies (ii)), but it follows from the other direction ((ii) implies (i)) that we cannot get from the analyticity of w(x,y,s) more information than that contained in (5.2). Note also that in the second order case, discussed in Section 2, we have used the fact that w(x,y,s) = 0 for Is1 < d(x,y) (actually we used the vanishing of the even part of w ) . Thus the Fourier transform (1.4) has a gap, or de(x,y,h2) possesses a spectral gap. It i s well-known [ Z O ] that a s ectral gap implies that (the even extension to X < O of ) the measure de(x,y,h ) is highly oscillating. Here we use the weaker "analytic gap".

5

Weaker oscillation results (rapid decrease instead of exponential decrease) can be obtained if the coefficients of A are only C" (see [151).

6. OPEN PROBLEMS I n this section we suggest certain possible directions for further research along

the lines outlined in the preceding sections.

6.1. Small parameters. Consider the function u(x,y,t;~) fundamental solution of the differential equation

defined as the

The asymptotic behavior of u(x,y,t;~) , when both t and E tend to 0, t , € > O , has been studied via probabilistic methods [16]; to the best of my knowledge the problem has not been approached in a purely analytic fashion. The transmutation method requires knowledge of the fundamental solution of the hyperbolic equations

130

Y. KANNAI

Singularly perterbed hyperbolic equations, such as (6.2), were not investigated ( s o far) in detail in the literature. 6.2. Degenerate equations. What happens if we remove the assumption that A elliptic and require instead only that Zy,j=l aij(x)SiSj

(6.3)

>, 0

for all

x E M

,5

is

,

E Rn

(i.e. we replace a positive definite form by a positive semi-definite one)? Using probabilistic methods, it was shown [7] that the operator still exists in a certain sense, and in several cases information can be obtained on the kernels u(x,y,t) . An approach like the one taken in Section 2 of the present paper necessitates the study of the degenerate hyperbolic equation (2.3), where we assume only that the principal part of a(x,a/ax) satisfies (6.3). 6.3. Local analyticity o f Fourier transforms in several variables. No fully satisfactory extension of Theorems 4.1 and 4.2 to n dimensions have been found till now. If we take for p in (4.3) a Gaussian measure (this case is not covered precisely by Theorem 4.1(ii), but the difference is not crucial, see Remark 3.3 in [15]), then (4.7) reads - m r m e - i x ~ - xw(x)dx 2~ = O(e -€') as 5 - t - , (6.4) and the existence of E > 0 such that (6.4) holds is equivalent to (0,l)fWFA(w). For n-dimensions, it was essentially proved in [2] that ( 0 , t ) f WFA(W) if and only if ther.e exists a positive E such that

.

.

in the set O S p L X Here < , > denotes the inner product in Rn One of the difficulties in extending part (ii) of Theorem 4.1 to n-dimensions lies in the fact that behavior of the Hilbert transform differs makedly, depending on whether or not n = l Thus, let w(z) be holomorphic in Im z > O and let w(z) have sufficiently nice boundary values on the real line. Then the origin is a removable singularity for the function w(z) if and only if the origin is a removable singularity for the function w+(z) defined by

.

w+(z)

=

w(x)dx !7 , Im z > 0

.

But in two variables, there exists a function w(zl,z2) i = 1 , 2 , such that the function

holomorphic in

might be extended analytically from Im zl,Im 22 > 0 to being a removable singularity for w where ul (z l ) u2(z2)

.

Im z i >0,

C2 without the origin

For example, let w(zl,z2)

=

u1(z1)u2(z2),

is holomorphic in Im z l > 0 and has a singularity at

zl=O

, and

is an entire function such that u2 (XI

r---x- z

dx : 0 for

r=l .

Im z > O

.

Note that the symbol of the pseudo-differential operator mapping w equal to if n 2 2

H(Ei)

to w+

If n = 1 then this symbol is smooth in T*(R1) ' 0

then the symbol is discontinuous in the set

U~=l{S : 5,

= 01

.

is

, but

Note that

131

PARABOLIC EQUATIONS

the symbol is (micro) elliptic in the conic set get analytic hypoellipticity in this set.

‘

{ S : S i > O , lCiiln}

Similarly, it is not known whether the operator P (Pu)(x)

=

Rn/e

i<x,S>-<x,x> 5

,

, and we can

defined by

‘j(S)dt

mapps only functions which are holomorphic near the origin into such functions. ix 5 - x ? ~ j does have this property.) e j j (It is clear that the symbol

n;=l

’

REFERENCES Babich, V.M. and Rapaport, J u . O . , Short time asymptotic behavior of the fundamental solution of the Cauchy problem for a second order parabolic equation. (Russian) Problems in Math. Phys. 7 (Izdat. Leningrad Univ., Leningrad, 1974) 21-31. Bros, J. and Iagolnitzer, D., Local analytic structure of distributions I Generalized Fourier transformations and essential supports, reported by D. Iagolnitzer in Hyperfunctions and Theoretical Physics, Lecture Notes in Math. 449 (1975) 121-132. Cohen, J.K. and Lewis, R.M., A ray method fot the asymptotic solution of the diffusion equation, J. Inst. Math. App. 3 (1967) 266-290. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 11 (Interscience, New York, 1962). Duistermaat, J.J. and HGrmander,L., Fourier integral operators 11, Acta Math. 128 (1972), 183-269. Duistermaat, J.J. and Guillemin, V.W., The spectra of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975) 39-79. Friedman, A., Stochastic Differential Equations and Applications, Vols. I and I1 (Academic Press, New York, 1975). Hadamard, J . , Lectures on Cauchy’s problem in linear partial differential equations (Dover, New York, 1952). Hersh, R., The’method of transmutation, Partial Differential Equations and Related Topics, Lecture Notes i n Math. 446 (1975) 264-282. HGrmander, L., Pseudo-differential operators and hypoelliptic equations, Amer. Math. SOC. Symp. Pure Math. 10 (1966) Singular integral operators, 138-183. HGrmander, L., Fourier integral operators I, Acta Math. 127 (1971) 79-183. II8rmander, L., Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971) 671-704. Iagolnitzer, D. and Stapp, H.P., Macroscopic causality and physical region analyticity in S-matrix theory, Comm. Math. Phys. 14 (1969) 15-55.

1 Kannai, Y., Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. P.D.E.

2 (1977) 781-830.

132

Y . KANNAI

[15] Kannai, Y., Local analyticity of Fourier transforms, J. D'Analyse Math. 34 (1978) 162-193. [16] Kifer, Ju.I., On the asymptotic behavior of transition densities of processes with small diffusions, Theor. Probability Appl. 2 (1976) 513-522. [17] Maslov, V.P., Theorie des Pertubations et Methodes Asymptotiques (DunordGauthier-Villars, Paris, 1972). [18] Molchanov, S.A., Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975) No.1, 1-63. [19] Seeley, R., A sharp asymptotic remainder estimate f o r the eigenvalues of the Laplacian in a domain of R3, Advances in Math. 29 (1978) 244-269. [20] Shapiro, H.S., Functions with a spectral gap, Bull. h e r . Math. SOC. 79 (1973) 355-360.

