Spectral Methods and Their Applications

Spectral Methods and Their Applications Guo Ben-Yu Shanghai University P R China ,.p World Scientific I Singapore· N...

Author: Ben-Yu Guo | Pen-Yu Kuo

105 downloads 1202 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Spectral Methods and Their Applications

Guo Ben-Yu Shanghai University P R China

,.p World Scientific I

Singapore· New Jersey· London· Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd.

USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661

POBox 128, Farrer Road, Singapore 912805

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data

Spectral methods and their applications I Guo Ben-yo.

Guo, Ben-yo, 1942p.

cm.

ISBN 9810233337 (alk. paper)

Includes bibliographical references. 1. Spectral theory (Mathematics) QA320.K84

I. Title.

1998

515'.7222--dc21

97-44642

CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any in/ormation storage and retrieval system now known or to be invented, without written permission from the Publisher.

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to

For photocopying of material in this volume, please pay a copying fee through the Copyright photocopy is not required from the publisher.

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

PREFACE In the past two decades, spectral methods have developed rapidly. They have become important tools for numerical solutions of partial differential equations, and have been widely applied to numerical simulations in various fields. The purpose of this book is to present the basic algorithms, the main theoretical results and some applications of spectral methods. Particular attention is paid to the applications to nonlinear problems. The outline of this book is as follows. Chapter 1 is a colloquial introduction to spectral methods. In Chapter 2, we discuss various orthogonal approximations in Sobolev spaces, used in spectral methods. We also discuss the filterings and recov ering the spectral accuracy. Chapter 3 is a survey of the theory of stability and convergence. Chapter 4 consists of two parts. In t "e first part, we present some basi c spectral methods with their applications t o nonlinear problems. In the second part, we consider the spectral penalty methods, the spectral viscosity methods and the spectral approximations of isolated solutions. Chapter 5 is devoted to the spectral approximations of multidimensional and high order problems. T he spectral domain decomposition methods and the spectral multigrid methods are also introduced. We consider the mixed spectral methods for semi-periodic problems in Chapter 6, and some combined spectral methods with their. applications in Chapter 7. T he final chapter focuses on the spectral methods on the spherical surface. I am gratef ul to Prof. Wong, R. S . C. and Dr. Dai, H. H . . T hey organized the Workshop at City University of Hong Kong, which motivated me to write this book. I thank Dr. Caldwell, J. for his help in polishing up the manuscript. I also thank Misters Chan,oH. C. and Cha, K. H. who typed the manuscript. Guo Ben-yu Shanghai University City University of Hong Kong December 1996

v

CONTENTS

PREFACE CHAPTER CHAPTER

2.1 2.2 2.3 2.4 2.5 2.6 2.7

1.

2.

v

1

INTRODUCTION ORTHOGONAL APP . ROX�MATIONS IN SOBOLEV SPACES

Preliminaries . . ; . . . Fourier Approxima tions Orthogonal Systems of Polynomials in a Finit e Interva l Legendre Approximations .' . . . . . . . . . . . . . . . Chebyshev Approximations . . . . . . . . . . . . . . . Some Orthogonal Approximations in Infinite Intervals . Filterings and Recovering the Spectral Accuracy . . -

CHAPTER 3 . STABILITY AND CONVERGENCE

3.1 Stability and Convergence for Linear Prob lems . 3.2 Generalized Stabili ty for Nonlinear Problems . 3.3 Initial-Value Problems . . . . . . . . . . . . . CHAPTER

4.1 4.2 4.3 4.4 4.5 4.6

4.

10

10 '16 24 31 43 56 66

82

82 86 94

SPECTRAL METHODS ANR) PSEUDOSPECTRAL METHODS 100

Fourier Spectral Methods and Fourier Pseudospectral Methods Legendre Spectral Methods and Legendre Pseudospectral Methods . Chebyshev Spectral Methods and Chebyshev Pseudospectral Methods Spectral Penalty Methods . . . . . . . . . . . Spectral Vanishing Viscosity Methods . . . . . Spectral Approximations of Isolated Solutions Vll

100 120 143 152 164 181

Contents

viii

CHAPTER' 5. SPECTRAL METHODS FOR MULTI-DIMENSIONAL AND HIGH 'ORDER PROBLEMS 188

5 . 1 Orthogonal Approximations i n Several Dimensions .

188

5.2 Spectral Methods for Multi-Dimensional Nonlinear Systems .

196

5.3 Spectral Methods for Nonlinear High Order Equations

210

5.4 Spectral Domain Decomposition Methods .

231

5.5 Spectral Mu ltigrid Methods . . . . . . . .

248

CHAPTER 6. MIXED SPECTRAL METHODS

257

6 . 1 Mixed Fourier-Legendre Approximations .

257

6.2 Mixed Fourier-Chebyshev Approximations

265

6.3 Applications . . . . .. . . . . . . . . . . .

273

CHAPTER 7.1

7.

COMBINED SPECTRAL METHODS

280

Some Basic Results in Finite Element Methods . .

280

7.2 Combined Fourier-Finite Element Approximations .

284

7.3 Combined Legendre-Finite Element Approximations .

289

7.4 Combined Chebyshev-F inite Element Approximations .

295

7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . .

300

CHAPTER

8.

SPECTRAL METHODS ON THE SPHERICAL SURFACE 313

8.1 Spectral Approximation on the Spherical Surface

313

8.2 Pseudospectral Approximation on the Sph erical Surface .

318

8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . .

322

References

331

Spectral Methods and Their Appli�ations

CHAPTER 1

INTRODUCTION Spectral methods have developed rapidly in the past two decades. They have been applied successfully to numerical simulations in many fields, such as heat conduction, fl uid dy namics, quantum mechanics and so on. Nowaday s, they are some of t he powerful tools as well as finite difference methods and finite element methods, for numerical solutions of partial differential equations. Th e main f eat ure of th e spectral met hods is t o take various orthogonal sy st ems of infinit ely differentiable global functions as trial functions. Different trial functions lead to different spectral approximations. For instance, trigonometric polynomials for periodic problems, Legendre polynomials and Chebyshev polynomials for non periodic problems, Laguerre polynomials for problems on the hal f line, and Hermite polynomials for problems on the whole line. The fascinating merit of spectral methods is the high accuracy, the so-called con vergence of " infinite order" . For example, we consider numerical solution of the Laplace equation on a cube, and divide the domain into N3 uniform subcubes. If we use linear finite element methods, then th e error in the L 2 -norm between the geiLUine solution and the numerical one is of order N-:- 2 , no mat ter how smooth t he genuine solution is. If we use the standard " nine points" difference scheme with the uniform mesh size if, then the error is of the same order. However, the smoother the genuine solut ion, the higher the convergence rate of the numerical solution by spectral meth ods. I n particular, when the genuine solution is infinitely differentiable, the numerical one converges faster than N-a , a being any positive constant. The basic idea of spectral methods stems from Fourier analysis. In 1 820, Navier 1

Spectral Methods am; Their-"Applications

2

considered a thin plate problem governed by the equation 84ij' 84 U 84 U 2 2 2 + f, + 0 < x, y 8 x8 y 8y4 = 8x4 with the boundary conditions 82 U x = O, 7r , U 8x 2 = 0, 82 U y = O, 7r. U 8y 2 = 0, Suppose that f( x, y) can be expanded as 00

f(x, y) =

0, U(x, O) = Uo(x), O:S;x:S;7r 00

{

00

Chapter

1.

Introduction

3

where Uo(x) is a given continuous function and vanishes at

x =

0, 71'. Let

U(x,t)=L U,(t) sin Ix 00

{

1=1

dU, = z 2 [/;I, dt U,(O) = Uo,,, 1 = 1 , 2, . . . ,

( 1 .7)

and substitute this expansion into (1 .6), we find that -

( 1 8) .

where UO,I is the Fourier coefficie nt of the initial state Uo(x),

2

UO, I = 71'

( 1 .9 )

0

1" Uo(x)sinlxdx.

By ( 1 .8) ,

We now truncate the series ( 1 . 7 ) and denote the corresponding one by UN ( X , t), N UN ( X , t) = L U,(t) sin Ix. 1=1

This is an exceedingly good approximation to ( 1 . 7 ) for any t > O. In fact, IU(x,t) - UN ( X , t)1 =

�

I I=f:N+1 Uo,le-I't sin lx l

2 max lUo(x) I iNroo e-tll'dy; 0$'"$"

Thus the error goes to zero very rapidly as N -+ 00 . There are several kinds of spectral method s, called spectral methods, pseudospec tral methods and Tau methods, respectively. All of them can be derived from the method of weighted residuals. Let n be a spat ial domain with the boundary an, and n = n u an. We consider here an initial-boundary value problem as follows

{

xE n, t > 0, LU=J, xE an, t > 0, BU=O, U(x,O)=Uo(x), xEn

( 1 . 10)

where L is a differential operator and B is a linear boundary operator, J(x,t) and Uo(x) are given functions. Assume that some conditions are fulfilled for ensuring the

Spectral Methods and Th�ir..¥\.pplications

4

existence, un iqueness and regularity of � olution of ( 1 . 10). The method of w eighted res iduals is to find an approxi� ate solution of the' form N (1.1 1 ) UN(X,t) = UB(X,t) + 2:>N,I(t)dN=(Uo, ¢>,)N'

o � I � N, t > 0, o � I � N.

( 1 . 18)

We now turn to the Tau methods. Suppose that ¢>/(x), 0 � I � N are orthogonal in L2(fl), but do not fulfill the boundary conditions. The component UB ( X, t) i n. ( 1 . l l ) is of the form N+k UB(X, t)= L UN,/(t)¢>/(x) /=N+l

Spectral Methods and Their ?ipplications

6 where

k

rPj (x), O

is t he number of independent. bound� y conditions. We take :::; I :::; N. Then i� rea. d as

( 1.12) ), rPl ) = (f(t ), rPl ) , 0 :::; I :::; N,t > 0, { (LUN(t (UN(O), rP') = (Uo, rP, ) , 0 :::; I :::; N

k

'ljJj (x) = ( 1.19)

with additional eq uations given by the boundary conditions. The Tau methods can also be descri bed by a projection operator, namely

{ UN(PNLuN(X, t) = PNf(x,t), X, O) = PNUo(x)

k

t > 0,

( 1.20)

with additional equations on an. In practice, mostly used methods for problems with non-homogeneous boundary conditions are some combinations of pseudospectral methods with Tau methods. It means that partial differential equations are approximated by collocation approach in which 1:::; N, do not satisfy the boundary conditions, and the boundary conditions , are treated as in Tau methods. When we solve initial-boundary value problems nUlIl erically, we need to discretize the derivatives of unknown functions with respect to time Denote by [Tl the integer part of any fixed positive constant Let be the mesh size for the variable and

rPl (X), O :::;

R.,.(T)

T.

r

= {t = kr I k = 1, 2, . . . , [�] } ,

t.

f4(T)

= Rr(T) U { o} .

t ( 1.21)

v ( x, t) = 2"1 v (x, t + r) + 2"1 v (x, t - r), D.,. v (x, t) = �(v (x,t + r) - v (x, t» ) , D.,.v ( x, t) = �(v ( x, t) - v(x, t - r ) ) , D.,. v(x, t) 21r (v(x, t + r ) - v(x, t - r)) , DTTV (X, t) = :2 (v(x, t + ) - 2v(x, t) + v(x, t - r)) . D.,. v and D.,. v approximate �; at time t with the trunc ation error of order O(r), provided that I �:� I is bounded. D.,. v approximates �; with the error O(r2 ) , when We use the following notations

r

Chapter 1. Introduction

I �:� I 04V as that I at4 l is bounded.

7

DT�V approximates 0ot22 V with the error O( r 2), as long . ov at t + T WIt' the truncat'Ion But DTv approxImates ot

is bound�d. Similarly,

error O(

r 2). It can be verified that 2 ( DTv (t), v(t ) ) 2 ( DTv(t), v (t)) 2 ( DTTV(t), D Tv(t))

{

11

'2

DT l v (t) 1 I 2 - r I D'Tv(t ) 1 I 2 , D'Tll v(t ) 1 2 , D'T I D'Tv (t) 11 2 .

( 1.22) ( 1 .23) (1.24)

As an example, we consider the problem at

au = a2u + I, O<xI)

=

(21 02UN (t + r ) + 21 02UN (t) ox 2 ox2 0� I � t E Ji, (T), +I(x,t + � ) ' ¢>I) ' 0� I� ( Uo, ¢>I) , N,

=

N.

( 1 .26) Further, by the orthogonality of trigonometric functions, we obtain from ( 1 .26) that

{ UN,D'TUN,I(O)I(t)= Uo,= -�/ , UN,, (t +. r ) - �UN,,(t) + II (t + t) ,

where

� 1� N, t E R'T(T), � 1� N

o o

UO,I is the same as in ( 1.9) , and II (t + �). = � f I (x, t + �) sin Ix dx.

i1'


8

Finally

Therefore we can evaluate the coefficients UN,I(t), 0 :5 1 :5 N, t E .R,.(T) explicitly. This is indeed another advantage of spectral methods. Conversely, if a finite difference scheme of Crank-Nicolson type is used for (1.25), then we have to solve a linear algebraic system for the values of unknown function at each time step. It is also true for any finite element scheme of Crank-Nicolson type. On the other hand, the matrices occurring in finite element methods may weaken the stability of computation. Sometimes, the "lumping" technique is used to overcome this trouble, but it usually lowers the accuracy. Even though the spectral methods have many advantages, they were not used widely for a long time. The main reason is the expensive cost of computational time. However, the discovery of the Fast Fourier Transformation (FFT), see Coo ley and Tukey (1965), removed this obstacle. Let N be the number of terms in one-dimensional Fourier expansion. Then FFT reduces the total number of opera tions from O ( N 2 ) to O ( N log 2 N ) . Especially, FFT saves a lot of work for multi dimensional problems, since trial functions in several dimensions are the products of several trial functions in one dimension. For example, we consider the summation

N N N

� �o NE =

} {. (Nl+P 1 + + 1 + Nnr) +1 . (N . (N ))

a',m,n

� � �e +1 1=0

exp 2?r�

'"' 2NmQ' � e +1 m=O

""'

mq N

�a" m,neN+1 n=O

�

.

If we use the usual method to evaluate the left-hand side of the above formula, then the total number of operations is O ( N6 ) . In contrast, it is reduced to O ( 3N2 1og 2 N ) , if the right-hand side is calculated by F FT. Since Chebyshev polynomials can be changed into trigonometric polynomials by a transformation of independent variables, FFT is also available for Chebyshev spectral approximations. Furthermore, the coming of the Fast Legendre Transformation (FLT) , see Alpert and Rokhlin (1991), and the , possible parallelization of FFT and FLT make the spectral methods more efficient.

Chapter

1.

Introduction

9

The first serious applic ation 'of the spectral methods to partial differential equ a tions was due to' Silberman (1954) . However, they have become practical only af ter Orszag (1969, 1 970) , and Eliasen, Machenhauer and Rasmussen ( 1970) devel oped transform technique for evaluating convolution sums arising from quadra tic non-linearities. The colloc ation approach was first used by Slater ( 1934) and Kan torovich (1934) in specific applic ations, and the foundation of orthogonal colloc a tion was laid by Lanc zos ( 1938). But the earliest applic ation of the pseudospec tral methods to partial differential equations was made by Kreiss and Oliger (1972) , and Orszag (1972) . Lanczos (1938) also developed the Tau methods. Gottlieb and Orszag ( 1977) summarized the state of art in the theory and applic ation of spectra l methods. They provided numeric al analysis for linear problems. H ald (1981 ) , Ma day and Quarteroni ( 1981 , 1982a, 1 982b), and Bernardi, Canuto and Maday ( 19 86) considered nonlinear problems. While Guo (1981a, 1 985) and Kuo (1983) devel oped spectral methods for nonlinear problems with numeric al analysis independently. There have been numerous introductory and review articles, e.g. , Mercier (1981 ) , H ussaini, Salas and Zang ( 1985), Gottlieb (1985) , and Quarteroni (199 1 ) . Some new developments in this field are contained in the books by Canuto, Hussaini, Qu arteroni and Zang (1988) , by Bernardi and Maday (1992a) , and by Guo (1996). On the other hand, applic ations of spectral methods in meteorology have been covered in the review by Jarraud and Baede (1985), and in the book by H altiner and Williams ( 1980) .