A N A L Y T I C A L AND NUMERICAL APPROACHES T O ASYMPTOTIC PROBLEMS Ih' A N A L S S I S S . A x r l s s o n , L . S . F r a n k , A . v a n d c r Sluis ( e o s . ) d North-Hul l a n d P u b l i s h i n g C o m p a n y , 1981

DIFFERENCE APPROXIMATION FOR A SINGULAR PERTURBATION PROBLEM WITH TURNING POINTS Bruce Kellogg

R.

*

I J n i v e r s i t y of Maryland C o l l e g e P a r k , Maryland

The numerical s o l u t i o n of t h e sing111.ar p e r t u r b a t i o n problem pu' + qu = f , -1 < x < 1 , w i t h u ( + l ) g i v e n , i s cons i d e r e d . We suppose t h a t q ( x ) > 0 , and t h a t p ( x ) v a n i s h e s a t a f i n i t e number of p o i n t s , t h e s t a g n a n t p o i n t s . C e r t a i n o t h e r h y p o t h e s e s a r e made. An e r r o r e s t i m a t e i s g i v e n f o r t h e Southwell-Allen e x p o n e n t i a l d i f f e r e n c e scheme € o r t h i s problem. The e r r o r e s t i m a t e i s uniform i n E

-EU"

+

.

INTRODUCTION

We a r e i n t e r e s t e d i n d e v e l o p i n g n u m e r i c a l schemes f o r s o l v i n g s i n g u l a r p e r t u r b a t i o n problems which do n o t r e q u i r e mesh r e f i n e m e n t s n e a r t h e boundary l a y e r . Such methods have been d e v i s e d f o r one d i m e n s i o n a l s i n g u l a r p e r t u r b a t i o n problems witho u t t u r n i n g p o i n t s [1,2]. I n t h i s paper we a p p l y one such scheme, t h e SouthwellA l l e n d i f f e r e n c e scheme, t o a problem w i t h t u r n i n g p o i n t s . We s t u d y t h e d i s c r e t i z a t i o n e r r o r a s s o c i a t e d w i t h t h e a p p r o x i m a t i o n , Under c e r t a i n assumptions we o b t a i n a n e r r o r e s t i m a t e t h a t i s u n i f o r m l y v a l i d f o r a l l E €(0,1]. Our p r o o f s , a s i n [l], u s e ,comparison f u n c t i o n s and t h e maximum p r i n c i p l e . Very l i k e l y our r e s u l t s a r e n o t b e s t p o s s i b l e , e i t h e r a s r e g a r d s t h e r a t e of convergence i n t h e e r r o r e s t i m a t e , o r t h e h y p o t h e s e s t h a t we make.

We c o n s i d e r t h e boundary v a l u e problem -Eu"

+

pu'

+

qu

given

u(+l)

(1.1)

,

f

=

-1 < x < 1 ,

.

We suppose t h a t p ( x ) , q ( x ) , f ( x ) a r e smooth f u n c t i o n s i n -1 < x < 1 , and t h a t W e a l s o suppose t h a t p(+l)-# 0-, and t h a t E > 0 and q ( x ) > 0 , -1 5 x 5 1 p(x) v a n i s h e s o n l y a t a f i n i t e number of p o i n t s i n (-1,l) We c a l l t h e z e r o s of p ( x ) s t a g n a n t p o i n t s of t h e problem. I f p(x*) = 0 , we suppose t h a t We s a y t h a t x* i s a n a t t r a c t i v e ( r e p u l s i v e ) s t a g n a n t p o i n t i f p'(x*) # 0 p'(x*) < 0 (p'(x*) > 0 )

.

.

.

.

I t i s n o t h a r d t o s e e t h a t t h e r e i s a unique s o l u t i o n of t h e problem (1.1). Physic a l l y , u ( x ) may r e p r e s e n t t h e c o n c e n t r a t i o n of a s u b s t a n c e which i s d i f f u s i n g and c o n v e c t i n g i n a f l u i d . With t h i s i n t e r p r e t a t i o n , E i s t h e d i f f u s i o n c o e f f i c i e n t , p ( x ) i s t h e f l u i d v e l o c i t y , f ( x ) i s a n e x t e r n a l s o u r c e of t h e s u b s t a n c e , and qu r e p r e s e n t s a l o s s t e r m . Our i n t e r e s t i n t h e problem i s when process. may b e shown t h a t a s s o l u t i o n u of t h e reduced problem

It

*

E E

i s s m a l l and c o n v e c t i o n dominates t h e flow 0 , t h e s o l u t i o n u converges t o t h e

-+

Work s u p p o r t e d i n p a r t by t h e N a t i o n a l I n s t i t u t e s of H e a l t h . I33

134

R.B.

pu'

+

$(-l) ;(1)

=

KELLOGG

qu = f

- l < x < l

= u(-l)

if

p(-1)

u(1)

if

p(1) < 0

> 0

.

The problem ( 1 . 2 ) c o n s i s t s of a f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n , and 0 , 1, o r 2 boundary c o n d i t i o n s , depending on t h e s i g n of p ( + l ) N e v e r t h e l e s s , i t may b e shown t h a t ( 1 . 2 ) h a s a unique c o n t i n u o u s s o l u t i o n .

.

PROPERTIES OF THE SOLUTION For our e r r o r a n a l y s i s we r e q u i r e e s t i m a t e s f o r t h e d e r i v a t i v e s of t h e s o l u t i o n of (1.1) t h a t a r e uniform i n c One may show t h a t a t each p o i n t x € (-1,l) such t h a t p ( x ) # 0 , t h e d e r i v a t i v e s of u s a t i s f y

.

I u ( ~ ) ( x ) I5 c ( x , k )

(2.1)

u

.

where c ( x , k ) depends on p. q , and f , b u t does n o t depend on E The p o s s i b l e p o i n t s of d i f f i c u l t y a r e t h e end p o i n t s , x = +1 , and t h e s t a g n a n t p o i n t s . I f p ( 1 ) < 0 , t h e r e i s a boundary c o n d i t i o n a t x = 1 f o r t h e reduced problem, and i t may be shown t h a t ( 2 . 1 ) h o l d s a t x = 1 . . I n t h i s c a s e , t h e r e i s no boundary I f p ( 1 ) > 0 , t h e arguments i n [l] g i v e a d e q u a t e i n f o r m a t i o n layer a t x = 1 t o bound t h e d e r i v a t i v e s . A similar t r e a t m e n t may b e given a t x = 0 I t remains t o c o n s i d e r t h e s t a g n a n t p o i n t s . I f x* i s an a t t r a c t i v e s t a g n a n t p o i n t , i t may a g a i n b e shown t h a t ( 2 . 1 ) h o l d s a t x = x* I f x* i s a r e p u l s i v e s t a g We s h a l l assume t h a t n a n t p o i n t , we s e t a = - p ' ( x * ) / q ( x * ) > 0 , and B = l/a

.

.

.

.

a > 1 f o r each r e p u l s i v e s t a g n a n t p o i n t

(2.2)

.

We then have Lemma 1. Let x* be a r e p u l s i v e s t a g n a n t p o i n t . c o n t a i n i n g X* and c o n s t a n t s Ck > 0 , k = 1 , 2 ,

There i s an open i n t e r v a l I t h a t , i f u solves(1.2),

..., such

We s k e t c h t h e i d e a s i n t h e proof of t h i s lemma. Suppose, f o r s i m p l i c i t y , t h a t t h e s t a g n a n t p o i n t i s a t x = 0 . The dominant b e h a v i o r of t h e s o l u t i o n n e a r x = 0 i s given by t h e s o l u t i o n of t h e homogeneous e q u a t i o n -Ez"

- p'(0)xz'

+

q(0)z = 0

o r , a f t e r d i v i d i n g through by

q(o), and r e d e f i n i n g

(2.3)

-62''

- axz'

+

E

,

z = 0 ,

.

Suppose z ( x ) solves ( 2 . 3 ) i n ( - l , l ) , w i t h z ( t 1 ) g i v e n independent of E Set t = x ( a / ~ ) l / ,~ z ( x ) = Z ( t ) Then Z ( t ) = e x p ( - 1 / 4 t 2 ) z ( t ) , where z i s a 1/2 We a l s o have z ( + ( a / ~ ) l / ~= ) p a r a b o l i c c y l i n d e r f u n c t i o n of o r d e r R i(tl) exp ( h a / € ) R e p r e s e n t i n g z ( t ) i n terms of t h e s o b a s i c p a r a b o l i c c y l i n d e r f u n c t i o n s , a s given i n [ 3 , 19.31 and u s i n g t h e p r o p e r t i e s d e s c r i b e d t h e r e , we a r e led to the desired inequality.

.

.

+

.

THE DIFFERENCE APPROXIMATION

Let N be a p o s i t i v e i n t e g e r , and l e t t h e r e be g i v e n a uniformly spaced c o l l e c t i o n of mesh p o i n t s on [-1,1], w i t h mesh s p a c i n g h = N - l The mesh p o i n t s a r e We u s e t h e d i f f e r e n c e o p e r a t o r s D ' u i = ( u i + l - u i - l ) / Z h , D 2 U i i h , -N 5 i 5 N u i - l ) / h 2 , and we d e f i n e y ( t ) = t c o t h t . We s h a l l c o n s i d e r (ui+l - 2ui t h e Southwell-Allen d i f f e r e n c e o p e r a t o r Lh , d e f i n e d by,

+

.

.

135

SINGULAR PERTURBATION PROBLEM

where p i = p ( i h ) , q i = q ( i h ) , e t c . The o p e r a t o r Lh i s of p o s i t i v e t y p e , i n t h e f o l l o w i n g s e n s e : i f U and V a r e 2 mesh f u n c t i o n s such t h a t U+N 5 V ~ N, and LhUi t h e n U i LVi f o r -N < i < N sequence, t h e mesh f u n c t i o n U i

5 LhVi ,

.

-

N ,

-N < i

(The proof of t h i s i s given i n [l].) d e f i n e d by

A s a con-

(3.1) h a s a unique s o l u t i o n . We a r e i n t e r e s t e d i n e s t i m a t i n g t h e e r r o r e i = ui - Ui , where u i s t h e s o l u t i o n t o (1.1) and U i s t h e s o l u t i o n t o (3.1). For t h i s , w e s h a l l e s t i m a t e t h e t r u n c a t i o n e r r o r ~i = Lh(u-U)i = Lhui - f i , and we s h a l l i n t r o d u c e comparison f u n c t i o n s t o c o n v e r t e s t i m a t e s f o r ~i i n t o e s t i m a t e s f o r ei * To e s t i m a t e ~i we u s e t h e f o l l o w i n g i n e q u a l i t i e s , which s h a r p e n t h e correspondi n g i n e q u a l i t y i n [l, Lemma 3.31:

x. X.

X.

1- 1

) I u " ( t ) Idt

1-1

1-1

xi+l ( X i + p ) lu"(t) d t ;

+ ch-llPilj xi

xi+l

We s h a l l u s e Lemma 1 t o e s t i m a t e ~i n e a r a r e p u l s i v e s t a g n a n t p o i n t . s t a t e an e a s i l y proved i n e q u a l i t y . Lemma 2 .

Let

u > 0 , and l e t

.

g ( x ) = (x+l);'

I

Ig(x+h) - g(x-h) Ig(x)-ll

Then f o r some

,

< ch(x+l)-'-'

5 c(x+xbg(x) ,

c > 0

We f i r s t

,

o.h5x/2, x > 0

.

To o b t a i n s u i t a b l e bounds on t h e t r u n c a t i o n e r r o r , we a l s o must assume:

(3.4)

each r e p u l s i v e s t a g n a n t p o i n t i s a l s o a mesh p o i n t .

Using t h i s assumption we have Lemma 3 . L e t x* b e a r e p u l s i v e s t a g n a n t p o i n t . There i s a n i n t e r v a l I cont a i n i n g x* and a con t a n t c > 0 independent of h and E s u c h t h a t , i f x i E I , and i f d = , and z i = \xi-x*l ,

(3.5)

(3.6)

bil I c

h6+h2-BdB + c z (h+6)2

'

{h6'+h1+' (h+6)2

+

(h+6)2-B

1

,

zi = 0 , h

.

R.B. KELLOGG

136

~

Proof.

zi

If

2 2h , we

u s e Lemma 1 and Lemma 2 t o e s t i m a t e xi+h zi+h

5

Iu"'(t)Idt

E

~5

(t+A)8-3dt

x.-h

z .-h

< c5 2 [(zi-h+6)B-2 - (zi+h+5)@-']

-

=

2 .

c@[(1+

5

2.

1-hp-2 -

($+1+h)B-2 5 1

6

< c6B -

. 1?6 .

2.

(++ 1)@-3

< ch6 2 (z~+LF)'-~

-

.

This bounds t h e f i r s t term on t h e r i g h t s i d e of ( 3 . 2 ) . ed s i m i l a r l y , g i v i n g ( 3 . 5 ) . Suppose now t h a t

xi =

X*

+

h

.

The second term i s t r e a t -

Then we have

xi+l l u " ' ( t ) l d t 5 c5

E

( t + ~ S ) ~ - ' d t< c(h6+h2-B6B)(h+6)B-2

,

X. 1- 1

0

'i-1

.

+

which proves ( 3 . 6 ) , i n t h e c a s e xi = x* h I f x = x* - h , a s i m i l a r argument a p p l i e s . I f xi = x* , pi = 0 , and t h e remainfng term i n ( 3 . 2 ) i s e s t i mated a s above, f i n i s h i n g t h e proof of t h e lemma.

A COMPARISON FUNCTION By a comparison f u n c t i o n we mean a mesh f u n c t i o n V such t h a t L Vi > 0 , -N < These f u n c t i o n s a r e u s e d , t o g e t h e r w i t h tke maximum p r i n i < N , and VtN > 0 It is evident t h a t the function c i p l e , t o c o n v e r t bounds on T t o bounds on e E d e f i n e d by E . 3 1 i s a comparison f u n c t i o n . I n [l] w e c o n s t r u c t a comparison f u n c t i o n t o handle t h e boundary l a y e r . I n t h i s s e c t i o n w e g i v e a comparison function t o handle t h e e r r o r a t a repulsive stagnant point.

.

.

Suppose, f o r s i m p l i c i t y , t h a t x* = 0 i s a r e p u l s i v e s t a g n a n t p o i n t , a n d l e t a > 1 be t h e number a s s o c i a t e d w i t h t h i s p o i n t . L e t a > a , B = l/a < B , and let P > 0 Set -

.