CHAPTER 2 ORTHOGONAL APPROXIMATIONS IN SOBOLEV SPACES

The remarkable convergences of spectral methods owe much to the rapid conver gences of expansions in series of orthogonal systems of smooth functions. We present a summary of the relevant theory in this chapter. Fourier approximation, Legendre approximation, Chebyshev approximation, Laguerre approximation and Hermite ap proximation are considered. The error estimations of different projections in Sobolev spac es are discussed. We also introduce the filterings which are used to improve the stability of pseudospectral methods for nonlinear partial differential equations. In order to recover the spec tral acc uracy of orthogonal expansions for discontinuous functions, some essentially nonoscillatory approximations and reconstruction methods are used.

2.1.

Preliminaries

Let IRn be the n-dimensional Euclidean space and x = (Xl,X 2 , , xn ) E IRn. Denote by n a bounded open domain in IRn with the boundary an. If for any x E on, . there exists a system of orthogonal coordinates Y = (yI , Y2,... ,Yn), a hypercube U'" = n;=l [-aq,aq] and a Lipschitz continuous mapping �'" from n;;} [-aq, aq] into [-!an, !an ] such that . • •

nnu", annu",

{y E U"" Yn > �'" (Yl, ... ,Yn-l)} , {y E U"" Yn = �x (Yl,' .. ,Yn-l)} , 10

Chapter 2. Orthogonal Approximations in Sobolev Spaces

11

th en we say that an i s Li pscni tz continuous. Further, let m b e a non-negative integer. A relati� ely open part r E a n i s sai d to be of class cm,1, if for any x E r, we can choose such a mapping �'" that its deri vatives up to the order m are Lipschitz continuous. In thi s book, we assume that an i s at least Li pschi tz continuous, and n = O U ao. Let V(O) be the space of infini tely differenti able functions wi th compact supports in 0, and Coo (n) be th e space of i nfinitely differenti able functions on n. The dual space V'(O) of V(O) i s the space of di stributions on O. By the theory of di stributions (see Schwarz (1966)) , for any v E V'(O), the associ ated parti al derivatives of any order a , and \l = (a1, a2, . ., an). Let kq, 1 � q � n exist. The notati on aq stands for aXq be non-negati ve integers. For any n-tuple k = (kl,k2,· . ., kn), Ikl = k1 + k2 + ... + k n etc . . The symbol a; = a�la;2 ... a�n. When n = 1 , a", = a1. In addition, at = We now introduce the Sobolev spaces. For any real number p, 1 .� p � 00, let .

£P(n) = {v I

with the norm

•

:t '

v

is measurable and IivIiLP(o)

(10

Iv(x}IPdx Iivliv(o) = Iivliv(o) = ess sup lv(x) l,

1

y,

0, i n C (0) for W ,P(O) i s continuously imbedded in Lq(O) for � � P

=

q

p

0, and in Lq(O) f or m � and any q, 1::; q < 00. p We now turn to the trace theory. Let ( VI," " n ) be the uni t outward normal n If 00 i s of class cm - l,l , then for a relatively open vector to 00, and v

-

part

=

m

,

(2.10)

Vw E B.

For simpli ci ty, the inner products (',')£2(0), (',')W(O) and the norms 11,11£2(0), II· Ilw(o), II'IILP(O), 11'llwm,p(o) will be denoted by (' , .), (', .).,11·11, 11·11., II·IILP, 11'llwm,p respectively. Similarly the spaces C (a, bi Hr(o) ) , C (a, bi Wm ,P(O ) ) , H'Y (a, bi Hr(o) ) and H'Y (a, bi Wm ,P(O)) will be denoted by C(a, biHr) , C(a, biwm ,p) , H'Y(a , biHr) and H'Y (a, bi Wm,P), etc . .

2.2.

Fourier Approximations

We consi der the Fourier approximations in thi s secti on. Let n =1 and A = (0, 27!'). The set of functions eilx , 1=0,±1, . . . , is an orthogonal system in L2(A). The Fouri er transformation of a function v E L2(A) i s

Sv=

00

1=-00

L

vleilx

(2.11)

where VI i s the Fourier coefficient, VI = -

11 -ilx d 1=0,±1,.... 27!' A V ( )e According to ll iesz Theorem, Sv converges to v in L2(A), and the Parseval equality •

x,

x

holds, namely IIvl1

2=27!'

00

L IVt!2. 1=-00

00

(2. 12)

latl2 < there exi sts a uni que function in L2( A) such that its Fourier coefficient VI = al for alI I . Thus we 1=-00

Conversely for any complex sequence { al } such that L

can write

00 " ilx v= L vie 1=-00

00 ,

Chapter 2. Qr,t!! ogonaJ Approximations in Sobolev Sp aces

- -

17

A general result i s the Young-Hausdroff inequality, see Hardy, Li ttlewood and P6lya (19 52) . It states t hat for 2 $ p < 00 , 1 + 1 = 1 , p

q

( 2� L Iv(x)IPdX) $ (J�:oo IVt!q) , (too IVt!p) $ ( 2� L Iv(x)lqdX) ; . 1

"

�

(2. 13)

1

P

(2. 14)

The convergence in L2(A) does not imply the pointwi se convergence of Sv to v at all poi nts of A. However, Carleson (1966) proved that Sv converges to v for all x outsi de a set of zero measures in A. Some other results are as follows,

(i ) if v i s continuous, periodi c and of bounded vari ation on A, then Sv i s uniformly convergent to v on Ai

(ii ) if v i s of bounded vari ation on A, then !v(x-) for any x E Ai (iii ) if v i s continuous and periodic,

x E A.

Sv

Sv is convergent-pointwi se to !v(x+) +

does not necessarily converge at every point

-

For di scussion of differenti ation, let m be a positive integer and fI';(A) be the subspace of Hm(A) whi ch consi sts of all functions who�; first m 1 derivatives are periodic. In thi s space, it i s permissible to differenti ate termwi se the Fourier series m times, in the sense of square mean. In deed, o:r:ei/:r: = ileil:1 and so . U:r:V 1= (!lu:r:v,eil:r:) 2 7r (!'}-)

Therefore, for any v E

iI�(A),

=

-(

o:r:v(x)

!l eil:r:) V,U:r: 00

=

I:

1=-00

= Z

·Z (v,eil:r:)

ilVlei1:r:.

By repeating the above procedure, we obtain 00 o:;v(x)= I: (ilrvleil:r:. 1=-00

=

' 2 7rZ'ZVI.

Spectral Methods and -r:itir Applications

18 Consequently, the norm

For r = m + u and 0
E VN and DN(t) Ntl DN(t) sin2 SIn2. 2 t =

Then

1>( X)

2 t

.!. f 7r JA

=

19

be the Dirichlet kernel, =

-21 + I:/=1N cos It.

(2.18)

1>(t)DN(X - t ) dt

and thus by the Cauchy inequality,

Furthermore,

1> 2 E V Ell.

� 2

1 1>

1 IIL � �

and so

C�: 1 ) :; 1 1>11 · 1

(Np;7r+ 1 ) ' 1 1>�1 I � (Np;7r+ 1 ) ' 11 1>11 �111>lllp, .

I

1

111>�IIL� �

Hence

p

Since �

q,

.

1

we get

•

Another kind of inverse inequalities is called the Bernstein inequalities. Some of them can be found in Butzer and Nessel

(1971).

Theorem 2 . 2 .

1> E VN,

Let m be a non-negative integer and 1 � p

Also for any r ;:::: 0,

11 1>11 , � cN'II1>II·

�

00.

Then for any

20


Proof.

Let DN(t) be the Dirichlet kernel. By differentiating Ox IILP:5 cN-mIl8;;'vIlLP,

� for

1:5 p:5 00.

p = 00. Also, we have

with u(p) = 0 for 1 < p < 00 and u(p) = 1 for p = 1, 00. Anyhow PNV approximates v in the D'-norms with the same order as the best approximation, and IIv-PNvIILP-+ o as N -+ 00. In many applications, the numerical algorithms based on Fourier transformation cannot be implemented precisely. So we prefer to use discrete Fourier transformation. Let AN be the set of interpolation points,

AN = {xU) I xU) = � 2N +l'

J'

-

- 0"

1 ... , 2N} .

(2.19)

The discrete Fourier transformation of function vEe (X) is

INv(x) = L: Vleilx III$N

such that

INv (xU)) = V (x(j)) , 0:5 j :5 2N.

if p = (2N +l)q, q is an integer, otherwise,

Due to the orthogonality, i.e. ,

{ 2N0, +1,

�eiPX(j) =

we have

(2.20)

j=O

� v (x j) ) e-ilx(j) . VI = -2N 1+1 L...J ;=0

_

The coefficient VI can also be expressed in terms of the coefficient

1 � VI - _ � 2N +1 ;=0 _

_

_

(� vpe ) L...J p=-oo

A

i(P-I)XU»

VI. In fact,

Chapter 2. Or't�ogona1 Approximations in Sobolev Spaces =

1_ t _ 2N + 1 p=-oo 00

23

(fj=O vpei(p-I)"'W)

( 2.21 )

1: VI+p( N+l ' p=-oo 2 )

The interpolation IN c an be regarded as an orthogonal projection upon VN with respect to the discrete inner product. To do this, we define the discrete inner product and the norm as

( 2 . 22 ) Then and by

(2 . 20),

VvE C (X) ,'E VN. ( 2 . 23 ) This shows that INv is the orthogonal projection of v for the inner product ( ·)N. /

-,

As in the c ase of Fourier transformation, the following results hold, (i) if v is continuous, periodic and of bounded variation on uniformly on Xj

X, INv

tends to

v

(ii) if v is of bounded variation on X, INv is uniformly bounded on X and converges to v at every continuity point for Vj

(iii) for any integer I ::f and any po sitive N such that N > I I I , let VI = vf ) be the N I-th Fourier «oefficient of INv. If v E c;:" (X), then vI ) decays faster than cl-m , m being any positive integer, uniformly in N. If v satisfies the hypotheses N for which VI = " (l- m ), then Vf ) possesses the same asymptotic behavior.

0,

I I

We now deal with the global error between v and INv. Theorem

2.4.

Ifr > ! and 0:51l:5 r, then for any vE H;(A),

N

24

Spectral Methods and Their Applications -

I ENVIl� c IL�N ( 12Y' I VI iid2 cN2P. I LI �N II:VI+P(2N+l,12 :::; cN2P. II:�N (L I I p(2N 1) 1-2r) (L p(2N 1) 12r 1VI+P(2N +ll I 2) :::; cN2P. L (N-2rIPI-2r IqI:N Iql2r IVqI2) :::; cN2p.-2r L Ipl-2rlvl; :::; cN2P.-2rlvl; which together with Theorem 2.3 imply the desired result. :::;

1+

#0

#0

•

The following result provides the estimate in Loo-norm, Kenneth ( 1978) studied the estimates in em -norms. =f. Finally unlike But for any E

oxINv INoxv

2.3.

PN.

v

H;(A) with r > �,

Orthogonal Systems of Polynomials in a Finite Interval

w(x)

Let be a non-negative, continuous and integrable real-valued function in the interval A = ( - 1 , 1 ) . The associated weighted space of real-valued functions is L� equipped with the norm

I (x), I = 0, 1 , . . , b e an orthogonal system of algebraic polynomials with respect to the weighted inner product ( , ')w' By Weierstrass Theorem, such a system is complete in L�(A). The formal series of a function v E L�(A) in terms of {1>I } is '

'

'

.

'

00

1=0

Sv =I:VI1>I (x) with

VI =

1 1 1> II �

�

(2.24)

L v(x)1>I(x)w(x)dx.

By the completeness of the system {1>I } ,Sv is convergent to v in L�(A). The rate of convergence depends on the choice of the weight function w(x). The different choices correspond to different spectral methods. Denote by the set of all algebraic polynomials of degree at most N in A. In this set, several inverse inequalities are valid. We first consider the relation between 111>IIL� and 1 11> I IL� ' 1 � P � q � 00. Let L be a linear operator defined on L is said to be of (p, q) type, if there exists a positive constant d depending only on p,q and N such that II L1> IIL� � dll1>IILE for any 1> E According to Riesz Thorin Theorem, we know that if L is of both (PI, ql) type and (P2, q2) type for

IPN

IPN.

IPN.

Spectral Methods and THeir Applications

26

Pl ;P2 I L� $ d "I 4>II L�" 1$

(2 . 25) 1= 1 , 2, then

(2.26) where

c (Pl ! P2 )

Theorem 2 . 5 .

number 6,

then for any 4>

PI and P2 . If 4>1 E y"(A) for alii, and for certain positive constant and real

is a positive constant depending only on

Co

E

(2.27) IPN

and aliI $ P $ q $00,

where u(N ) = N2 6+1 for 6 >

6 < -!.

-!,

u(N) = In N for 6 = -!, and u(N) = 1 for

Proof. Let us consider, for example, the case 6 > - ! . By (2.24 ) and (2.27), (2.28)

d CN2 6+I .

Clearly , ) type with the constant = Thus the identity operator is of ( 1 00 for any r = � , 1 $ is of ( r, r ) type with = 1 . For 1 :5 < 00 , we set = 1 , = 00 , = = r and = � i n (2.25 ) . Then (2.26) implies the desired result. On the other hand, we know from (2.28) that is also of ( 1 , 1 ) type and 00 ( ,00 ) type. The proof is complete. •

PI

qI

L p:5 q:5 oo, L P2 q2 9

d

P

L

We now turn to another kind of inverse inequalities. The first result in the fol lowing theorem is due to M arkov (1898), e.g. , see Timan (1963). Theorem 2.6.

For any 4> E IPN,

Chapter

2.

27

Orthogonal Approximations in Sobolev Spaces

If, in addition, then

for

Proof.

all 2 � p �

(2.29) 00 ,

Set x = cos y ,

(2.30)

vex) = v(cos y).

From the proof of Theorem 2.2 , I Ox¢>(cos y) sin yl::; N O:O;v$ max" 1¢>(cos y) 1 2 and so (2.31 )

Next let ¢>N be an algebraic polynomial of degree N, and distinct points in A. Set

Then we have the Lagrange interpolation as N- l Ox

IPN is such a mapping that for any

34


or equivalently,

N PNV( X) = L v,L,(x). '=0 We now estimate the difference between v and PNv. L'emma 2 . 2 . For any v E Hr(A) and r � OJ Proof. Assume at first that r = 2m, m being an integer. We have I v - PNv l 1 2 = '=LN+l vr ll Ld l 2 • Let Av = -all' ((1 - x 2 ) a.,v). B y (2.44) , V, I L� 1I 2 i v (x)L, (x) dx = 1 ( 1 1\ I L1I1 2 i v( x)AL,( x) dx Av(x )L, (x) dx. l(l 1\ I L 1I1 2 i 00

+

+

Iterating this procedure yields

and thus

which is the desired result for space interpolation as before.

r = 2m. For any r � 0, we deduce the result from the •

a.,PNV( X) '" Piva., v(x ). But we have the following result . Lemma 2 .3 . For any v E Hr(A) and 0 ::; ::; r - I J Usually

j.l

Chap ter 2. OIlthogonal Approximations in Sobolev Spaces

35

Proof. According to the space interp olation, it suffices to consider the case r � 1 , I-' = O. Assume first that O",V is continuous and so (2.51) is valid; Also let

Then by

N N-1 o",PNv(x) = L w/L/ (x), w/ = (21 + 1 ) L vp. /=0 p=l+l p+1 odd

(2.51 ),

if 1 + if

Accordingly,

PNO", V

( X ) - u",PNv ( x ) .

where

£l

_

{

N

is odd ,

1 + N is even.

1 . ( 1) ", I ( ), v V. (1) ",0 ( ) 1 N 'f'N X 2N i: 3 N+ 1 'f'N X I V. ( 1 ) ",0 ( ) v. ( 1 ) ",1 ( ) , 2N + 3 N + 1 'f'N X 2N + 1 N 'f'N X 2N

I

+ +

+

¢>�(x) = L (2p + l)Lp(x) , ¢>}.(x) O

0, we have

H ! -' (A) .

Ilv - INv l 1' ::; I v - PNvl 1' + cN21' I PNv -INvl cN"(I',, rJl l vl r cN21'( l v - PNvl I v - INvl ) cN"(I' r) I vl r + cN21'- r+ ! Ilvl r.
II L:';

=

G)

�

1 :::; p :::;

IW IILP(A') '

(2.74 )

00.

Then the desired result comes from Theorem 2. 1 .

•

Theorem 2 . 14. Let m be a non-negative integer and 2 :::; p :::; I/> E IP N ,

00.

Then for any

Also for any r � 0, Proof. Let

�/

and

�il)

respectively. By (2.72 ) ,

(C/�P») 2

:::;

be the coefficients of Chebyshev expansions of I/> and ox rP

( P�' p2) ( P�' ��)

4

p+r odd

Accordingly,

Il ax l/>II�

=

N-l

2 /=0 ( �P » )

?!:. L 2

q

:::; N ( N + 1 ) (2 N + 1 )

�

p+l odd

:::;

N-l N ?!:. N (N + 1 ) (2N + 1 ) L -1 L 3

/=0 c/ /=0 ��

:::;

��r 4N4 1 1 rP ll�

from which, Theorem 2. 6 and a repetition, the first conclusion follows. ,We now prove the second one. If r

=

m+

(J'

and m is a non-negative integer, 0

an inequality parallel to (2. 15 ) ,

< (J'

�

and 0 :5

J.t

:5 r,

Proof. We use the transformation (2.73) and denote by IN the corresponding in

terpolation associated with the interpolation points in A· . It is easy to see that (IN V r = INv· and thus Hence th, result for J.t = 0 comes from Theorem 2.4 immediately. For J.t > 0, we have from Theorem 2.14 that

Then we reach the aim, using Theorem 2.15 and the previous result.

•

Theorem 2.19 is cited from Canuto and Quarteroni (1982a) . Another estimate is h t at for r > �, We also refer to Bernardi and Maday (1989), and Maday (1990) . Finally, by Theo1 rem 2.19 and (2.42) , for any v E H�(A) , r > 2 and ifJ E lPN, (2.87)


55

In the end of this section, we state a close relation between Legendre transforma tion and Chebyshev transformation . Let r(y) be the Gamma function and 1/1 (y) = (y + + 1 ) . Denote by and a pair of (N + 1 ) x (N + 1 ) mat ices with the elements and

r

! ) r - I (y

{I,

AN,i,k =

{

AN,i,k � 1/1 2 (�) ,

� 1/1

0,

AN BN,i,k ,

(9) (�) , 1/1

BN

if 0 = j � k < N + 1 and k is even, if 0 < j � k < N + 1 and j + k is even, otherwise,

BN,i,k = 2",(;- k)(i, + �) 1 ( k-i - 2 ) 1 ( k+i-I ) ' (k+i+1 ) (k -i) 2 2 �

. ,

. ,

'I'

'I'

0,

if j = k if 0 < j

;

= 0, = k < N + 1,

if 0 � j < k < N + 1 and j + k is even, otherwise. -

Now, suppose that v(x) has a finite Legendre expansion of the form v(cos y)

N

= L a / L / ( cos y) . /=0

Then it also has a finite Chebyshev expansion of the form

where

a

= ( ao, . . . , aN)

v(cos y ) and

b

N

/=0

= L btTt(cos y)

= ( bo, . . . , bN)

(2.88)

( 2 . 8 9)

are related by the equation

Conversely, if v is a function given by (2. 89) , then it may be expressed in the form of (2.88 ) , where Based on the above fact , Alpert and Rokhlin (1991) developed the Fast Legendre

values at N + 1 Chebyshev points in A with a cost proportional to (N + 1) In(N + 1 ) .

Transformation (FLT ) . For a Legendre expansion of degree N, FLT produces its at N + 1 Chebyshev points. The cost of this algorithm is roughly three times that of Similarly, FLT produces the Legendre expansion of degree N from the values of v

FFT of length N + 1 , provided that the calculations are performed to single precision

56

Spectral Methods and Their Applica tions

accuracy. In double precision, the ratio is approximately 5.5. FLT saves a lot of work in Legendre spectral methods and Legendre pseudospectral methods , and makes them more efficient and applicable to numerical solutions of partial differential equations.

2.6.

S om e O rt hogonal Approxim at ions in Infinit e Int ervals

A number of physical problems are set in unbounded domains . The first remark with spectral methods on these problems, is that polynomials are not integrable on unbounded domains. So the idea is to work with weighted Sobolev spaces where the

( )=

weight must be of exponential type in order to avoid restrictions on the degree of spaces Lr, (A) , p � 1 , W':'I'(A) , H: (A), C ( a , bj W':'I'(A)) , H" ( a, bj W':,P(A)) and their the polynomials. We first consider the case A = (0, 00) and

w x

e -X •

Define the

semi-norms and norms in the same way as in Section 2.3 . Also we set the inner product ( . ·) w of L! (A) following the same lines as in that section. Clearly for any ,

bounded interval A*

c

A , the restrictions to A* of all functions in H: (A) belong to

H: (A* ) . We quote some essential properties of H: (A) . They state that (i) for any r � 0, the space Coo (ii) for any r � 0, the mapping: onto. H' (A)j (iii) for any 0 :::;

(iv) for any r

II

< r and 0
� , the space H: (A)

is contained in C

(1\.) , and for any v E H� (A),

For technical reasons , we also need another Sobolev space associated with a positive integer

a,

This space is provided with the associated natural norm

II v llr,w,a = II v( l + x) � Ilr,w .


57

The Laguerre polynomial of degree I is

' (x ' e-") . £ , (x) = .!.e"o l! "

(2.90)

It is the I-th eigenfunction of the singular Sturm-Liouville problem

= I. Clearly £oCx) = l ' £ l (x) = 1 - x, and they

related to the I-th eigenvalue A, satisfy the recurrence relations ( I + 1)£ ' +1 (X)

=

1 - x)£,(x) - (1 - l )£ ' _l (X), I � 1, £,(x) = o,,£,(x) - 0,, £ 1 + 1 (x), 1 � O. It can be checked that

(21 +

(2.91) (2.92)

£,(0) = 1, 0,,£,(0) = I for 1 � 1, and 1 £,(x) 1 � e � ,

x E A.

(2.93)

The set of Laguerre polynomials is the L�-orthogonal system on the half line A, i.e. ,

By integrating by parts, we deduce that

1 o,,£,(X)O,,£m (X)XW(X) dx = IO m ' "

(2.9 4 )

The Laguerre expansion of a function v E L�(A) is 00

vex) = L: v,£, (x) '=0

with

v, =

lPN

1 v(x)£,(x)w(x) dx.

(2.9.5 )

Now let be the space of restrictions to A of polynomials of degree at most N. Several inverse inequalities exist in this space. To do this, we need some preparations. Let A* = ( a, b), - 00 � a < b � 00, and L (A') be the space of functions, p -th power integrable with the weight x(x) � O. We define the norm II ' IIL� in the usual way. Let

�

58


'lh(x), 1 = 0, 1 , . . . , be an orthogonal system such that 'lp, E L � ( A* ) for alI I ::; p ::; and suppose that there exist positive constant Co and some real C such that

00,

(2.96)

For any w E L � (A* ) , its formal expansion in terms of 'lh( x) is 00

w (x) = L WI'lh(x) 1=0

with

WI =

w (x)tPl (x)X (x) dx . IItP li2)( l .

�

Consider the Cesaro mean of order k, k being some non-negative integer. Set A� (,%t�)! . Then the Cesaro mean of the function w is of the form

The system { tPl } is said to be regular, if for Bome k* , there exists a constant such that for any w E L � and 1 ::; p ::; 00 ,

Cl

=

>0

Denote by QN the span { tP l (X) 1 0 ::; I ::; N}. Nessel and Wilmes (1976) proved the following result. Lemma 2 . 9 . Let {tPl } be regular and (2. 96) hold. Th en for any tP

p :5 q :5

E QN and 1 :5

00 ,

where u(N) is the same as in Theorem 2. 5. Proof. For example, we consider the case

C

>

-�. By (2.96) , (2.97)


A(X)

Let be a smooth function on A* such that for x � 2. Then the mean

A(X)

=

1 for 0 :::;

59 x

:::; 1 and

A(:C)

=

0

possesses the following properties:

LNW E Q 2N for all W E L� (A·) j (ii) LNtP tP for all tP E Q N j (iii) for any W E L� (A·) and 1 :::; p :::; I LN wI l L� :::; clllwl I L� { I y,\:o a;o+I A( Y ) I dyj (iv) for any w E L� (A* ) , (i)

=

00 ,

Indeed, the results ( i ) and (ii) are obvious. The result (iii) follows by a multiplier criterion as given in Butzer, Nessel and Trebels (1972) . The result (iv) is an immediate is of types (1, 1 ) , ( 1 , 00 ) and ( 00 , 00 ) . So consequence of (i) ,(iii) and (2.97) . Thus the classical Riesz-Thorin convexity theorem implies that is of type (p, q ) for all 1 P :::; q :::; 00 . In view of (ii), we get the desired conclusion.

LN

:::;

Theorem

Proof. Let

2. 20.

a =

For any E

0, b =

00

LN

lPN and 1 :::; p :::; q :::;

and x( x )

==

00 ,

1 . Consider the functions

1 = 0, 1 , . . . .

By (2.93) and the orthogonality of C/(x), we know that (2.96) is valid and 6 Since {lPl ( x ) } is regular, Lemma 2.9 leads to the conclusion of this theorem. Theorem

2.21.

For any

E

lPN,

=

O. _


60

Proof. By (2.92) ,

and so l I a3;q)I I � =

This gives the result.

� C�l �lr

The L�-orthogonal projection PN ; L�(A) v E L� (A) , or equivalently, PNV (X) = 2. 1 0 .

any v E H�(A, a ) ,

L emma

Let r � 0 and

a

�

E �� ( N

--+

-

P

)' •

IP N is such a mapping that for any

N

Ev,C,(x), 1=0

be the largest integer for which

Proof. We have

a

< r + 1 . Then for

00

EN l vr

I= + According to the Sturm-Liouville equation, we set

Il v - PNv l l � =

It is a continuous mapping from HJ, + 2(A, (3 + 2) into HJ,(A, (3) where ,, (3 � 0 and , is an integer. When r = a is an even integer, we have from integration by parts that

whence II v - PNv l l � � N-ct

A� V (X ) C l (x ) e-3; dx ) 2 ( r iA +l

f:N

I=

< N-ct I I Af v l l � � N -ct l l v ll ;,w, ' ct

Chapter 2. Orthogonal Approximations in Sobolev Spaces z _ a;,

LA a;' v (x).c, (x ) e-'" dx i{A a", (A a-' v (x) ) a",.c, (x)xe - X dx .

61

When r = a is an odd integer, we have from integration by parts again that z - !ttl 2

VI By (2.94) ,

I l v - PNv l l � ::; N-CX

::;

a ' 2 a", (A ; v (x) ) ax.c, (x)xe - '" dx ) f: ( 1 A I=N l

(L (ax.c, (X) )2 xe-x dxr 1 a ' N -cx ll a", (A ; v ) x � I I � ::; N - cx ll v l l � ,w,cx ' +

The general result follows by an interpolation argument between the spaces a + 1). 1 ) and

H�( A,

Theorem

2. 22.

For any

vE

H;(A, a )

and

wh ere a is th e larg est integer such that a p.

We observe from (2.92) that

Thus

N

(v, - i, (x' , A")

)

" e "

(2.116)

+ f (x , x", A") .

(2.116) yields an essentially nonoscillatory approximation provided that the approx

imations for the location x' and the jump A are quite accurate. Indeed, it is shown that V (vN ) :5 V(v) + c

ln N

N

+ cN ln N I x ' - x" 1 + e ln N IA - A" I ,

Il v - VN I Ll :5 c l�� + e ln N l x' - x' l + c I A - A" 1

where. V ( v) is the total variation of v . Moreover, both I x' - x" 1 and I A - A" I are of if x ' and A" are determined by the following system the order of

0 (�2 ) '

{

-i(N+1).,* A* i(N+1) e i(N+ 2 )"'* A* i(N+ 2 ) e

= =

VN+1 , •

•

VN+2 '

A more efficient approach is related to the idea of Harten, Engquist, Osher and Chakravarthy ( ) Let xU) be the same as in and vex) have only one discontinuity at x'. Define a polynomial interpolant of degree m for vex) at point xU) , i.e. , Qm xU) ; v = v xU» , 0 :5 j :5 2N, Qm(x ; v) = qm .j+ � (x; v ) , for x(j ) :5 x :5 xU + 1) ,

1987 .

(2.19)

( ) ( )


73

qm ,i + � (Xj V ) interpol �tes v(x) at (m + 1 ) successive p oints x ( ,x), A m (j) � A � A m (j) + m. The stencil of these (m + 1 ) points will be chosen according to the smoothness of the data v ( x ( ,x» around xU). A recursive algorithm to define A m (j ) starts with

)

(2. 1 1 7) It means that ql,j + � (x) is the first degree polynomial which interpolates v( x) at x (i ) and xU + I ) . Assume that qp ,i + � (x) i s the p-th degree polynomial which interpolates v (x) at the points (2.1 18) Then we need one additional point for determining qp+ l ,j + � (X) . That point may be the nearest one to the left of stencil (2.1 18), i.e., x (,x"U) - l ) , or the nearest one to the right of stencil (2. 1 18), i.e. , x (,x"U) +P + I ) . The choice will be based on the absolute values of the corresponding (p + l )-th order divided differences, namely

[

if v x (,x,,(i ) - l ) , . . . , x (>',,(i ) +p) otherwise.

] < v [x (>',,(i)) ,

•

• •

]

, x (>'"U) +P+ I ) ,

(2.1 19) The piecewise polynomial Q m (Xj v) gives uniform nonoscillatory approximation to vex) up to the discontinuity. In fact, it can be shown that for 0 � k � m, (2. 120) except for the cell containing x'. Using sub-cell resolutions in Harten ( 1989), this is also correct in the shocked cell. Cai and Shu (1991) provided a uniform approximation by combining the idea in the previous paragraph and the filtering (2. 1 10). Assume that the discontinuity has been detected within an interval [x; , x�] . Denote all the points inside the interval [x;, x�] as xU'), . . . , x (ir) . Then we define a piecewise m-th order polynomial ¢(x) which interpolates the function v( x) at the points xU) , jl � j � j. ,

[

if x E x (i) , xU + I ) if x E [0, x;] , if x E [x� , 2'1l'] ,

]

n

[X I x�] for some j, '


74

{

where qm ,i + t (x) is defined by (2. 1 1 7) - (2. 1 1 9 ) , and PI (X) and Pr (x ) are both m'-th order polynomials on the interval [0, x l ] and [x � , 2 11" ] respectively, m' = 2 m + 1, and for 0 $ k $ m, a!PI (x l ) = a! qm,il - t (x l ) , a!Pr (x � ) = a! qm,ir+! (x � ) , (2. 12 1 ) a!PI (x � - 2 11" ) = a! qm,ir+! (x � ) , a!Pr (X l + 2 11" ) = a!qm,il - ! (x l ) . This condition ensures that 'I/;(x) is at least globally em continuous. There are

2m + 2 = m' + 1 constraints on the m'-th polynomials PI (X) and Pr (x) respectively. Therefore, they are uniquely defined. By (2.120 ) ,

( ) j, $ j $ j. , m 'I/; (x) - v(x) 0 ( N- - 1 ) , x E [x ; , x �l . ( )

'I/; xW = v xCi » ,

(2. 122) (2. 1 23)

=

Now we consider the difference w(x) = v(x) - 'I/;(x) . w(x) is a e m function in A. except at x', and [w]." = 0 (N- m -l ) . Moreover w xCi » = O, j, $ j $ jr ' Let O'N, I

=

where

exp ( - 0 1:k1 2'l)

( )

and

-- L: ( ( ) ( L: ( ( )

( ))

2N 1 v xCi » - 'I/; x U » e - il.,w 2N + 1 i=O 1 v xCi » - PI xCi » e- il.,W + v x Ci » - Pr x U» __ 2N + 1 iir

( ))

Finally we define the uniform approximation by

L: ( (

)

( )) e- il.,(j)

PN v(x) = 'I/;(x) + WN,q ( X ) .