S ( x ) = (1x1

+

.

& @

We s h a l l c o n s t r u c t o u r comparison f u n c t i o n u s i n g t h e mesh f u n c t i o n

We s h a l l assume t h a t (4.1)

a < a*

=

1.035

.

We c o n j e c t u r e t h a t t h i s assumption i s n o t e s s e n t i a l .

Si = S(xi)

,

SINGULAR PERTURBATION PROBLEM Lemma 4.

For

xi

0

i n a neighborhood of

(4.2)

xi # 0 ,

and

’-’

-

I I

5

LhSi

,

c1 (a xi h+c) ( X ~ + E ’ ] )

0 , /Lhsi/

For x i o u t s i d e any neighborhood of a r e independent of E and h

.

137

.

5 c2

. The c o n s t a n t s

c 1 , c2

.

P r o o f . We s h a l l prove ( 4 . 2 ) f o r x > 0 Suppose, w i t h o u t l o s s of g e n e r a l i t y , t h a t q ( 0 ) = 1 . S i n c e S ( 4 ) ( x ) < 0 , we have from [1,(3.7)],

.

D2S(xi) < S ” ( x i ) Since

y(t)

,

c3(l+ltl)

with

c3

.654 , t h i s g i v e s

=

To h a n d l e t h e f i r s t d e r i v a t i v e term, w e s t a r t w i t h t h e i n e q u a l i t y

(l+z)B - (1-2)B

< -

202

where

+

c 22,

0
0

.

arrive a t the inequality

x piBSi

5 -1

-

B*

We s e t

= .965

1-1 = 2/(2-8)

!j c4pih

:(

-

x + E ’ ) ~ ~. ~

.

cZ(1) = -.6137, w e s e e t h a t t h e r e i s a -

suffices.

,

’0

,

B”5Bil

This proves ( 4 . 2 ) .

and u s e t h e i n e q u a l i t y (x+e’)B-2 2 c ( x e-z+€z)-i

Then we may f i n d c o n s t a n t s (4.5)

=

,

f o r x i > 0 , xi n e a r 0 For x i s u f f i c i e n t l y c l o s e Using t h i s , and combining ( 4 . 3 ) and ( 4 . 4 ) , we o b t a i n

B(l-B)c3 - C 4 G )

In f a c t , l y proved

< 1

.

c 4 = c4(B) = 28 - 28 Setting

2

c5 > 0

Lh(S+cgE)i

B* < 1 s u c h t h a t

.

The rest of t h e lemma i s r e a d i -

.

such t h a t

2

I I

c 6 ( a xi h+E )

2-B+€2

+ c 7 , . i # 0 .

X.

This i n e q u a l i t y g i v e s t h e d e s i r e d comparison f u n c t i o n t o h a n d l e t h e r e p u l s i v e s t a g nant point. ERROR ESTIMATE

We now s k e t c h t h e d e r i v a t i o n of uniform e r r o r e s t i m a t e s f o r o u r d i f f e r e n c e approximation. For t h i s , we c o n s t r u c t a comparison f u n c t i o n V , and we show t h a t t h e r e I ~ i l5 Ch’LhVi From t h i s and t h e a r e p o s i t i v e c o n s t a n t s 0 and c such t h a t l e i 1 5 cheVi , which i s f a c t t h a t Lh i s of p o s i t i v e t y p e , i t t h e n f o l l o w s t h a t t h e d e s i r e d e r r o r e s t i m a t e . The comparison f u n c t i o n i s chosen t o be a p o s i t i v e l i n e a r combination of t h e f u n c t i o n s S , one f o r each a t t r a c t i v e s t a g n a n t p o i n t , t h e f u n c t i o n E , and ( i f n e c e s s a r y ) t h e comparison f u n c t i o n s used i n [l] t o hand l e t h e boundary l a y e r . I n t h i s p a p e r , w e d i s c u s s o n l y t h e e r r o r a r i s i n g from a r e p u l s i v e s t a g n a n t p o i n t , a s t h e e r r o r a r i s i n g from t h e bounda’ry l a y e r i s known t o be O(h)

.

.

138

R.B. KELLOGG

To o b t a i n o u r e r r o r e s t i m a t e s , w e s h a l l r e q u i r e an i n e q u a l i t y r e l a t i n g geometric programming and l i n e a r programming. Lemma 5. L e t Bi 5 0 and e i j 2 0 , b e g i v e n , where Suppose t h e s e t of l i n e a r i n e q u a l i t i e s

2

n

, 15

j

5m

.

lziLrn,

y a e . < e i ,

(5.1)

15 i

j = l j i~ -

m

has a solution.

For p o s i t i v e q u a n t i t i e s

Hence, s e t t i n g

,

wi

w.= J

t o prove ( 5 . 3 ) , i t s u f f i c e s t o prove n Bi n w.1 i= 1

Since

kn

and numbers

n

n

< c

cii

satisfying (5.2),

eij

w i=l i

-

w i E (O,l),

.

1

(5.3)

Proof.

Then t h e r e i s a c o n s t a n t c > 0 such t h a t f o r a l l n 8. m n 8.. n Wil 5 c n WilJ i=l y = l i=l

’

m

ci.

.

n w J j

j=1

i s a monotone f u n c t i o n , i t s u f f i c e s t o p r o v e , f o r some n m eiLn wi 5 a.knW. + c = a j e i j e n wi , -i=l j=1 J J i,j

1

1

1

or C(ei-

i

Since

kn wi

5 0 , t h i s inequality

l a j e i j ) k n wi j

5c

.

i s i m p l i e d by ( 5 . 1 ) , proving t h e lemma.

We now s t a t e Lemma 6 . L e t x* be an a t t r a c t i v e s t a g n a n t p o i n t . Then t h e r e a r e c o n s t a n t s 8 > 0 and c > 0 , independent of h and E , such t h a t f o r a l l x i i n a neighborhood of x*, x i # x*, 5 Ch%h(SfE)i The c o n s t a n t 6 may be t a k e n t o be any number < ( 1 / 2 ) 6

.

ITI~

.

The proof r e q u i r e s a d e t a i l e d examination of t h e bounds f o r -ii Lemma 3 . I n t h i s , Lemma 5 i s a u s e f u l t o o l . The r e s t r i c t i o n 0

,

a s g i v e n by

0

I

5

ch

0

,

xi

i s independent of

# a repulsive stagnant point, h

and

E

.

REFERENCES [l]

Kellogg, R.B. and Tsan, A . , A n a l y s i s of some d i f f e r e n c e a p p r o x i m a t i o n s f o r a s i n g u l a r p e r t u r b a t i o n problem w i t h o u t t u r n i n g p o i n t s , Math. Comp. 32 (1978) 1025-1039.

[2]

Berger, A . E . , Solomon, J . M . , Ciment, M . , , L e v e n t h a l , S.H. and Weinberg, B . C . , G e n e r a l i z e d O C I schemes f o r boundary l a y e r problems, t o a p p e a r i n Math. Comp.

[ 3 ] Abramowitz, M. and Stegun, I . A . ,

Handbook of Mathematical F u n c t i o n s (U.S. Government P r i n t i n g O f f i c e , Washington, D.C. 1 9 6 4 ) .