The derivatives of v(x) will be approximated by those of PNV(X), i.e. ,

It is proved that

)

(2.124)

.


75

which establishes the uniform (m + 1 Hh order convergence of the approximation P;'v to v in L 2 -norm. In the regions outside [xl, x�l, (2.124) has the convergence rate of "infinite order" . Another uniform approximation is based on the one-sided filtering. Suppose that the function v(x) is analytic over [0, 271") , but v (O) #- v(271"). Vandeven (1991) consid ered the two-sided filtering (2. 1 10) with (2p I ) ! i. I f y ( 1 - y )I P- 1 dy , p = Ni , 0 < c < 1 , UN,1 - 1 «p 1 )!) 2 io -

-

_

(2. 125)

and proved that where 8 = Ne - l , and A is a certain constant independent of N. Cai, Gottlieb and Shu (1992) defined the one-sided filtering as RNfjI ( x) = L: N �

1 1 1� N

where

u ,I l eil:r: ,

V fjI E VN

(2. 126)

m L...- ( -l)pH p I(. m ! p) 1. exp (,.p iNe -l ) , m = Ni , UN,1 = UN, 1 � = p1 *

_

and UN, 1 is given by (2.125). They proved that if v(x) is periodic, analytic in [0, 271"), but v(O) #- v(271"), then for any 0 < c < � ,

where 8 = 2N� e- l . The above filtering i s naturally labelled "right-sided" filtering �ince it can recover the spectral accuracy up to the discontinuity x = 0 from the right. The "left-sided" filtering can be constructed similarly. But a very large N is needed for the reasonable size of m in (2. 126). Cai , Gottlieb and Shu (1992) used a least square procedure to create uN, 1 for 8 � N � 32. The numerical experiments showed the advantages of this approach. Recently Gottlieb, Shu, Solomonoff and Vandeven (1992) developed a new method for recovering the spectral accuracy. The main idea of this method is to reconstruct

76

Spectral Methods an.d Their Applications

the Fourier expansion by using Gegenbauer polynomials. The Gegenbauer polynomial of degree 1 with parameter >. � ° is given by

>. CI,>.(X) = GI,>. ( 1 - x 2 ) � - OxI ( ( 1 - x 2 ) 1 + >'-� ) where

(-l) l r (>. + �) r(l + 2>.) GI , >. = 1 2 l!r(2).)r ( 1 + >. + �) , (_1) 1 y0r for 1 � 1 G 1,0 - 2 1 1 1r - (1 + �) ,

for >. >

(2.127)

0,

and

Go,o = 1. B y this standardization , Co,o( x) = 1 and for 1 � 0 , CI,o (x) = I2 TI(x),

(2.128)

The Gegenbauer polynomials satisfy the recurrence relations Ox ( ( 1

- x 2 ) >'-� CI,>.(X) ) = G Gl ',.\ 1 ( 1 - x 2 ) >'-� CI+1, >. - 1 (X) 1+1 >'-

(2.129)

and

(2.130)

It is shown that CI,>.( l ) = and I CI,>.(x) 1 � CI,>.(l) for I x l � 1. Let A = (-1, 1) 2 and w>.(x) = (1 - x ) >'-� . The set of Gegenbauer polynomials i s the L� A -orthogonal system in A, i.e. ,

��g�)

(2.131) where for >. >

0,

There exists a constant

Co

hi, >. =

y0r CI,>.(l)r (>. + �) (l + >.)r(>.)

.

independent of 1 and >. such that


77

v L�, (A) is vex) I: VI,>, CI,>. (X)

The Gegenbauer expansion of a function E =

00

with the coefficients

VI,>.

=

(2.132)

1=0

I hI ,>' JAr V(X)CI, >. (X)W>.(x) dx ,

-

1 = 0, 1,

.. .

.

The corresponding orthogonal projection is =

N

VN, >. (X) I:VI, >. CI,>. (x) . 1=0

(2.133)

Now, we consider an analytic, but non-periodic function in A. If it is extended periodically with the period then has a discontinuity at = ±l . The Fourier coefficients of are defined as

vex)

2,

v( x)

VI

=

� r

vex)

x

2 JA v(x) e -il1l"x dx.

The traditional truncated Fourier sum is =

V (X) I: vl eil1l"X . Let (x) be the Bessel function and WI , >. be the coefficients ofGegenbauer expansion of VN(X). Then they can be expressed as >' VP' WI,>. OI,OVO + + >.)r(>.) I: JI+>.(p7r ) (2) p7r The reconstructed expansion of v (x) is v'N (x) I:WI , >. CI,>. (x). (2.134) N

1>.

=

Il i �N

il ( l

O< lpl � N

=

M

1=0

c( p) depending only on p ::::: 1 such that max l a;v(x)1 � c( p) k� . (2. 135) P This is a standard assumption for analytic functions. The quantity p is the distance Assume that there is a positive constant Ixl � l

from A to the nearest singularity of

v( x) in the complex plane.

Gottlieb, Shu,

78


Solomonoff and Vandeven (1992) proved that if (2.135) holds, A {3 < then max Iv(x) - vN(x) 1 :5 bI N2 qf + b2qf

f,;1re,

1",19

where

bl and b2

are certain positive constants related to 0$/$00 max

ql =

( 21re )

27{3 f3

-

q2 =

< 1,

( )

= M =

{3N and

l tid , and

27 f3 < 1. 32 p

The meanings of ql and q2 are explained i n their paper. (2. 134) gives a uniform approximation to v(x), including the discontinuity, and recovers the spectral accuracy. We also refer to work in Gottlieb and Shu (1994) . All of the methods in the previous paragraphs are also available for Legendre approximation and Chebyshev approximation. For instance, let v E IP N and be the Chebyshev coefficients of v(x) , 0 :5 :5 N. Guo and Li (1993) offered a filtering. Ma and Guo (1994) considered RN (a, {3) and R�(a , {3) for ¢> E IP N , defined by

ti,

1

RN (a, {3)¢> (x) R� ( a, {3) ¢>( x)

� (l - l � rr �/TI (X) , � +

where the constants

( 1 1 rr �,T,(x) a N-ITN-I (X) aNTN (x),

N-2

1-

N

a, {3 � l,

(2. 136)

(2. 137)

a N-I and a N are determined by

R� (a, {3) ¢>(-l ) = ¢> ( - 1 ) , Let

+

a, {3 � l ,

�(a, {3) ¢> (1) = ¢>(1).

w(x) = ( 1 - x 2 ) - ' . It can be checked that 1

( RN (a, {3)¢>, 1/Jl.., = ( ¢>, RN (a, {3) 1/Jl.." V ¢>, 1/J E IP N , ( RN (a, {3) ¢>, tPl.., = (¢>, R� (a , {3) 1/JL , V ¢> E IP N-2 , 1/J E IP N . Theorem 2 . 27. For

any ¢> E IPN and 0 :5 r :5 a,

(2. 138) (2.139)


Proof. We have

-2 (32 L:1=1N Il-N 1 2'" ¢; :::; c(32N-2r L:1=1N [2r ¢; 2 < c(32 N -2 r t [2 r ¢; :::; c(32 N -2 r [' l o� ( L: ¢ l l I eiI Y ) 1 dy 1=0 I I I $N 2 :::; c(32N-2r L: l o� (� ¢cT! (COsy )) 1 dy :::; c(32 N - 2r ll v ll �,w ' < 1

79

7r

- 7r

Theorem 2 . 28.

For any ¢>

with

E

lP N }

b", ,/3 - ( _

If in addition ¢> with Proof. Let

E

lP � }

then

d"',

(

_

/3

-

+ l) f ( �) (3 ( + 1 + �)

)

2

+ (�) ( + 1 + �)

)

:;

f (2 (3 of 2

f (2 (3 2 ) f of 2 (3

ox RN( O, (3 ) ¢> ( x ) =

al = -C2I

Then

and thus I RN( 0 ,

(3)¢> I � ,w

•

L: p

p=l+l p+ l odd

1

1

L:- acT!(x) .

N 1 1=0

(1 - .I -NP 1 "') /3 ¢>1' A

/3 1 ¢d ) 2 I 1 [ � (� ( _ J"') < cN � [ 2 ( 1 I � I"') 2/3 � ¢; :::; cN3 � 1 � 1 2 (1 _ I � I"') 2/3 � ¢; :::; cN4 11¢>1 I � l y ( 1 - y"' ) 2/3 dy CN4 b!,/3 I ¢>I I � . % �1 ci a ; :::; cN _

2

=


80 The second result can be proved similarly.

_

A general form of filtering for Chebyshev approximation is

N L: = UN, ,�m(X),

V ¢> E IPN 1=0 where 0 � UN,I � 1 , UN,I "" 1 for I � N and UN,I � 1 for I "" N. Weideman and Trefethen (1988) used this approach for evaluating eigenvalues of matrix related to the second order differentiations of Chebyshev expansions, and took

RN ¢> (X)

UN,I =

{ I, (_ (l=ilL)4) a

exp

N - 1o

o � I � 10 , '

a

> 0, 10

respectively. For any defined as

Let L be an operator from operator equation

v

Bl to B2 , and f be a given element in Bz . Consider the LU = f·

( 3.1)

Bl , B2

In order to solve ( 3 . 1 ) approximately, we first approximate the spaces and be finite dimensional Banach spaces L, f. Let N be any positive integer, and with the norms II . IIB " N ' q = 1 , 2. For any v E " N and R > 0, the balls are defined

Bq , N B

82

Chapter 3. Stability and Con,vergence

by

83

S

= {w E Bq ,N I li w - v I i B.,N < R} , q = 1 , 2. Assume that Dim(Bl ,N) = D im( B2,N ) , and Bq,N approximates Bq , q = 1 , 2. Let Pq,N q , N ( v , R)

be continuous operators from Bq to Bq,N such that .

Ji� I i Pl ,N v Ii Bl.N = I lv I i B" . 'V v E Bl , 2,N , 'V v E B2 · = Nlim ......oo I lp v Ii B. . N I i v I I B.

(3.2) (3.3)

Let qN be a continuous operator from B2,N to B2 , satisfying (3.4) We denote by B3 another Banach space with the norm II · Ii B3 ' UN is a continuous operator from Bl ,N to B3 , while w is a continuous operator from Bl to B3 • Suppose that there exist positive constant Co and positive integer No such that for all N � No, (3.5) and for any v E Bl ,

= UN ,N Nlim -+oo Ii Pl V - wv l i B3

O.

j

(3.6)

In this section, we assume that all operators L, Pl ,N , P2,N , qN, UN and w are linear.

B'

/

�

L

B',

' B2

---

L-l

"'oN

LN

"" N,

qN

B1-,N----------------L-7l----------------N

Figure 3 . 1 .

Let L N b e an invertible operator from B1 ,N t o B2,N. We approximate (3. 1 ) by (3.7)

Spec tral Methods and Their Appli cations

84 The approximation error is defined as

(3.8)

f

Let D be a subset of solutions of (3. 1 ) , and F be the set of all related to U E D. Assume that F is dense in The approximation (3.7) is said to be consistent with (3. 1 ) , if for . any v E D ,

B2 •

(3.9) IT for all U, the solutio�� of (3. 1 ) and holds

UN, the corresponding solutions of (3.7) , there

then we say that (3.7) is convergent. Usually the data possesses certain error, denoted by It induces the error of denoted by If there exists a positive constant Cl such that for any j and

UN ,

UN.

N 2: No ,

f

1.

(3.10) then we say that (3.7) is stable. Since

LN and UN are linear, it is equivalent to (3. 1 1 )

Since

UN L'ilP2 ,Nf , i t i s also equivalent to =

(3. 12) Theorem

3. 1 .

If L - 1

exists, and (3. 7) is consistent with (3. 1), then the stability is

equivalent to the convergence. Proof. IT (3.7) is convergent, then for all

l I uNLf/P2 , N f Il B3

f,

Thus is bounded uniformly for N. By the Uniform Boundedness Theorem, (3.12) is valid.

.

Chapter 3. Stability and Conyergence

Now assume that (3.7) is stable. For any mate solution

UN,

85

U E D and the corresponding approxi

The first term on the right-hand side tends to zero by (3.6) . Since (3.4) implies

B2 ,N ,

IIUNUN - UNP1 ,N U I B3

IIUNLiV1 p2 ,NI - uNLiV1 p2 ,N QNLNP1 ,NU I B3 I uNLiV1 p2 ,N (LU - QNLNP1 ,NU) I B3 ::; IIUNLiV1 p2 ,N I I RN(U) I B2 '

By (3.9) and (3.12), the above right-hand side tends to zero as N any E D,

U

U

LNP1 ,NU

-+

00.

E

(3. 14)

Hence for (3.15)

If is a solution of (3. 1 ) , but UED, then there exists a sequence to the data such that

{ /(I) }

{ U (I) } E D, related (3.16)

ulj}

Let be the corresponding approximate solutions of (3.7) with the data we have

1(1) . Then

IIUNUN - WU I B3 ::; I UNUN - uNuIj}I I B3 + l I uNulj} - wU(l) I B3 + I w U (I) - WU I B3 ::; I uNLiV1 p2 ,N I I J - 1(1) li B. + IIUNulj} - WU(l) I B3 + I w 111l L - 1 11111 - 1 ( I) II B2• (3. 17) By (3. 12) , (3. 15) and (3. 16), the above right-hand side goes to zero as N, This completes the proof.

l -+

00. •

Theorem 3 . 1 can be found in Guo (1988a) . We now turn to the convergence rate. Theorem 3 . 2 . II

L- 1

exists, and (3. 7) is stable and consistent with (3. 1), then

Spectral Method� and Their Applications

86

Proof The conclusion comes immediately from (3.13), (3. 14) and (3. 17).

•

If w is an isomorphism between and then the above estimate also holds for IJW- 1 UNUN - II B, . = = qN, UN and w are the identity op If = 1 , 2, then erators. If in addition D = then Theorems 3.1 and 3.2 turn to be the Basic Convergence Theorem . of Kantorovich (1948) . Besides, N could be replaced by any real parameter. It is also noted that in spectral methods, are certain subspaces are some projections from = 1 , 2, usually. In this case, to of and = we can take and II · IIB'o N = II · liB. , = 1 , 2. Therefore UN and w are the identity operators, and so

Bl

U B3 Bq Bq , N , q

Bq ,

3.2.

B2 , Pl,N , P2 ,N ,

B1 ,

Bq Bq ,N, q

Pq , N B3 B1

Bq , N

q

qN,

Generalized Stability for Nonlinear Problems

In this section, we deal with nonlinear problems. We still consider problem (3. 1 ) , but have the same meanings as and may be nonlinear. The notations and could be nonlinear. Let before. They also fulfill (3.2) and (3.3) , but and approximate and respectively. Consider the approximate problem

L

LN

IN

L

I

Bq, Bq,N Pq,N Pl,N P2 ,N

(3.18) The approximation error is defined as = Let D be the same as before. If is a solution of (3. 1 ) , then The approximation (3.18) is consistent with (3. 1 ) , if for any v E D and I E F ,

U

If for all

RN(U) IN-LNPl,N U ,

U, the solutions of (3. 1 ) and UN, the corresponding solutions of (3. 18) ,

Chapter 3. Stability and Convergence

87

then we say that ( 3.18 ) is convergent. How to define the stability is a problem. The simplest way is to follow ( 3.10 ) . E and It means that there exists a positive constant Co such that for any N � No, ( 3.19 )

v, w B1,N

Such kind of stability is called linear stability. It is an alternative expression to the stability in the previous section. But nonlinear problems usually do not possess this property. Indeed, this stability is very strong so that it is essentially suitable for linear problems and some specific nonlinear problems, such as sine-Gordon equation. Firstly, we see that the constant Co is uniform for all whereas the corresponding constant depends on the exact solution of ( 3.18 ) for most nonlinear problems. Next, ( 3.19 ) is However, the feature of nonlinear problems is that valid for any only when it lies in certain region, the difference between and is controlled. But may grow rapidly, once exceeds some critical bound. Thus ( 3,19 ) cannot describe the essence of nonlinear problems properly, Finally, Clearly this is not possible for nonlinear problems E ( 3,19 ) holds for all with several solutions, These facts motivated us to look for new definitions of stability, Strang ( 1965 ) served the weak stability for linear problems, which is also used in spectral methods, see Gottlieb and Orsag ( 1977 ) . Stetter ( 1966, 1973 ) developed the nonlinear stability applied to initial-value problems of nonlinear ordinary differential . equations, while Guo ( 1965, 1982 ) provided several kinds of generalized stability used for numerical solutions of nonlinear partial differential equations, All of these works are suitable only for problems with unique solutions, Keller ( 1975 ) generalized the stability in a different way and proposed the local stability, which has been applied successfully to boundary-value problems of nonlinear ordinary differential equations. In 1985, Guo unified the above work, see Guo ( 1988a, 1989 ) . We shall present this general result below, We first int roduce some terminologies. The operator is said to be stable in the ball R) , if there are positive constants R and CR such that

UN ,

I LNv - LNw I i B2 ,N '

I LNV - LNW I B2 ,N

I l v - wI i B1 ,N

v

w

v, w B1 ,N '

Sl (v ,

L

( 3.20 )

Spectral Methods and Their A,pplications

88

L

If U is a solution of (3.1) and there exists a positive constant R such that is stable in 81 (U, R) , then we say that U is a stable solution. Obviously such a solution is ( unique in 81 (U, R) . Assume that the data in (3. 1 ) possesses the error - which induces the error of. solution, denoted by If the operator is linear and stable in - f. 0 and U' - = R $ the ba1l 81 (U, R); then • . Indeed, let Then

I I U�II B, $ R and

II U IIB, cR ll f I lB

U.

f

L ( U + U- ') = f + II URlB, . II I U' II B, - cR1l�JIU II1BIIB2 ,

By (3.20) ,

L II U I B,

f

U · U II IIB,

max ( s, 0), then •

o as

Cl(Pl,N U, N) I RN(U) 1 I 2 ,N

N -+

I RN(U) I ! B2 .N :::; c2 (U)N- q , q I l uN - Pl,N U I B, . N :::; c2 (U)N-Q .

3.3.

Initial-Value Problems

We focus on initial-value problems in this section. Let Bl and B2 be two Banach and B � B2 . spaces with the norms and are two differential operators with respect to the spatial variables, mapping v E B into B2 . For simplic ity, assume that is a linear invertible operator. Denote by B3 the third Banach whose element is a mapping from the interval (0, T] to space with the norm B T> Let J(t) be a given element in B3 and be a given element in B1 • describes the initial state. Consider the operator equation

I . I BI

Q

2,

O.

I . l i B" l

I . I B3'

Q

L l

Uo

{ 8U(O)t (Q U(t))Uo.

=

=

L U(t) + J(t),

0 < t :::; T,

Uo

( 3 .31

)

Chapter 3. Stabilit,y and Convergence

95

We define a genuine solution of (3.31) as a one-parameter family U(t) such that U(t) is in the domain of Q and L for each t E [0, TJ , and

· I � (QU(t + s ) - QU(t)) - LU(t) - f(t) t,

�

0,

as s --+ O.

Let Dl and D3 be such subsets of Bl and B3 that for any Uo E Dl and f E D3 , (3.31 ) has a unique solution. Suppose that Dl and D3 are dense in Bl and B3 respectively. The corresponding subset of solutions is denoted by D 2 • Now let N be any positive integer and Bq,N be finite dimensional Banach spaces with the norms II · II B " N ' approximating the spaces Bq , q = 1 , 2, 3. In addition, Dim(B1,N) = Dim(B2 ,N) . Denote by Pq,N the continuous operators from Bq to Bq ,N, satisfying the following conditions: (i) for all v

E Bl , Pl,NV = P2 ,NV i

(ii) for all v

E Bq , I l pq,Nv ll

--+

IIv ll q as N --+

00, q

=

1 , 2, 3.

Next, let r = r (N) > 0 be the mesh spacing of the variable t, and Dr v be the forward difference quotient with respect to t, defined in Chapter 1 . We now approximate QU(t), LU(t) , f(t) and Uo by QNUN(t) , LN(UN(t) , UN(t + r)) , fN(t) and UN,O respectively, and then consider the approximate problem

{

Dr (QNUN(t)) UN(O) = UN,O.

=

LN (UN(t) , UN(t + r ) ) + fN ( t),

(3.32)

Suppose that for each t, UN(t) is determined uniquely by (3.32) . The simplest case is that Q ,/ exists and LN (UN(t) , UN(t + r)) is independent of UN(t + r) . The approx imation error for t > 0 is defined as RN(v(t))

=

- P2 ,N 8t Qv(t) + DrQNPl,NV(t) + P2 ,NLv(t) - LN (Pl,NV (t) , Pl,NV (t + r)) + P2 ,Nf(t) - fN(t) .

If U(t) i s the solution of (3.3 1 ) , then RN(U(t)) i s reduced to

Spectral Methods and Thejr, Applications

96

The approximation (3.32) is said to be consistent with (3.3 1 ) , if for any fE ::; T, ( and 0

::; t

D 2 , D3

If for all

Uo E. D1, U E

U(t), the s�lutions of (3.31) and UN(t), the corresponding solutions of (3.32) , I p1, N U(t) - uN(t) I B1 ,N - 0 , 0 ::; t ::; T, as N -+ 00,

then we say that the approximation (3.32) i s convergent. We now turn to the stability. We first consider the linear stability. It means that there is a positive constant eo(T) such that for all N � and 0 ::; T,

uW

No

::; t

u��o

fiJ ) , q =

where are the solutions of (3.32), corresponding to and 1 , 2. Following the same line as in the proof of Theorem 3 . 1 , we can prove the following Lax Theorem ( see Richtmyer and Morton ( 1 967 ) ) . Theorem 3 . 7 . If (3. 31) is a well-posed linear problem and (3. 32) is consistent with

(3. 31), then the linear stability is equivalent to the convergence. Moreover for all

0 ::;

t ::; T,

We now consider nonlinear problems, e.g. , see Guo ( 1 994 ) . If there are positive constant N, T) such that for all N � N, T) and non-negative constant C1 No, the inequality

eo( uW ,

(u�) ,

I uN,(1 )O - uN,(2 )O I B" N + Il fN(l ) - fN(2) I Ba ,N ::; eo (UN(1) , N, T)

(3.34)

implies

l I u�) (t) - u�) (t) I Bl N ::; C1 (uW, N, T) ( 1 I u��o - u��o I l Bl,N + Il f� ) - f� ) I Ba , N ) ,

o ::; ::; T,(3.35)

t

Chapter 3. Stability and Convergence uW (t) being the solutions of

97

( 3.32 ) related to u ��o and j o and s < then ( ) is convergent with the accuracy 0 (N - q ) . I n derivation of generalized stability, we often meet some nonlinear discrete in equalities. Let R-r(T) be as in ( )

q,

,

=

=

3.32

=

q

1.21 .

Lemma 3 . 2 .

Let E(t) be a non-negative function and pet) be a non-decreasing func tion on R.,. (T) . The constants d > 0, A > 0 and c � o. Assume that


98 (i) if E (s)

:::; d

for all s E

R,.(t - r ),

then

FE (t - ,r ) FE ( E (O) , E ( r ), · · · , E(t - r )) :::; cE(t - r ), GE (t) GE ( E(O), E ( r ), · · · , E (t)) � A E(t)j =

=

GE (t) :::; P(tl) + r L: FE (s)j .•

Then for all t E

Proof. Let

.,p(t)

ER� (t-'T)

R,.(tl), be such a non-negative function that

�

.,p(0) P l) =

A.,p(t) P(tl) + cr L: .,p(s) .

and for all t

:::; tl ,

=

•

Then

ER�(t-'T)

�

.,p(t) :::; P l ) est :::; d. Clearly E (O) :::; .,p(0). Assume that E (t) :::; .,p(t) for t E R,.(tl t :::; tt ,

) Then for any

r .

A E (t) :::; P(tl) + cr L: E (s) .E� (t-'T)

:::; p(tl) + CT L: .,p(s) A.,p(t) =

.ER�(t-T)

which implies the conclusion. In many cases,

GE (t) = E (t)

FE ( t ) = Co In addition, if for

_

and

(1 + � NaIEbl (t)) E (t) + H(E (O), · · · , E (t)) ,

s :::; t , E ( s) :::; CNd

and c >

0, then

H(E(O), · · · , E (t)) :::; o.

hi �

O.

Chapter 3. Stability and Convergence

Therefore, if

99

(

)

p ( t 1 ) e cott(M+ l ) O. Moreover Uo a.n d f possess the period 211' for the variable x, a.nd so does the solution U. Let A = (0, 211') a.nd denote the space C(O, Tj H;(A)) by C(O, Tj H;) for simplicity. Also 11 · 11 . = 1I ' lIw ( Al etc .. It ca.n be verified that if Uo E H�(A) and f E L 2 (0, Tj L�) , then (4. 1 ) has a unique solution U E LOO (O, Tj H�). It is commonly admitted that a reasonable numerical algorithm should preserve the features of the genuine solution of the original problem as much as possible. In fact, the solution of (4. 1 ) possesses some conservations, such as

l U(x, t) dx l Uo(x, t) dx + II f (x s) dxds, =

II U(t) 11 2 + 81 U(t) l�

,

=

Il Uo l l 2 + 81 Uo l� + 2

l ( f (s), U(s)) ds.

(4.2) (4.3)

We are going to construct the Fourier spectral scheme. Let N be any positive integer a.nd VN be the subspace of all real-valued functions of VN by (2.17) . Denote by PN the L 2 -orthogonal projection from L�(A) onto VN . For convenience of analysis, let 1 1 (4.4) J(v(x) , w(x)) = 3 w(x)0",v(x) + 3 0", (w(x)v(x) ) . If v, w, z E

H�(A), then (O",(wv), z) + (wo",z, v) (WO",v, z) + (O., (wz) , v)

0, =

O.

Hence (J(v , w), z) + (J(z, w) , v)

=

O.

(4.5)

In particular, (J(v, w), v) = O.

(4 . 6)

Spectral Methods and Their ,Applications

102

On the other hand,

(J(v, w), w) (J(v, w ), w)

= =

1 1 "32 (w2 , f}:r; v) - "31 (wf}:r;w , v) , "3 (w2 , f}:r;v) + "3 (w f}:r; w , v)

from which (4.7)

¢i, t/J E VN, (4.8) (PNJ(¢i, w), t/J) + (PNJ(t/J, w), ¢i) = 0 , (4.9) (PNJ(¢i, w), ¢i) = 0, (4.10 ) (PNJ(¢i, t/J), t/J) = � (f}:r;¢i, t/J 2) . For discretization in time t, let T be the mesh size and use the notations in Chapter 1, such as Rr (T), R.,.(T) and DT v(x, t), etc . . Let UN be the approximation to U, and be parameters, 0 :5 :5 1 , 1 = 1 , 2, 3. A Fourier spectral scheme for (4. 1 ) is to UN(X, t) E VN for all t E RAT) such

In addition, for

{

that

find

17/

17/

DT UN(X, t) + u1 df}:r; U N( X, t + T) + ( 1 - (1 ) df}:r;UN(X, t ) + U2 PNJ (UN(X, t + T), UN(X, t)) + (1 - (2 )PNJ (UN(X, t), UN(X, t)) - {jDTf}� UN(X, t) = FN(X, t), 1 x E lR\ t E Rr (T), UN(X, 0 ) = UN,O (X), x E lR

(4. 1 1 )

The first formula of (4. 1 1 ) can be rewritten as

DT uN( X, t) + df}:r;U N(X, t) + U1 dTDTf}:r;U N(X, t) +PN J(UN (X, t) + U2 TDTuN (X, t) , UN (X, t)) - {jDTf}�UN(X, t) = FN (X, t) . (4.12) If 171 172 = 173 = 0, then (4. 12) is the explicit spectral scheme in Guo and Manoran jan (1985) . If 171 =f 0 and 172 = 0, it becomes an implicit scheme. But by the =

Chapter 4. Spectral Method� and Pseudospectral Methods

103

orthogonality of Fourier system, we .can still evaluate the, coefficients of UN(X, t) ex plicitly at each time t E R.,. (T). This is t.he advantage of using spectral methods. In (4. 1 1 ) , the nonlinear term is approximated by partially implicit approach and so we only have to solve a linear system for the Fourier coefficients step by step, even if 0"10"2 =f. O. Indeed (4 . 1 1 ) reads UN(X, t) + 0"1dr8",UN(X, t) +0"2 rPNJ (UN(X, t) , UN(X, t - r)) - 08�UN(x, t) "'; GN(x , t - r)

(4.13)

where the term GN(x, t - r) depends only on u N(x, t - r) and rFN(x, t - r ) . We take the inner product of both sides of the above equation with UN(t), i.e. , multiplying UN(X, t) and integrating the resulting equation over A. Then we obtain from (4.9) that If GN(X, t - r) == 0 for x E A, then UN(X, t) == 0 for x E A. Since (4. 13) is a linear system, the above fact ensures the existence and uniqueness of UN(X, t) for all t E R,. (T). We check the conservations of the solution of (4. 1 1 ) . We first take 0"2 = 0, integrate (4 . 12) over A and sum the result for t E fl.,. (T). Then it follows that Next, we let 0"1 = 0"2 = l , take the inner product of (4. 1 1 ) with UN(t + r ) + UN(t) and sum the result for t E R,. (T). We obtain from (4.9) that l I uN(t) 1I 2 + O IUN(t) l �

=

lI uN,oll 2 + O IUN,ol � + r

E

(FN( S ) , UN( S + r) + UN( S ) ) ,

.eR,. (t-.,.)

Evidently the above two equations are reasonable analogies of the conservations (4.2) and (4.3) . We now turn to analyze the errors. Suppose that UN,O and FN have the errors UN,O and FN respectively. Then we get a disturbed solution corresponding to UN,O + UN, O and FN + FN, denoted by U N + UN. For simplicity, we denote the errors UN, UN,O and

Spectral Methods and Their' Applications

104

FN by U, Uo and F . They satisfy the error equation as follows D.,.u(x, t) + do",u(x, t) + U1 dTD.,. O",U(x, t) + PNJ (UN(X, t) + U2 T D.,. U N(X, t), u(x, t)) + PNJ (u(x, t) + U2 TD.,. u(x, t), UN(X, t) + u( x, t)) (4. 14) -oD.,. o;, u(x, t) = F (x, t), x E IR? , t E 1l.,.(T). We take the inner product of (4. 14) with 2u(x, t) and then obtain from ( 1 22 ) and .

(4.9) that

D.,. l I u(t) 1 1 2 +oD.,. l u(t) I � -T I D.,. u (t) 11 2 -oT I D.,. u (t) I � + ;=E3 1 Fj(t) = 2 ( F (t), u(t) ) -�

(4.15)

where

F1 (t) = 2U1 dT (D.,.o",u(t), u(t)) , F2 (t) = 2 (J (UN(t) + U2 TD",UN(t), u(t)) , u(t)) , F3 (t) = 2U2 T (J (D.,. u (t), UN(t) + u(t)) , u(t)) .