This Page Intentionally Left Blank

A N A L Y T I C A L AND NLI.YZRICAL APPROACHES TO ASYMPTOTIC PROBLEMS I N A N A L Y S I S S . A x e l s s o n , L . S . F r a n k , A . van der S l u i s (eds.) @ N o r t h - H o l l a n d P u b l i s h l n y Company, 1981

STABILITY AND CONSISTENCY ANALYSIS OF DIFFERENCE METHODS FOR SINGULAR PERTURBATION PROBLEMS Jens Lorenz Fakultat fur Mathematik Universitat Konstanz Konstanz Bundesrepublik Deutschland The well known principle "stabilty and consistency imply convergence" is often used to study the behaviour of discretisation processes. In this paper we consider singular perturbation problems and ask for convergence uniformly in the perturbation parameter e . The principle "stability uniformly in E and consistency uniformly in E imply convergence uniformly in E " will be used to study difference methods for a mildly nonlinear two-point boundary value problem of singular perturbation type * INTRODUCTION Consider a "continuous" problem depending on a parameter e E (1)

(Oreo]

T u = 0, u < U c l R R &

where u is an unknown function out of some function space U . As a specific example we shall take the boundary value problem (2a) T u = - E U " + a(x)u' + b(x,u) = 0, X E [0,1], e 2 (2b) u E U = { V E C [0,11: v(0) = v(1) = 0). Assume that (1) has a unique solution called u e E U , and let (3)

Telhu = 0, u E U h

=

lRRh

be a discretisation of (1) with a unique solution called u E r h E U h . Q h c Qdenotes a finite mesh with the discretisation parameter h. We use the notation of finite difference methods, but the general ideas also apply to finite element equations with a slight change of terminology. If E is kept fixed, then the discretisation ( 3 ) is called (asymptotically) stable with respect to norms I1 11, I1 11' if there exists ho(c) > 0 and CO(&) > 0, such that the following stability inequality holds (4)

IIU-~IISC~(E) IITc,hU-TE,hVIIt Vu,vEUh,

(The norms I1 11,

11

VhE (OfhO(&)l.

formally depend on h, but we suppress this 141

142

J . LOREN2

notationally. ) Let uE l h E lRRh = Uh denote the restriction of the continuous solution u to the grid ah; thus TErhuClhis the vector of & truncation errors, and the discretisation ( 3 ) is called consistent of order p (for E fixed) if (5) IITCrhu&IhII' s C 1 (c)hp. Evidently, we can conclude from ( 4 ) and ( 5 )

I I U ~ ~ ~ - U ~ , IITt,h~cIhII' ~ ~ I ~ C ~ ~ ( C~o ( ) & ) C 1 ( ~ ) hfor P

O
-(l-y(P,pt)t). P

d.

It is sufficient to prove the inequalities (34), (35) for 0 < r s p , 0 < t s 1. Since the function p + y ( p ,pt) decreases, the estimate (34) follows from q(rt) 5 1 - (l-l/q(r))/y(r,rt), and this last estimate is easily proved. To show (35) notice that 1 r+-(1-l/q(r)) is decreasing. Thus it is sufficient to prove (35) for r r = p, and again the estimate l / q ( p ) 0 )

the estimate (36) is proved in [131. All functions listed in [191 fulfill (37). 4. CONSISTENCY AND CONVERGENCE Consider an equation (2) under the general conditions of section 1, and let G be a grid 'as above. In section 3 we have described operators

152

J

I R ~ TE ,G: which are T& u = (L = L E IG

. LORENZ

+ x~m , T C l G u = Lu + MbGu, u E discrete analogues of the different a1 operator cu" + a(x)u' + b(x,u) = L u + b(x,u and M = M are (m1m+2)-matrices.) The truncation error is -

E

,G

TE,GUCIG Rm' Now let u & = g, + 1,. We have in mind to split up u in a layer& function l& and a function g with harmless derivatives; see the & theorems 4 and 5. Then 0 = TEIGUClG= LuelG + MbGUEIG = LuclG + M(-L u ) I G f thus 0 = qelG -

&

(38)

rl =

{Lg,lG - M(L,g,) lG)

&

+ (L1,IG - M(Lcl,) lGl.

Therefore the discussion of the truncation error can be split up into two parts, namely an estimate for the harmless part g and an estimate & for a pure layer-function 1 . Such a split up is trivially possible & for linear problems, but, in general, seems to be impossible for nonlinear equations. We indicate the proofs of the following two results [131, where an equidistant grid G = Gh = {O,h,...,mh,ll is assumed. u E Rm+2denotes the discrete solution, and C is a generic constant c,h independent of h and E . Theorem 10 Let a E C2 [0,11, a(x) t a > O , b E C 2 ([O,l]xR). If

1 - a

E

ih

is computed with

IInII1

and thus with Y&I

U

IILV, h "1

38) fol ows

'

using standard arguments. It is just needed that

STABILITY AND CONSISTENCY ANALYSIS

=

&(h-2(-l,2,-l)gd

+

+

g''(xi))

+ a1. (-(-l,O,l)gd 1 2h

153

- gC(xi))

E(O(O~)-I) h -2 (-1t2r-l)gd.

Using standard estimates for difference formulas we find 1 Ig"(s) Ids + &pih-1 J Ig;(s) Ids, liil s T E J Ig;'(s) I ds + piJ & where the integrals are taken over [xi-l,xi+l]. Here p = aih/(2c) , i and summing up we find m 1 1

Now ( 3 9 ) immediately implies I1 5

Ill

5

Ch.

c. The estimate IILcpEIhlll5 C h is trivial if a(x) is constant, but requires care otherwise. We omit the proof here, see [13]. Theorem 10 generalizes results given in [10,11,151 for linear problems. formly in

E ,

For the Mehrstellen-scheme quadratic convergence, unican be stated under special assumptions [131.

Theorem 1 1 Let a(x) = a E I R , a + O , b ( x , u ) = b ( x ) u

-

c(x), 0 , c € C5 [0,1]. If u

is the solution obtained with the Mehrstellen-scheme, then

E

th

A proof follows similar lines as above where again corollary 1 establishes stability uniformly in c . The solution u is spl.it up & using theorem ,5. The boundary layer term does not produce a truncation error here since a(x) is constant. In theorem 10 and 1 1 we stated convergence, uniformly in E , for methods on equidistant grids. The difference formulas were exponentially fitted: this is important for a uniform estimate of the consistency error of the layer-term. (Compare [I41 for conditions which are necessary to obtain uniform convergence on equidistant grids.) We now assume a E C 2 [O,ll, b E C 2 ([O,llxIR), a(x) > a > a ' > O , and consider a nonequidistant grid G with hi+l 5 hi Let TE,G:IRm+2+IRm

(i=l,. . . ,m).

be defined as in (28) with help of the function

u 2 given in (19); thus

li =

-

+ aiD1, p i = aih/(2c). &(1+pi/(l+pi))D2 2

D2 and D 1 are defined in (15). Theorem 12 Under the above conditions holds

154

J . LOREN2

xi

with

= exp(-(l-xi+l)a'/c).