� be an undetermined positive constant and take the inner product �T D.,. u (x, t). We have from (4.9) that 6 �T I D.,.u(t)1 I 2 + �oT I D.,.u(t) l � + jE= Fj(t) �T (F (t), D.,. u (t) ) (4. 16) 4

Furthermore, let of (4. 14) with

=

where

F4 (t) = d�T (o",u(t), D.,.u(t)) , F5(t) = �T (J (UN(t) + U2 TD.,. UN(t), U(t)) , D.,. u (t)) , F6(t) �T (J (u(t), UN(t) + u(t)) , D.,.u(t)) . =

Let

e be a positive constant. Putting (4. 15) and (4. 16) together, we obtain that D.,. l I u(t) 1 2 + oD.,. l u (t) l � + T(� - 1 - e) ( 1 I D.,.u(t) 1 2 + o I D.,. u (t) I D (4.17) + E1=1 F;(t) :5 l I u(t) 1 2 + (1 + :rf) II F (t) 1 I 2 . We begin to estimate I Fj(t) l . It is easy to see that 1 F1 (t) + F4 (t) 1 = I dT(� - 2ud (o",u(t), D.,.u(t)) I T � (e : 2(1 ) 2 I u (t) I � . :5 eT I D.,. U (t) 1 2 + (4.18)

Chapter 4. Spectral' Methods and Pseudospectral Methods

Let r >

105

�. By (4. 10) and imbedding theory,

Similarly Further, by Theorem 2 . 1 ,

an d s o b y (4.8) ,

I F3 (t) + F6(t) 1 I T (e - 20'2 ) (J (u(t), UN(t) + u(t)) , Dru(t)) 1 :::; c; T I Dru(t) 11 2 + CT(e � 20'2 ) 2 (1 I uNllc(O,T ;W) l I u(t) lIi + N l u(t) 1 2 I u l i ) .(4.21 ) B y substituting (4. 18) - (4.21 ) into (4. 17), we find that

where

C ( 1 + ( 1 + � (e + (e - 20'2 ) 2 ) ) I UN ll c(O ,T ;W) ) l I u(t) 11 2 + c; ((e - 2 0' }2 + (e + (e - 20'2 ) 2) II UN ll c(O ,T;W) ) l u (t) l � + CTc;N (e 20'2 ) 2 1 I u(t) 11 2 I u (t) l� + ( 1 + ::2) I F (t) 11 2 . 1 Now let e > 1 , 0 < qo < � - 1 and c; 4' (e - 1 qo). Then we deduce that R(t)

t

_

=

-

For describing the average error of numerical solution, we define E(

v, t) Il v(t) 1 2 + 8 I v (t) I� + qOT 2 ER2:=t r (l I Drv(s ) 1 2 + 8 l Drv(s ) I i) . s (- ) =

\

,

Spectral Methods and Tbeir Applications

106 Also for the average error of data, set

p( v , w ,t) = l I v Il 2 + h'lvl� + T t E(u, t) � cp(uo , F, t) + c L:

L:

seR(t-T)

I w(s) 1 I 2 .

By summing (4.22) for E R... ( T) , we obtain r

seR. (t-T)

[(1 +

Il uN llc(O,T; W ») E(u ,

s

) + TN (� - 20'2 ) 2 2 u , s J .

E( )

(1'2 > �, then we can take e = 2(1'2 and so E( u , t) � cp( uo , F, t) + cr E ( 1 + I UN ll c(O,T ;W» ) E(u , s).

In particulaJ, if

BeR. (t-T)

Finally we use Lemma 3.2 with following result.

E (t) = E( u , t) and p (t) = p( uo , F, t), and obtain the

Theorem 4 . 1 . There exist positive constants b1 and b2 depending only on d, h' and

I UN I C(O,T;W) , r > �, such that if for certain tl E R,.(T) , p( uo , F, t 1 ) � :�, then for all t E R... (t 1 ) , E( u , t) � cp( uo , F, t) eb2t • If in addition 0'2 > !, then the above estimate holds for all t E R ( T ) and any p( uo , F, t). T

Theorem 4.1 shows that scheme (4. 1 1 ) is of the generalized stability with the index s 1 0 (N-q ) . If 1, then s O. In this case, provided that the computation is stable when the average error of data does not exceed a critical value cb1 • If then there is no restriction on It means that scheme (4. 1 1 ) possesses the index of generalized stability s We know from the above' analysis that the suitable choice of parameter in the approximation of nonlinear term can improve the stability essentially. We next deal with the convergence of (4. 1 1 ) . Let and 0 for

= - q,

0'2 > !,

T=

q=

=

p( uo , F, t). = - 00 .

UN = PNU

0'3 =

Chapter 4. Spectral" Methods and PseudospectraJ Methods

0

1 7

( ), DTUN(X, t) + doxUN(x, t) + U1 dT DToXUN( X, t) +PNJ (UN(x, t)s + U2 T DTUN(X, t), UN(X, t)) - 8DTo;UN( X, t) = FN(x, t) + L G;(x , t), x E lR? , t E RT( T), ;=1 UN(x, 0 ) = PNUO (X), x E lR \

simplicity. By 4 . 1

where

G1 (x , t) = DTUN(X, t) - OtUN(X, t), G2 (x , t) = U1 dT DToXUN(X , t), G3 (x , t) = PNJ (UN(x , t), UN(x, t)) - PNJ (U(x , t) , U(x , t)) , G4 (x, t) = U2 T PNJ (DTUN( X, t), UN( X, t)) , Gs(x , t) = 8ot o;UN(x, t) - 8DTo;UN(X, t). Further, let UN(X, t) = U N (X, t) - UN(X, t) , also denoted by U(x, t) for simplicity. By (4. 1 1 ) , it follows that

DTU(x, t) + doxU(x, t) + U1 dT DToXU(x , t) + PNJ (UN(x, t) + U2 T DTUN( X, t), U(x, t) ) +PNJ (U(x, t) + U2 T DTU(x , t), UN(X, t) + U(x , t) ) - 8DTo;U(x, t) = - j=Ll G;(x , t). ( 4.23) In addition, U(x, O) = O. By comparing this error equation to (4. 14) , we find that it suffices to estimate the terms Gj (x, t), 1 � j � 5. Since T (4.24) Otv(x , t) - DTV( X, t) = - -Tl 1.t t+ (t + T - s)o; v(x , s) ds, S

the property of Bochner integral leads to

T ER(tL T I G1 (s) 1 2 � CT 2 11 U 1 �2 (O,t ;£2 ) ' s -) Clearly,

T ER(tL T I G2 (s)1 1 2 � CT2 1 1 U 1I �' (O,t;H' ) ' s -)


108

Moreover for any theory,

v E H; (A) with r > �, we deduce from Theorem 2.3 and imbedding

I l v8",v - PNv8",PNvi l Thus

I v8",v - V8",PNVi l + I v8",PN V - PNv8",PNVi l :5 I v il Lool v - PNv i l + 1 8",PNV il Lool i v - PNvi l l :5 cN - r l v ll� ·
, w) . This technique i s called skew symmetric decomposition of nonlinear convective term. We shall see below that it enables the numerical solution to keep the conservations. On the other hand, the nonlinear computation may destroy the stability. Mainly, this trouble is due to the higher frequency terms. To remedy this, we can use the filtering RN( CY., (3) in (2. 107) , denoted by RN for simplicity. The use of filtering for nonlinear terms brings some difficulties in error analysis. To simplify the statements, we introduce the circle convolution as (4.31) ¢> * ¢ = :L :L �p "j; , _ pe i'x , V ¢> , ¢ E VN .

J(

It is easy to see that ¢> * ¢ = ¢ * ¢>.

111� N lp l� N

Lemma 4 . 1 . For any ¢>, ¢ E

VN and X E VN ,

IN( ¢>¢ ) = ¢> * ¢,

( ¢> * x , ¢ ) = ( ¢>, X * ¢ ) ·

Proof. For the first conclusion, i t i s sufficient t o prove that

In fact,

� �1 eipx(J) � .7. i( l -p)x(i) � o/l - p e � p

L �, ipx

Ip l:S N

e

)¢

111� N

Ip l� N

(j

(xU»)

=

¢> (xU») ¢

(x U») ,

and so the desired result follows. Next, by (4.28) and the first conclusion, we deduce that

Lemma 4 . 2 . For any ¢>, ¢

•

E VN, W E H; (A)

and r

> �,

112

Spectral Methods and THeir Applications

{,

Proof. Let WI be the Fourier coefficients ofw. Define 1 - p + 2N + 1 , if / - p < - N, ( 4.32 ) if 1 / - p i � N, '"(I,p = 1 - p, l - p - 2N - 1 , if / - p > N. Clearly '"(- I , -p = -: '"(I ,p . We have (8x 4> * RN"p, w) = (8x¢> * RN.,p, PNW) = 27ri L it, L ( 1 - I � l a r '"(I,p J,-p¢P , .(¢> * RN8x.,p, w) = (¢> * RN8x.,p, PNW) = 27ri IL5N it, pLl 5N (l - I �D,B pJ, -P ¢P · 11 I �

m�

;

��

Putting these two formulas together, we get that

(8x¢> * RN.,p, w) + ( ¢> * RN.,p, w) = 27ri IL5N (1 + I / l ) it, pLl 5N ).. " p�,-p ¢p I 11 where

O" p = (l - I�D (p + '"(I,p ) ( 1 + 1 1 1 ) -1 . Set f(x) = ( 1 - xa ) ,B - a,8(l - x). Clearly f( l ) = 0 and for 0 � x � 1 , 8xf( x) = a ,8 ( 1 - xa- 1 (1 - X a ) ,B- l ) � o .

Let

Therefore for 0 and s o for

� � 1, x

I I I , Ip i � N,

(1 - xa l

� a,8(1 - x),

We now estimate only. If 1 1 - �

I O"p l . Since O_ I, _p = - O" p , it suffices to consider the case 0 � p � N p i N, then I O',p l = I I I 1( 1+-I IN)I � 1 . If 1 - p -N and s o 0 � N - p - I I l l , then (N - p)( 2 N + 1 + 1) 1 / 1 (2N + 1 - I l l ) 1 0 I ,p I = N(1 + I l l ) - N(1 + I l l ) - 2.
�, such that if for certain tl E R,. (T) , p (uo, P, t 1 ) $ �, then for all t E R (td,

If in a ddition 0'2 > 21 ' then the above estimate holds for all t E RT (T) and any p (uo, !, t) . We now turn to the convergence of (4.33). Let UN PNU, U UN - UN and Us = 0 in (4.33) . We obtain the error equation as follows DTU(x, t) + dRN 0", U( x, t) + 0'1 d7 RND.,.o",U(x, t) + RNJN (RNUN(X, t) + 0'2 7RND.,.UN(X, t), U(x, t» ) + RNJN (RNU(X, t) + 0'2 7RND.,.U(x, t), UN(X, t) 6 (4.38) +U(x, t » ) - 6D.,. 0� U(x, t) - j=1 L Gj (x, t), where G1 (t) and Gs(t ) are the same as in (4.23), and G2 (t) 0'1d7 RN D.,. 0", UN (x , t ), Gs = RNJN (RNUN(X, t), UN(x, t)) - PNJ (U(x, t), U(x, t)) , G4 (t) 0'2 7 RNJN (RNDTUN(X, t), UN(x, t)) , G6 PNf(x, t) - RNINf(x, t). =

=

=

=

=

=

Chapter 4. Spectral Methods and Pseudospectral Methods

117

B y Theorem 2.4, Theorem 2.26, Lemma 4.4 and imbedding theory, we obtain that for a 2: r > � ,

T ERrL T) I G2 (s) 1 2 :s CT2 1 U 1 ;'1 (0,t ;Hl) , s (tT L I G3 (s) 1 2 :s cN2-2r l U l t 0, T ; W ) ' ( sERr (t-T) T L T) I G4 (s) 1 2 :s CT 2 1 U 1 �(0,T;W) I U I ;'1 (0,t ;Hl ) , sERr (tT SERrL T) I Gs(s) 1 I 2 :S cN2 - 2r llfll�(0,t ;W_l) . (t-

Finally by an argument as in the proof of Theorem 4.2, we obtain the following result.

Theorem 4.4. Let (j3 0 in {4.33}. If U E C(O, T; H;) n H2(0, T; HI ) , Uo H; ( A) , f E C (O, Tj H;-I ) , a 2: r > � and T :S bs N -!, th e n for all t E flT( T ) , =

where

E

are positive constants depending on d, 11 U 1 I C(o,T;H; ), I U I H2 (O,T;Hl), I l Uo ll r and Il f llc(o,T; W -' )' If (72 ) ! , then the conclusion is valid for any O. b6 - bs

8, a ,

(3,

T>

0

' Scheme (4.33) with (71 = (j2 = (j3 = was investigated in Guo and Cao (1988) . We have seen in the derivation of (4.37) and (4.38) that since the nonlinear term the effects is approximated by a skew symmetric operator of the leading nonlinear error terms in both (4.37) and (4.38) are cancelled, i.e. ,

UoxU

RNJN (RNUN, UN),

(RNJN (RNU(t), u(t)) , u(t)) = 0, (RNJN (RNU(t), U(t)) , U(t)) = o. Otherwise, we shall require that p( uo . io , td :S bs /N in Theorem 4.3.

It indicates that the suitable approximation to the nonlinear term, not only enables the scheme to keep the conservations, but also strengthens the stability and raises the accuracy.


118

' It also shows that there is a close relationship between the conservations and the stability. This idea was first proposed for finite difference methods by Guo (1965) . Finally, we present som� numerical results. The meaning of E(v, t) is the sanie as before. We take the test function (4.26) with A = B and w = 0.5. In calculations, 0'1 = 0'2 = 0'3 = 0, fJ = 1 , N = 8 and T = 0.01. We check the effect of the filtering RN (a, 1 ) , and fi� d the following phenomena. (i) If the vibrations of genuine solution of (4. 1 ) is small (e.g., A = 0 . 1 ) and d = 0, then the effect of RN(a, 1 ) is not clear, see Table 4.2. ,

(ii) If 8 is large (e.g. , 8 = 1), then the filtering is not important even though the vibration of genuine solution is large, see Table 4.3. (iii) If the vibration of genuine solution is large and is small (e.g., A = 1 , 8 10-4 , 10-6 ) , then the effect of filtering is clear, see Tables 4.4 and 4.5.

8

=

(iv) The smaller the parameter 8, the more important the linear term do",U. In this case, the filtering plays an important role. Tables 4.2 and 4.5 tell us that its effect is not clear for d = 0, A = 0 . 1 , but very clear for d = 2, A = 0 . 1 . (v) The value of a i n the filtering must b e chosen suitably. For every large a, the filtering is weak. But for too small a, the accuracy is lowered. So far, there is no criterion for the best choice of a. Numerical experiments show that a better choice is 3 :::; a :::; 10. But the best ones depend on N. The best one in Table 4.5 is a = 3. Table 4.2. The errors a=5 8 = 1.0 8 = 10- 2 = 10-4 = 10-6

8 8

2.36857E-3 2.29817E-3 2.28399E-3 2.28242E-3

E ( UN, 5.0), d = 0, A = 0.1. a = 10

a = 50

2.36493E-3 2.29622E- 3 2.28281E-3 2.28203E-3

2.36577E-3 2.29651E-3 2.28183E-3 2.28193E-3


119

Table 4.3. The errors B ( U N , 5 0) , 6 = 1 , A = 1 .0. a=5 a = 50 a = 10 d = O 2.31011E-3 2.15183E-3 2 . 16537E-3 d = 2 1 .441 1 8E-3 1 . 18726E-3 1 . 15333E-3

Table 4.4. The errors E(UN , 1 .0 ) , d = 0 , A = 1 . 0 . a=5 0 a = 10 a = 50 6 = 10-4 3.29499E-3 3.55039E-3 2.35286E- 1 6 = 10-6 2.83234E-2 1 . 24055E-2 1 . 70894E- 1

Table 4.5. The errors B (U N , 2.0) , d = 2, A = 0 . 1 . 6 = 10-4 6

=

10-6

a=3 1 . 55333E-3 1 . 56715E-3

a = 50 a = 10 a=5 1 .63698E-3 2.43057E- 2 7.02116E-1 > 10 1 . 86465E-2 1 . 1 1043E-2

Scheme (4. 1 1 ) and scheme (4.33) are of first order in time. In order to match the spectral accuracy in the space, we should use more accurate temporal discretizations as discussed in Gazdag (1976) . The Fourier spectral methods and the Fourier pseudospectral methods have been widely used for numerical solutions of various partial differential equations. We refer to some early work. For instance, Shamel and Elsasser (1976) , Fornberg and Whitham (1978) , and Abe and Inou ( 1 980) used them for Korteweg-de Vries equation (KdV equation ) , while Guo (1981a) proposed the spectral schemes for KdV-Burgers equa tion with numerical analysis. Pasciak (1982) considered Fourier spectral scheme for generalized BBM equation. Kriess and Oliger (1979) applied them to hyperbolic equations with filterings. Cornille (1982) used Fourier approximation in numerical study of shock waves. Canosa and Gazdag ( 1977) used it for KdV-Burgers equation and found shock-like waves.


120

4.2 .

Legendre Spectral Methods and Legendre Pseudospe O. Since (Drr U(t), t - ! (u� (t), a",(1>w)) + IL ( a", U N(t), a,,, ( 1)w)) (J (t), 1>t , V 1> E lP�, t E Rr(T), UN( r) UN,! , UN(O) UN,O =

=

( 4.65)

=

The scheme (4.65) stands for

which is easy to be implemented in practice. (4.65) is an implicit scheme. At each time E we need to solve a linear system

t Rr (T), F(t)

uN(t - r), uN(t - 2r) UN(t)

i(x, t).

and where depends on By Lemma 2 . 1 , it possesses a can be calculated explicitly as explained above. unique solution. Furthermore, In actual calculations, we can take the basis functions as

( ) { T/(x) T/(x) - To(x), - Tl (X),

1>/ x

=

l is even, I is odd.

But these functions lead to a linear system with a full matrix. A better choice ( see Shen ( 1995)) is to take


Then

{

and

145

l = m, I = m - 2 or m + 2,

otherwise,

21r ( m + 1 )( m + 2 ) , l = m , (o:z: '1h , O:z:( 1/Jmw)) = 41r ( m + 1 ) , I = m + 2, m + 4, . . . , 0, 1 < m or 1 + m odd. We now consider the gener alized stability of (4.65 ) . Let uo , U l and J be the errors of UN,O , UN,l and PN j , which induce the error of UN, say U. From (4.65 ) , (b'T u(t), L - ( RN INi(t), t/» N,w ·

r ::::: 1. We have I G1(t) 1 $ I (b., U(t) - atU(t), t/» w ! + l (b., U(t ) , t/>L - (b., U(t), cf» N.w ! + I (b.,U(t) - b., UN (t), cf» N,w ! ·

According to Theorem

2.18 and ( 2.87),

with Moreover, '(

We have

l: A( s ) $ c'(4 I J U I �3 (O ,T ; L� ) + cN -2r I J U II�1 (O ,T ; H.!; ) · ·.ERr(t-.,)

15

�d

0

Spectral Meth s and Their Applications

By Theorems 2.19 and 2.27,

where

d4 (v) is a positive constant depending only on I v l c(O,T;H�) ' We also have

Besides,

iI U(O) i I w :::; cN-r l Uo l r,w , I U(r) 1 ! w :::; cN-r ( I U(O) l I r,w + I U(r) l I r,w + r l otU(O) l I r,w ) + cr 2 11 U Il c2 (O,T ;L� ) '

Finally, we apply Theorem 4.12 to ( 4.74 ) .

E

n

n

E

Let :::; bs N-i . If U Cl (O, T; HJ,w H�) H3 ( 0, T; L�), f C (O, T; H� ) and 1 :::; r :::; then E (UN - UN, t) :::; bg eb, o t (r4 + N-2r ) , bs , bg and bl O being positive constants depending on and the norms of U and f in th� spaces mentioned above.

Theorem 4 . 13.

r

a,

p.

In order to save work, we sometimes use explicit Chebyshev pseudospectral schemes = for ( 4.63 ) . For instance, let a, (3) , a, f3 ;::: 1 be the filter defined by ( 2. 137 ) , and consider the following Chebyshev pseudospectral scheme

R� R�(

DTuN(X, t ) + � INRNO:r: (RNINu'fv(x, t ) ) - p.INRN{J�R�uN(x, t) = INRNINf(x, t ) . ( 4.75 ) 4 0 then we can derive the same result of stability and convergence r (N- ), as in Theorems 4.12 and 4.13. The condition on r is much more severe than the condition for the stability in the corresponding Fourier pseudospectral schemes, in which r 0 (N- 2) . However, the usefulness of the filters RN and R� can improve If

=

the results in practical computations. We check the effects of the filters in ( 4.75 ) by numerical experiments. For description of the accuracy, let =

Chapter

4.

Spectral Methods and Pseudospectral Methods

151

We take f = 0 and so (4.63) has a solution

2x - t ) . U (x,t) = -21 (1 - tanh -81-'

Set I-' = 0.005, T = 0.0005 and N = 64. To test the effects of the filters, various values of a and /3 are carried out. For a = 00 or /3 = 0, (4.75) becomes the usual Chebyshev pseudospectral scheme. In this case, B2 ( UN , 0.25) = 1 . 302 and Boo ( UN , 0.25) = 3.907. The numerical errors with other choices of a and /3 are presented in Tables 4.10 and 4. 1 1 . Evidently, the filters improve the accuracy. For N =64, a better choice appears to be a = 6 '" 8, /3 = 2 '" 3. Table 4.10. The errors E2 a = 10 a = 8 /3 = 1 0.883 0.787 /3 = 2 0.393 0.016 /3 = 3 0.016 0.018

0.25 ) . a=6 0.023 0.018 0.021

UN ,

Table 4. 1 1 . The errors Boo ( UN , 0.25) . a = 10 a = 8 a = 6 /3 = 1 3.409 3.283 0. 102 /3 = 2 2.596 0.065 0.073 /3 = 3 0.065 0.071 0.082 The Chebyshev spectral methods and Chebyshev pseudospectral methods for gen eralized Burgers ' equation are given in Ma and Guo (1988). The filtered schemes can be found in Ma and Guo (1994) . The Chebyshev spectral methods and Chebyshev pseudospectral methods have been used successfully for various partial differential equations. For instance, Maday and QuarterOIii (1981) used it for steady Burgers' equation. Gottlieb (1981) first applied the properties of Gauss-type quadrature to studying the stability of Cheby shev pseudospectral methods for linear hyperbolic and parabolic equations. Gottlieb, Orszag and Turkel (1981) considered hyperbolic eq� ations with variable coefficients.

Spectral }.(etho:ds and Their. Applications

152

4.4.

Spectral Penalty Methods

We can deal with inhomogeneou s boundary conditions in several ways. One of them is the spectral penalty method in which the boundary conditions become a part of equation, see Funaro and Gottlieb (1988,1991) . In the last section, we introduced the Chebyshev pseudospectral method associated with the Chebyshev-Gauss-Lobatto interpolation points. Those points are chosen, because they allow the use of FFT in computations. For inhomogeneous boundary conditions, we can use Chebyshev penalty method as described below. Consequently, we adopt the weighted norm in numerical analysis. However, this is not a natural norm and complicates the analysis. A more natural way is to use Legendre-Gauss-Lobatto interpolation and the corresponding penalty method. As those points are not given explicitly, their evaluation for large is not robust due to roundoff errors. In this section, we focus on a new approach, implementing Legendre method on Chebyshev points, enjoying the advantages of both Legendre and Chebyshev methods. It is named the Chebyshev Legendre method by Don and Gottlieb (1994) . We begin with a simple problem. Let A = ( - 1 , 1 ) and consider the hyperbolic equation = E A, 0, (4.76) 0, = = E

N

{

8tU(x, t) - 8",U(x, t) f(x, t) , x t > t> U(l, t) g(t), Uo(x), x A. U(x, O)

x(j) and w(j) be the Chebyshev-Gauss-Lobatto interpolation points and weights. AN = { x U) I ° :5 j :5 N}. The usual semi-discrete Chebyshev pseudospectral scheme is to find UN(X, t) E IPN for t 2: ° such that 8tUN( X, t) - 8:cUN(X, t) = f(x, t), x E AN/{l}, t > 0, (4.77) uN(l, t) = g(t), t > 0, = UN(X, O) Uo(x), x E AN . Let

{

Funaro and Gottlieb (1988 ) proposed a penalty method in which, instead of the boundary condition in (4.77) , we require that

>.

where is determined from stability considerations. This approach is stable for the initial data as long as 2: Since are the zeros of the polynomial

>. � N2•

xU)


(1

{

- X 2) OxTN( x), this scheme can be written as >'(1 X)8 (x) 8tU � (x, t) - 8",UN(X, t) � "'r) (uN(I, t) - get)) 2 ",TN 1 +

153

I(x, t), x E AN , t > 0 ,

x E AN· (4.78) The main feature of (4.78) is that the numerical solution does not fulfill the boundary condition exactly, but only in the limit as N --. 00 . In fact, the boundary condition is a part of the equation. Another penalty method based on Legendre-Gauss-Lobatto interpolation was served by Funaro and Gottlieb (199 1 ) . Let be the and N nodes and weights of this interpolation. = y (i) = x ( -i) I 0 :::; j :::; N } . Since y (i) . are the zeros of the polynomial ( 1 the penalty scheme for (4.76) is to find t) E for t � 0 such that UN(X, 0)

=

Uo (x),

UN(X ,

{

lPN

8tUN(X, t) - 8",UN(X, t) +

. UN(X, O)

=

=

AN { x2) OxLN(X ),

>'(1 + x)8xLN(X) . (UN(1, t) - get)) 28", LN (I)

w (j )

x(i)

=

I(x, t), x E AN , t

-

Uo (x),

This scheme is also stable for the initial data when A � ! N ( N + 1 ) . The Chebyshev-Legendre penalty scheme for (4. 76) i s t o find t � 0 such that

{

� 0,

x E AN· (4.79)

UN(X, t ) E lP N for

8tUN(X, t) - 8xUN(X, t) +

>'(1 + x)8",LN(X) . (UN(I, t) - get)) 28xLN(I)

=

I(x, t), x E AN , t

-

� 0,

x E AN . (4.80) We now explore the relation between (4.79) and (4.80). The insight can be gained if one compares the differentiation matrix induced by (4.79) and the differentiation matrix induced by (4.80). As in the proof of Theorem 2.6, put UN(X, O)

=

Uo (x),

qc( x) = (1 - X 2) O",TN(X), q£C x) = ( 1 - x 2 ) O",LN(X) . Define the Lagrange polynomials

154

Spectral M�thods and Their' Applications

v C ( A ), the Chebyshev interpolant IN v and the Legendre interpolant iNV are N ( I» (4.81) INv(x) = L ',,;N0 > (X ) 91 (X), (l) (4.8 2 ) iNV( X ) = L 1=0 > (y ) h,(x). Denote by Bc the differentiation matrix in Chebyshev pseudospectral approximation, and by DL the differentiation matrix in Legendre pseudospectral approximation. It means that N (4.83) 8.,!NV (x (l) ) = m=L:O DC,l,m V (x (m» ) , 0 � 1 � N, N (4.84) 8.,iNv (y (l) ) = m=L:O DL,l,m V (y (m») , 0 � 1 � N. For E as follows

By virtue of (4.81) and (4.82),

DC,l,m = 8.,9m (X (I) ) , DL,l,m = 8., hm (y (l» ) , 0 ::; 1, m ::; N. Let A and B be the ( N + 1 ) x (N + 1 ) matrices with the elements A" m = hm (X (I») , B',m = 9m (y (l) ) , 0 ::; 1, m ::; N. Furthermore, denote by F the matrix induced by the differentiation and the penalty in (4.80), and denote by G the matrix induced by the differentiation and the penalty in (4. 79). We have the following results.

We have AB = I and Dc = ADLB. Proof. Since 9m(X) E IP N, it can be expressed as N 9m(X ) = j=O L:9m (y (j») hj (x) . Because 9m (X ( I ») = Ol ,m , N N L: A" j Bj,m O',m = j=O L:9m (y (j» ) hj (x (I» ) = j=O Lemma 4 . 1 5 .

(4.85)


whence

AB

=

155

I. Next, differentiating (4.85) yields

N 8.,9m(x) j=O I: 9m (y (j» ) 8., hj (x). =

Since

E

8., hj (x) lPN, it can be expressed as N 8., hj (x) k=I:O 8., hj (y (k» ) hk( x). =

Therefore

N N (j » I: I: 9m (y ) 8., hj (y (k» ) h k (x (l») j=ON k=O N I: I: A/ , k DL, k ,j Bj,m (ADLB) / ,m . j=Ok=O =

•

We have F AGB. Proof. The differentiation matrix G is essentially the matrix DL introduced in (4.84) ,

Theorem 4 . 1 4 .

=

modified to take into account the boundary conditions, imposed via penalty in (4.79). Thus

G/ ,m DL,/ ,m - J.. h/ ,o hm,o . =

(l» ) 8.,LN (x (I») F/ ,m Dc,/,m - J.. ( 1 + 2x8.,LN( hm,o . I)

In the same manner, we obtain from (4.80) that =

(4.86)

By Lemma 4.15, we deduce that

NN (ADL B ) /,m - J.. j=O I: kI:=O hj,ohk,o hj (x (l) ) 9m (y (k» ) Dc,/,m - J.. ho (x (I» ) 9m (y (O» ) . According to the definition of h o(x), a computation produces that (AGB}! ,m

=

=

(4.87)

Spectral Metljods and Their Appliqations

156

yeO) = x (O) = 1 , gm (y (O) ) 8m 0 . Hence " ( 4. 8 7) is equal to the right-hand side of (4.86 ). Then the desired result follows. We now turn to analyze the stability of scheme (4. 80 ). By the theory in Chapter 3, it suffices to verify that the norm of UN depends on the norms of data continuously. Let N 2 > (y (j ) ) W (y Ci) ) w (i) , (V , W)N = j=O and I v l i N = (v, v)Ar · Also set N N ( I) ) W (x ( m) ) x H" mV ( L I v liN = L I=O m=O where N H" m = j=O L g, (y Ci) ) gm (y Ci) ) w Ci) . For any if> E IP N, we have N if>(x) = INif>(x) = L=0 if> (x (I) ) g,(x) =

Since

,

•

1.

1

and so

(4. 88 ) 1 I if> I N = 11 if> I N . Theorem 4.15. If H1 + ) (N2 + N) ::; A ::; (N2 + N) nd > O J th en Il uN(t) ll � ::; en ( I UN(O) I � + � l ( I INf(s) l I � + 4c�l (s) ) dS) . Proof. Since UN(X, t) E IPN for t ;::: 0, (4.80) implies that for all x E A and t > 0, + X )O",LN( X ) OtUN(X, t) - O",UN(X, t ) + A(l 2o",LN(1) (uN(l , t) - get)) = INf(x, t). ( 4.89) By setting x y Ci ) in (4. 89 ) , multiplying it by 2UN (y Ci) , t ) w (i ) and summing the Consequently,

e

result for 0

::;

=

j

::; N,

we obtain that

Cl

a

e

Ot l uN(t) ll � - 2 (UN(t) , O",UN(t)) N + 2AW (0) U�(1, t) 2Aw (0) uN(1 , t)g(t) + 2 (INf(t), UN(t))N ' =


By

(2 . 37),

157

2 L UN { X, t)O",UN(X, t) dx u�(l, t) - u�( - 1, t)

and then

otlluN {t) l I � + (2 "\w (O) - 1 - e) u�(l , t) + u�( - 1, t) (.� (O» 2 :::; e I UN(t) l I � + -Ie1 I INf {t) l I � + we ) l(t). Sinc w (O) = : � N { + 1) ' we integrate the above inequality to obtain from which the conclusion follows. Theorem 4.15 shows the linear stability of scheme (4.80) . In particular, if 0, then

f(x, t)

•

=

g {t) = I l uN(t) I � + l (( N (�..\+ l) - l ) U�(l ' S ) + U�(-l, s» ) ds = l I uN(O) I � · Next, we consider the convergence of (4.80) . Let UN = iNU. We get from (4.76)

that

(4.90)

where

G1 (x, t) = iNo",U(x, t) - o",iNU(x, t), G2 (x, t) = iNf(x, t) - INf(x, t), G3 { x,t) = iNUO (x) - INUo (x).