Here C i s a c o n s t a n t i n d e p e n d e n t o f c

and t h e g r i d G. Proof: Theorem 9 i m p l i e s

1

1

-~ u & 1 , ~~ I L1I ,"t

~

hilqiil

i=1 = T c I G u c I Gd e n o t e s t h e t r u n c a t i o n e r r o r of t h e c o n t i n u o u s

if

s o l u t i o n . For f i x e d i E ( 1 , 2 , , ~d = ( u , ( x i - 1 )

-

let

uc ( x i ) r uc ( x i + l ) )

r

Then

I nil

. . ,m}

+

*

aiuk ( x i ) ) I

=

lliud

6

c l u r ] ( x i ) - D2udl + c p i l D j u d l / ( l + p i )

(-ELI:

2

hilu'&'

2

(El) I +

iEhiluz'

5 C

+

(xi)

T

(c3) I}

with

2

Using p i / ( l + p i ) 5 p i 5 C h i / c

Inil < C h i ( l + &- 2 x i )

ck

+ ailDlud - uk(xi)I

l/(l+pi)

5 xi+l.

w e f i n d w i t h theorem 3

and t h e a s s e r t i o n f o l l o w s . q.e.d. F i n a l l y , c o n s i d e r t h e same d i s c r e t i s a t i o n where a s p e c i a l g r i d G w i t h o n l y t w o s t e p - s i z e s H end h i s u s e d : L e t hl = = hL = H 2 h = h L + I = - hm+l.

...

, 0

H

,

x1

.*.

,

H

xL-l

. h .

x~

x

~

Theorem 1 3 3 3 L e t a E C [0,11, b E C ([O,llxIR),

.

h, 1

+xm ~

a ( x ) t a > a ' > O . There e x i s t s C

i n d e p e n d e n t of H , h , and c s u c h t h a t

I I u & I ~ - uc,GILsCH a s l o n g a s h s c n and lc:= $2 & l o g -15 1 &

-

x

L'

Proof:

W e u s e t h e same n o t a t i o n s a s i n t h e p r o o f o f theorem 1 2 and f i n d f o r i = L+l,

...,m:

lqil s C ( c h 2 ( l + ~ - ~ + h ~c-1h2(1+c-2Ai) ) I Ch 2 & - l ( l + & - 2 X i ) .

+

h2(l+c3hi)}

155

S T A B I L I T Y AND CONSISTENCY A N A L Y S I S

(Note that the applied difference formulas are equidistant and make 2 use of pi/(l+pi) ICh2c-2.) This yields

For i=l,...,L-I we have

A.

ie

and in addition holds

= c2

A L < & ea'h/E 0, and f o r all i n i t i a l guesses. 8. Using t h e s m a l l e s t p o s s i b l e number of boundary c o n d i t i o n s .

-

We s h a l l p r e s e n t t h e methods, give d e t a i l e d proofs of sane new theorems, and give computational r e s u l t s . I n s e c t i o n one we p r e s e n t t h e upwind d i f f e r e n c e algorithms f o r a s c a l a r conservat i o n laws i n one space dimensions. Connections with t h e s m a l l d i s t u r b a n c e equation of t r a n s o n i c flow were p o i n t e d out i n L2.1, [ 3 ] , [41. Results proven elsewhere w i l l be discussed here (as i n all s e c t i o n s ) . I n s e c t i o n two we p r e s e n t t h e upwind f i n i t e element algorithm f o r a m u l t i dimensional s c a l a r conservation law. Proofs o f s e v e r a l r e l e v a n t theorems w i l l be given i n Appendix one. I n s e c t i o n t h r e e we p r e s e n t t h e upwind f i n i t e d i f f e r e n c e algorithm f o r a wide c l a s s of s i n g u l a r l y perturbed s c a l a r two p o i n t boundary value problems. I n s e c t i o n four we p r e s e n t t h e upwind f i n i t e element algorithm which a p p l i e s t o a wide c l a s s of s i n g u l a r l y perturbed s c a l a r e l l i p t i c boundary value problems. Proofs of s e v e r a l r e l e v a n t theorems w i l l b e given i n Appendix two. I n s e c t i o n f i v e we p r e s e n t t h e upwind f i n i t e d i f f e r e n c e algorithm f o r g e n e r a l s t r i c t l y hyperbolic systems of nonlinear conservation laws i n one space dimension. I n s e c t i o n six we p r e s e n t t h e d e t a i l e d upwind a l g o r i t h f o r one important p h y s i c a l problem compressible i n v i s c i d gas flow i n Lagrange coordinates -and a l s o d i s c u s s t h e p o t e n t i a l equation of t r a n s o n i c flow and E u l e r ' s equations i n one and two dimensions.

-

* Research

p a r t i a l l y supported by NASA #INCA-2 -0R390-002.

- Ames 179

University consortium interchange

180

S . OSHER

I n s e c t i o n seven we discuss numerical r e s u l t s f o r problems of c m p r e s s i b l e i n v i s c i d gas dynamic.

I. UPWIND FINITE DIFFERENCE APPROXIMATIONS FOR A SCALAR CONSERVATION LAW IN ONE SPACE DIMENSION We consider t h e equation :

with

f

+ f(u)x

ut

(1.1) E

= 0

C2(R1). with - Zn-' ~ i = atp o

x . = jh, t

We s e t up a g r i d

J

+- uj

define t h e d i f f e r e n c e operators To o b t a i n f ' ( u ) 2 0,

=

un

approximating u(x

j

-

? ( u ~ + ~u.).

-

j'

t"), and

J

1 if

f h s t order d i f f e r e n c e approximation, we f i r s t l e t X(u) X(u) 0 i f f ' ( u ) < 0 and t h e n d e f i n e

OUT

f

=J

fJU)

0

f-(u) =

It i s c l e a r t h a t

f;(u)

s i n g l e minimum a t

-2 0,

U

JOU

5

f:(u)

X(s)fl ( s ) d s

(1 0.

- x(s))f~(s)ds Moreover i f

f"(u) 2 0

and

f

has a

u = u, t h e n ( 1 . 2 ) reduces t o : f (u) = f(max(u,u))

+

(1-3 ) We note t h a t

In general

f-(u) = f(min(u,u)).

f;(u) f + and

2 0,

f'-(u) 5 0,

f _ are

f ' ( u ) = fk(u)

and piecewise

C1

second d e r i v a t i v e s only at c r i t i c a l p o i n t s of

C2,

+

f;(u). having p o s s i b l e jumps i n

f.

Our b a s i c upwind d i f f e r e n c i n g i s defined as follows:

(1.4)

f(dx

--f

&+f

-(Uj

1 + A_f+(Uj 11.

For t h e time d i f f e r e n c i n g , we l e t

(1.5) and define

xn

= Atn/&.

The r e s u l t i n g scheme is

-

This scheme is f i r s t order accurate and monotone G i s a nondecreasing f u n c t i o n of i t s arguments i f t h e CFL c o n d i t i o n I h f ' (u)I 5 1 is_v a l i d . !thus, by t h e r e s u l t s of [l], [61, it follows t h a t f o r i n i t i a l d a t a i n L1 n ,L and of bounded

-

181

NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS

v a r i a t i o n , t h e approximate s o l u t i o n converges t o t h e p h y s i c a l l y c o r r e c t s o l u t i o n of (1.1) as

h, atn + 0.

We t u r n now t o higher order upwind approximations of (1.1). Consider a method of l i n e s approach t o

(1.7)

ut = aux

for a w constant

a > 0.

We l e t

uj(t)

approximate

g e n e r a l upwind d i f f e r e n t i a l - d i f f e r e n c e equation:

The scheme is of order of accuracy

u(x.,t) = u(j&,t) J

i f f o r smooth functions

q

via the

q(x),

[ ? I , [41.

Then we have t h e following theorem

Theorem (1.1). If (1.8)i s s t a b l e , t h e n t h e scheme has order of accuracy a t most 2. Moreover t h e r e do e x i s t s t a b l e second order approximations and t h e most compact ( s m a l l e s t p ) i s given by B, = 1, B2 = -1/2 f o r p = 2. Next we c o n s t r u c t a simple second order upwind approximation t o (1.1). Our f i r s t attempt i s t h e simplest p o s s i b l e second order analogue of ( 1 . 4 )

- 1/2 f):n

f(uIx

(1.10)

The r e s u l t i n g scheme w a s shown i n some overshoot for s t e a d y shocks. f(u) =

a(u

- u)’

with

a > 0,

-

(u.) J

+ (A- +

1/2 A2)f - +(u j ) I .

[41, both

a n a l y t i c a l l y and numerically, t o have For t h e important s p e c i a l c a s e

and


0 and f ’ ( u ) < 0. Then t h e monotone t r a v e l i n g wave s o l u t i o n s t o (1.6) f o r s h r a t i o n a l have t h e p r o p e r t y t h a t t h e r e e x i s t s

uR f o r j >- j,. ( S i m i l a r l y for s < 0 and f’(uL) > 0 ) . L R = 0 f o r convex f with f ’ ( u ) 0 > f’(u ), t h e n t h e r e e x i s t s L - R L j o such t h a t u = u , j < j,, u = u , j > j o + 1 with u E [u,u 1, j,

(b)

with

u

Let

s

j

j

j

j0

E

183

NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS

f - ( u j +1) + f ( u . ) = f - ( u L ) + f + ( u L ). 0 Jo F i n a l l y we a l s o have i n f i n i t e r e s o l u t i o n for s t e a d y shock s o l u t i o n s of t h e second order scheme (1.11)f o r convex f ( u ) .

U

t

j

[uR,u]

s u b j e c t only t o t h e c o n d i t i o n

+

meorem (1.3). The s t e m t r a v e l i n g wave s o l u t i o n s t o (1.11)f o r convex a r e t h e same as t h o s e f o r ( 1 . 6 ) mentioned i n p a r t ( c ) of Theorem ( 1 . 2 ) .

f(u)

These waves look l i k e :

UPWIND FIIYITE ELEMENT APPROXIMATIONS FOR MULTI-DIMENSIONAL

11.

CONSERVATION LAWS

SCALAR

We consider t h e s c a l a r multi-dimensional problem u

(2.1) t > 0,

for f

t

t

x

E

n,

where

C2(n) with values i n

Let

{$}

+ V.f(u)

N R .

is either

or an N

R"

dimensional t o r u s , and

n i n t o closed nonoverlapping polyhedra.

be a decomposition of

assume t h e followirg p r o p e r t y :

u ( x , o ) = o(x)

= 0,

h F(Tj)

if

(resp.

p(T:))

is t h e s m a l l e s t ( r e s p

t h e l a r g e s t ) diameter of t h e b a l l s c o n t a i n i n g ( r e s p . contained i n ) t h e r e e x i s t s a p o s i t i v e constant

j

We approximate

'I? t, hen

such t h a t

K-% 5 i n f p($)

(2.2)

(2.3)

K

We

5 sup ~ ( $ 1 5 a. j

u using t h e decomposition

u h ( x , t ) = ~ $ ( x ) p ~ ( t ) ,with

3

olhJ

$

l?

Ph a p o r t i o n of a hyperplane bounding and we d e f i n e J ,J J t h e u n i t normal. t o P ! p o i n t i n g out of J,l J We t h e n use t h e flux d e c m p o s i t i o n of t h e previous s e c t i o n t o d e f i n e t h e s c a l a r functions

Given

l?. J

t h e c h a r a c t e r i s t i c f u n c t i o n of

l?.

h

v.

J,J'

(2.4)

are, for fixed

t,

piecewise constant functions of

x

having d i s c o n t i n u i t i e s

184

S . OSHER

( i n general) a t t h e boundary of each

5. For J

t

f i x e d and f o r each

P

introduce t h e following piecewise constant f’mctions defined on

j,a’

we

J

h

We d e f i n e we l e t

h

u_ ( x , t ) = BJ. ( t ) ) .

(noting t h a t

vj)-

f(Eh(x,t))

and

f(u -h ( x , t ) ) . v j ) +

i n t h e same fashion.

Finally

be t h e volume of

Ah j

We f i r s t d e f i n e t h e semi-discrete approximation t o

J

with

Bh .(O)

=

J

h un(x)

N e x t we l e t B:(&).

(I T(x)dx)/A;. 8J

For

approximate

n = O,l,.

..,

h

u ( x , d t ) by l e t t i n g Bh

j,n

we l e t

approximate

(2.9) -h

with

h

f3j,o

h = B.(O). J

.1

We impose t h e CFL c o n d i t i o n

The main theorems, s t a t e d i n [lo] and proven i n Appendix 1 below, a r e : h h Suppose u ( x , t ) and v ( x , t ) Theorem (2.1). t h e n have t h e estimate f o r any t 2 0

a r e defined by ( 2 . 3 ) , ( 2 . 8 ) .

W e

185

NUMERICAL SOLUTION OF SINGULAR PERTURBATION PROBLEMS

Similarly, for n 1 0

h un(x)

that for

defined by (2.3), (2.9), we have f o r any

For a f’unction of the type

Definition (2.1).

where each

h and vn(x)

ThJ

is the centroid o f

x

8J j and

Theorem (2.2).

uh =

cj

c$(x)$

we define

and the sum is taken over all

j,J

such

have a comon plane boundary

!I$

Suppose

Ph Then we have: j,J* i s defined by (2.3), (2.8), or (2.3), (2.9) and t h e

h

u

sequence converges a.e. t o u as h + O . Moreover, suppose B(uh) i s uniformly bounded f o r all h and t . Then u i s a weak solution of ( 2 . 1 ) which s a t i s f i e s the entropy condition. Remark. For N = 1, Richard Sanders [ l ) ] has proven t h a t the assumptions made above are v a l i d for i n i t i a l . d a t a i n BV n L, n L1. 111. UPWIND FINITE DIFFERENCE APPROXIMATION FQR A NONLINm SINGVMLY PEBTURBED TWO POINT BOUNDARY VALUE PROBLFM

We consider the boundary value problem: (3.1)

y“

- a(y)y’ - b(x,y)

(a)

E

(b)

d-1) = A ,

= 0,

-1 L

X

51

Y(l) = B

where 0 < E 0 ) of (3.1) f o r t h e corresponding E. All limit s o l u t i o n s have v a r i a t i o n bounded by IBI + IA/. i s precompact i n

L1[-l,ll.

Thus any i n f i n i t e c o l l e c t i o n with

The one drawback of t h e e x p l i c i t time i t e r a t i o n is t h e r e s t r i c t i v e CFL c o n d i t i o n ( 3 . 3 ) ( d ) . Newton‘s method w a s t r i e d on (3.4) numerically with great success. It i s i n t e r e s t i n g t o note t h a t t a k i n g t h e d i f f e r e n t i a l of t h e nonlinear operator i n (3.4) at a s t a t e F = (7.] leads t o a l i n e a r t r i d i a g o n a l problem: J

(3.5)

( a ) F ( x . 1 = -b (x T )Y J Y j’ j j . ,+N 1 j = O,+l,.

.

-

- D - ( f+’ ( T j ) YJ. )

- D+(fL(T- j )Yj

+ E D+D-yj

-

187

NUMERICAL SOLUTION OF S I N G U L A R PERTURBATION PROBLEMS

which has a unique s o l u t i o n s a t i s f y i n g t h e e s t i m a t e

N

c lvjl j =-N

N-1

c

5

IF(xj)l

j =-N+1

y.

independently of E and t h e s t a t e Thus t h e i n v e r s i o n of t h e l i n e a r i z e d problem f o r Newton's method p r e s e n t s no d i f f i c u l t i e s . A suggested procedure i s t o use t h e a r t i f i c i a l time i t e r a t i o n on a coarse g r i d , t h e n use t h e r e s u l t i n g s o l u t i o n s u i t a b l y i n t e r p o l a t e d , on a f i n e g r i d (perhaps r e f i n e d near evident boundary and i n t e r n a l layers) as an i n i t i a l guess f o r Newton' s method. Preliminary c a l c u l a t i o n s i n d i c a t e e x c e l l e n t success with t h i s approach.

IV. UPWINE FINITE ELEMENT AF'PROXlMATIONS FQR NONLINEAR SINGULARLY PERTURBED DIRICHLET PROBLtENS We consider t h e boundary value problem

(4.1) with

UI

= $(x)

f

2

C

- V . f ( u ) - b(x,u)

nu

E

(d). The domain

t h e following p r o p e r t y :

For any

N

R c R

in

= 0

has piecewise

R

nonoverlapping union of closed r e c t a n g u l a r p a r a l l e l i p i p e d s p a r a l l e l t o t h e planes

x . = 0, J

property ( 2 . 2 ) above and t h a t of any p o i n t of

J

.. ,n.

j = 1,.

R CR

h

.

contains a boundary p o i n t of

T!

C2

boundary and has

h > 0 w e can f i n d a domain

Clh

with s i d e s

{ jo

J

uR

J

u (a), u. ( a ) f o r 0 5 j0 1o+l v e c t o r s s a t i s f y i n g t h e following: and

+

1

a r e a smooth one parameter f m i u of

sCY

L

u ( a ) i s connected t o u = u v i a a curve r J o defined as above b u t j0 j, jo jo-l for which o* rd ,rd-l ,...,:r a r e used; i . e . r jv0 = {uj 1, Y =

...,k

- 1,

= k,k

+ 1,...,d.

1,

j0

rY, Y

and moreover

X,(u(s)) 2 0

( a ) i s connected t o uR via a curve r

uj0+1 which only

j +2

j +2

j0+2

rko ,rk-l ,...,r 1

+ 1,..., d,

0

on each

jo+2

is used, ( i . e .

defined as above b u t f o r j +2

0

-- @jo+13, r v0 , V = j +2

,..., k. The curve r has t h e p r o p e r t y t h a t Xv(u(s)) 2 0 on each r v , j +1 Y = d,d - 1,..., k + 1; X,,(U(S)) 5 0 on each rYo , v = k - 1 , k - 2 ,...,1; j +1 j +1 0 and Xk(u(s)) on rko decreases monotonically for 0 5 s 5 s k , Y

= k

j +1

and moreover

havLng a s i n g l e zero at xk (u) = 0. The v e c t o r equation

(5.14)

s

=!

-s

Xv(u(s)) 5 0

on each

1,2 j +1 0

i n this c l o s e d i n t e r v a l , at which

-

u = u,

194

S . OSHER

i s s a t i s f i e d by every member of t h i s family. We next have t h e i r uniqueness (modulo t h e second (perhaps unnecessary) hypothesis below). Theorem (5.3).

Let

be a s t e a d y d i s c r e t e shock with

(u,]>

s m a l l , and with t h e p r o p e r t y t h a t i f

k (u ) k j of t h e form defined i n t h e previous theorem.

Vl.

5

0, t h e n

16 u 1 +j

sufficiently

Xk(uj+l) 5 0.

Then it i s

THE UPWIND DIFFERENCE ALGORITHM FOR SaME IMPCWTANT EXAMPLES

The equation (5.7) can be i n t e g r a t e d i n closed form f o r many important p h y s i c a l problems. Hence a t h r e e p o i n t scheme ( 5 . 1 0 ) involving various "switches", which depend on t h e s i g n of Xk(u), can be constructed and t e s t e d . This was done, with very s u c c e s s f u l numerical r e s u l t s , sane of which we r e p o r t i n t h e next s e c t i o n . The algorithms mentioned here a r e taken fTm [12] and [51. We begin with one dimensional gas flow for a non i s e n t r o p i c gas i n Lagrange coordinates. The equations a r e w r i t t e n i n t h e form

(6.1)

7

-vx=0

vt

+ p,

Et Here

Let

+

for

= 0

(PV),

= 0,

z i s s p e c i f i c volume, v i s v e l o c i t y , u = (T,v,E) T , f ( u ) = (-v,p,pv), then

0

and t h e eigenvalues of

k(u)

p = (y

y

Tt;

E

e -?

- 1) -

1.k

is energy,

-1

p

is p r e s s u r e .

O

1

are

(6.3)

f o r a l l values of u, t h e r e a r e no switches, and t h e 2 3 scheme becomes f a F r l y simple, once t h e i n t e g r a t i o n i n (5.7) i s performed exactly Since

h

1

< 0 e x < h,

NUMERICAL SOLUTION OF S I N G U L A R PERTURBATION PROBLEMS

- see

[121 f o r t h e d e t a i l s .

In p a r t i c u l a r

3 2

Tn

j -1

with

(6.5)

dJ. = 2 ( J T n ) ' d m

and (using t h e above d e f i n i t i o n s )

195

196

S.

OSHER

with

(6.7)

' k e n m e r i c a l approximation is now obtained by using (6.1), (6.4) and

(6.6).

I n [121 t h e algorithms were worked out for, among o t h e r s , t h e one dimensional i s e n t r o p i c gas dynamics equations i n E u l e r i a n coordinates, and a l s o t h e two dimensional v e r s i o n of t h e system (with t h e help of a dimensional s p l i t t i n g algorithm). For l a c k of space, we anit t h e d e t a i l s h e r e b u t p r e s e n t sane computational r e s u l t s i n t h e next s e c t i o n . One f u r t h e r example, from [51, i s t h e p o t e n t i a l f l a w equation i n one space dimension w r i t t e n &s a hyperbolic system

(6.8) with

c = c(p) =

= AypYm1 for

dP

m e eigenvalues of

&

-e

vectors

r1,2 =

(*.').

A > 0,

axe

=

X1,2

1