Put (;

= UN - UN. Then we have from (4.80) and (4.90) that for all x E A and t > 0, + X)O",LN{X) U(l, t) -G1 (x, t) - G (x, t). (4. 91 ) OtU(x, t) - o",U(x, t) + ..\ ( 1 2o",LN(1) 2 �

�

,

�

=

Spectral Methods and Thrur Applications

158

1\

.

'

Furthermore, by using (2.63), ,

Similarly,

IIG2 (t) l I � � cI I G2 (t) 1 1 2 � clliNf(t) - f(t )1 I 2 + cIl INf(t) - f(t) I �· By Theorem 2.19 and (2.63),

Also,

U(x, O) = - G3(x), and

Finally we get the following result by comparing (4.91) with (4.89).

Let ). + 1 ) and r > � . ffU E L2 ( O, Tj W+1 ) , Uo E H� ( A ) and f L2 ( 0, then for all t � T,

Theorem 4.16. � �N(N E Tj H� ) ,

where bl is a positive constant depending only on the norms of U, Uo and f in the mentioned spaces.

{

Now we apply the Chebyshev-Legendre penalty method to parabolic equations. Consider the problem

x E A , t > 0, 8tU( x, t ) - 8; U(x, t) = f(x, t), 0:8",U(1, t) + (3U(I, t) = g+ (t), t > 0, (4.92) -'"'(8",U(-I,t) + 8U(- I , t ) = g-(t ), t > 0, XEA U(x, O) = Uo(x), where 0:, (3, 8, '"'( � 0, 0: + (3 > ° and '"'( + 8 > 0. For the Chebyshev pseudospectral scheme, Gottlieb (1 981 ) proved the stability with 0: = '"'( = 0, and Gottlieb, Hussiani


159

and Orszag proved the stability with = = 0. In order to describe the Chebyshev-Legendre penalty method, we introduce the operator . as

{3 8

( 1984)

A

Av(x,t) = - o�v(x, t) + R(v, x, t) where

R(v, x, t)

Ao QO ( X ) (ao",v (1, t) + {3v(1, t) - g + (t)) + AN QN(X) (- ,o",v( - 1, t) + 8v( - 1, t) - g- (t))

with The Chebyshev-Legendre penalty scheme for such that

{ UN(X, OtUN(X, t) - O�UN(X, t) + R (UN, X, O) Uo(x),

E

(4.92) is to seek UN( x, t) lP N for t � °

t) =

f(x, t), x E �N ' t > 0, (4.93) x AN. lx:;) (O) and We first consider the stability. For simplifying the statements, let K = a c( a, b) = a�(O) (1 + 2 1C + 2JK2 + K) , d( a, b) = a�(O) (1 + 2K - 2 JK2 + K) . Lemma 4 . 1 6 . Let d(a, {3 ) :5 A o :5 c( a, {3 ) and d(,, 8 ) ::; AN :5 c h, 8). Then for all 0, the norm I I U( t) - uN ( t ) I I L1 (A) is bounded by Ve(N). Now, let A = ( - 1 , 1 ) , Q = A x lRt+ and consider the initial-boundary value problem of (4.96) with appropriate data g(t) E Hl� AlRt+ ) , prescribed at x = ± l . Let PN and IN be the L 2 -orthogonal projection upon IPN and the interpolation associated with Legendre-Gauss-Lobatto interpolation points x U) and weights Let v E L 2 ( A ) and v/ be the coefficients of its Legendre expansion. A viscosity operator q is defined as N

wei).

q v (x) L q/v/L/(x) /=0 =

with M = M( N ) and

{

q/ 0, M4 q'/ -> I - [4 ' =

for [ :5 M,

N. The free pair (e , M) will be chosen later, such for M < [ :5

Let e = e(N) be the small parameter. that e --+ 0, M --+ 00 as N --+ 00 . On the other hand, let

B (UN (t))

=

(A( t ) ( l - x )

+ j.t(t) ( l + x)) OxLN ( X ) .

Spectral Methods and Their Appiic.ations

174

The pair (J., ,, ) is chosen to match the inflow boundary data prescribed at = 1 whenever f ' (UN ( 1 , t)) < 0, and 'll.t = - 1 whenever f ' (UN( - 1 , t)) > 0. A Legendre vanishing viscosity penalty scheme for the initial-boundary value problem of (4.96) was given by Maday, Kaber and Tadmor (1993). It is to find UN(X, t) E IPN for t � ° such that ( Ot U N(t) + o",!N f (UN(t)) , ¢» N + c (q O", U N(t) , O"'¢» N = E IP N, t > 0, (UN(t)) , ¢» N ' (4. 1 12) U N (X, t) = at the inflow boundary points, t > 0, UN(O, t) = INUo(x), xE

{

x

x

(B

V ¢>

g(t),

A.

We shall establish some a priori estimates for U N. First of all, let that for any E IP N , N

¢>

u be a filter such

u ¢>(x) L, 1=0 ul �,LI (x). =

Also let

I v ll � = (uv , v).

Lemma 4.20. For

any

¢>

E

IPN,

N

2 l I o", ¢>I I ! ::; cN2 L, 1=1 IUdl¢>1 I •

.

Pro of Let

J', N

=

{j I I

+ 1 ::; j ::; N, 1 + j odd}.

Then by (2.51), N -I '(1 ) '(1) = (2 o", ¢> (x) = L, 1=0 ¢>, L,(x) , ¢>, 1 + 1) L, ¢>j . •

By (2.48),

2 � ul l �P) 1 2 I 1 LdI 2 � 171 (2 1 + 1) 2 Ij � �j 1 1 Ld 1 2 =

(uo:c¢> , o:c¢»

< < It is noted that when

N

1 ) 17 + 1 ) . 2: l�j I 2 I 1 LjI 1 2 . , 2: (2 1 1 2: I L 'I 1 2 1=0 (N N N 2 I LjI 1 2 L, IUI (N2 - 12) ::; cN2 L, ludl ¢>1I 2 . CL, �j I I 1=0 1=0 j=O

2

N -I

171

J EJ1 , N

3 E JI , N

3

• ==

1 , Lemma 4.20 is reduced to Theorem 2.8.

Chapter 4. Spectral Methods and Pseudospectral Methods Lemma 4 . 2. 1 . For

any

175

¢> E IPN }

= 1 - ii,. Then 0-, = 1 for I � M and 0-, � 'f.' for M < 1 � N. Since 2 = ., 8., 118 ¢>1 1 11 ¢>1 I � + 118., ¢>1 I � , it remains to bound 1 1 8.,¢>1 I � . To this end, let J = log2 � and decompose the function ¢> as M 2i M L ¢>(x) = L + = (x) ,(x) , L ¢>j(x), ¢>j L �,L,(x). � '=0 j=l '>2i-' M By Lemma 4.20, Proof. Let 0-,

J

Therefore

118., ¢>1 I �

� �

2 /1 8., � �'L{ + 2J � 118,, ¢>jll � 2 2CM4 1 � �'L' /l + 4cJM4 � II¢>jI l 2 � cM4 In N I ¢>1 I 2 .

•

To simplify the presentation, we shall deal with the prototype case where one boundary, say = is an inflow boundary, while is an outflow one. Then

x=1 B (UN(t)) = A(t)( I - x)8",LN(x). As we know, x(j ) are the zeros of 8 LN ( x ) 1 � j � N 1 and w ( O) = ;: N( + 1 ) ' N 8",LN( -1 ) � (_I) +1 N(N + 1 ) . Thus (B (UN(t)) , v)N 2( _ 1 ) N+ 1 A (t)V( -1, t). (4. 1 13) Let ¢> 1 in (4. 1 12). Since INf (uN(x,t)) E IPN, we deduce from (2. 37) that x -1,

"

,

-

=

=

==

Consequently,

(4.1 14)

Spectral Methods ' �d Their Applications

176 Further, set and assume that for all N,

' t/J(t) = l >.(s) ds, (4. 1 15)

Then for

t S; T, 1 t/J (t) 1 '(t)UN( - 1 , t) (8",!(UN(t)), UN(t)) - (8",IN!(UN(t)), UN(t)) N = - «I - IN)!(UN(t)), 8",UN(t)) S; ;1I 8",!(uN(t)) l i 1 8",uN(t)1 I S; � (1 I 8",UN(t) ll� + M4 InN l uN(t) I �) . Thus for any t S; T, Set if> =

==

By (4. 116),

il >.(s)g(s) ds i

i g (t)t/J(t) - g(O)t/J(O) - l 8.g(s)t/J(s) dsi < II g (t) II (1 I uN(t) 1 I + Co + CA) + l I 88g(s) 1 ( l I UN( S)1 I

+

Co + CA)

ds .

Chapter

4.

Spectral Methods and Pseudospectral Methods

177

Putting the above two estimates together, we have from (4. 1 1 7) that By using Lemma 4.21 and (4. 1 1 7) again,

(4.118 ) Next, we set rP OtUN in ( 4 . 1 1 2 ). By Maday (1991), I IN VI I � ci v i l for any v E Hl ( A ). So by (2.37) , (2.60) and (4. 114 ), I l otUN(t)1 1 2 + � Ot I l OXUN(t) l � � cA l oxUN(t) 1 2 + � I OtUN(t)1 1 2 + CA I Otg(t) 1 2 + CA . Temporal integration of the above inequality followed by ( 4.118 ) , implies I OtUN lllfoc (Q) � CA (Co + Il oxuN lllfoc (Q) + Il g l I �l�c (lRt) + 1 ) � c; (Co + Il g ll �l�c (lRt) + 1 ) . (4 . 1 1 9) We now prove the existence of weak solutions. Let D = A (0, T) and define (" ' )D and I · l i D as before. Let (V, W)DN, = Jo( T (V(t), W(t))N dt. For any W E HJ (D) , define WN(X, t) = 1: PN-lOyW(y, t) dy . Clearly WN E lP N and wN(-1,t) = O. Let 1 - q and rP = WN in ( 4 . 11 2 ) . We have from (2. 37) and (4.113) that s COtUN, Ox!(UN), W )D j=2:l Gj (w), V W E HJ C D ), =

x

u =

=

where

G1CW) G2 (w) G3 (w) G4 (w) Gs(w)

= = = = =

COtUN + Ox!(UN), W - WN)D, (Ox!(UN) - oxIN!CuN), wN)D, COtUN + oxIN!CuN), WN)D - COtUN + oxIN!CuN), wN)D,N , E( UOxUN, OxWN)D , - E(OxUN, OxWN)D.

Spectral Method; and Their Applications

1 78

By (4. 118) and (4. 1 19),

CA ll w - wN Ii D :s: CA "a"WID . I G1 (W) 1 :s: yg ygN

According to (2.63) and (4. 1 18),

CA ll a"WN Ii D . IG2 (w) 1 = 1 (f(UN) - IN!(uN), a"wN)DI :S: NC "a"!(UN)IDla,,WNI i D :s: ygN

By virtue of (2.64) and (4. 1 19),

We see from the proof of Lemma 4.21 that (4.120)

Clearly, The above statements with (4.1 17) imply that

at UN a,,!(UN)

It means that belongs to a compact subset of HI�� (Q). Furthermore + let be any strictly convex entropy and be the corresponding entropy flux. We have 5 (4. 121) = 2: +

E

F

(atE(UN) a"F(UN)' W)D j=l Gj (E' (UN)W) .

The previous statements tell us that

It Gj (E'(UN)W)D I

CA ( N� + M2 v'ln N + Vc") ( lI a,"UN I D ll w I LOO (D) + CA lla" w II D ) :s: CA ( I W II L OO ( D) + ( � + c:M2 v'ln N + Vc") lI a" W I D ) . N Thus the term at E(UN) + a"F(UN) can be written as a sum of two term� which :s:

c:

belong to a compact subset of HI�� (Q) and a bounded set of L�oc(Q). Then by Lemma 4.18, it belongs to a compact set of Hi;'� (Q) . Further, Lemma 4. 1 7 ensures


$

1 79

that for 1 Cf < 00 , UN converges to U·; a weak solution, in £foc ( Q) strongly. Finally, we check the entropy condition. Let a < 1 . It is easy to see that 3

L I G; (E'(UN)w) 1

O. J(x) is a. given function. Consider the problem

{

U8", U - 1-'8;, U = J, U = 0,

x E A, x = -I, 1.

( 4 . 122)

· Spectral. Method� and Their Applications

182

Set

a(v, w) = L o",v(x)o",w(x) dx, b(v, w, z) = ! L v(x)w(x)o",z dx,

v

v, w E H1 (A), V v, w E L4(A) , z E H1 (A). The solution of (4.122) is defined as a function U E HHA), satisfying ( 4.123) p,a(U, v) - b(U, U, v) = (f, v), V v E HJ (A). If (f, v) is a linear continuous functional in HJ (A), then by Fixed Point Theorem, . (4. 123) has at least one solution. Letting v = U in (4. 123) and f, v) l , sup 1( IlI f lll = 'EH�(A) Iv l1

we get that Wit :::; p,- l ll 1f lll . Assume that (4.123) has the solutions U and U· , and f) = U· - U . Then p,a (f), v) - b (f), u + f), v) - b ( U, f) , v) = 0, V v E HJ (A) . Putting v = f), we deduce that .,0.

We know that

II v I l L< :::; v'2 l v l 1 for any v E HHA) , and thus

Hence (4.123) has only one solution as long as 1 11 / 11 1 < If (4.123) has a unique solution, then it is easy to solve it by various spectral approximations as presented in Sections 2.4 and 2.5. Here we are more interested in some problems with multi-solutions. In this case, it is better to consider (4. 123) in an abstract framework. To do this, we need some preparations. For convenience, we shall adopt a unified notation as in Section 2.3, i.e. , == 1 for the Legendre · 1 approximation or (1 for the Chebyshev approximation.

p,2 .

w(x) = - x2) - '

Lemma 4.22. The imbedding

HJ,w CA)

w(x)

C

L�(A) is compact.

w(x) 1, the conclusion comes from the well-known imbedding theory. Now, let w(x) = (1 - x2 ) - ! . It suffices to verify that v E L�(A) vw! E L2(A) is Proof. For

==

-+


183

an isomorppism, and that E HJ,,,, (A) --+ W E HJ (A) , is a continuous mapping. The first fact is trivial. We now prove the second one. Let E HJ,,,, (A). We have • 1 1 1 = +

V�

v

v

a" (VW' ) a"vw' 2 xvw" Moreover by Lerp.ma 2.7, vw� E L2 (A).

a:rvw�

a" (vw� )

Clearly E L2 (A). E Hence l H (A). Next, the set V(A) is dense in H� (A) and so for any E HJ,,,, ( A), there exists a sequence E V(A), I = 1 , 2, . . , such that --+ in H�(A). Then is the limit in Hl (A) of which vanishes at x = - 1 , 1 . Therefore E HJ (A). It implies _ the continuity of the mapping E HJ,,,, ( A) --+ E HJ (A).

ZI

ZI v vw�

.

ZIW�

J.!

v

v

vw�

VW�

C

( 1 - x2 ) - ' . Firstly, let r = 1 , J.! = 0 , be a bounded sequence of H� (A). For each VI , there is E HJ,,,, ( A) and such that = + and

Lemma 4.23. For any 0 :$

< r, the imbedding H�(A)

Proof. It suffices to consider the case with W

and VI WI IP 1 E

=

H/:, (A) is compact.

1

v?

VI v ? W I

IP 1 ,

v?,m v?,m WI,m

By Lemma 4.22 and the equivalence of norms on we can extract subsequences and such that , --+ in L!(A) and --+ in LOO (A). Thus --+ + + in L!(A), and this completes the proof for r = , I and J.! = O. The general _ result follows from an inductive procedure and the space interpolation.

WI ,m VO W

WI ,m W

VI m v

Lemma 4.24. For any r > ! , H�(A)

c

LOO(A), and for any r

;:::

1 , H�(A) is an

Proof. We only have to verify the conclusions with = ( 1 - x2 ) -' . Since r 1 , H� (A) C H (A) C LOO (A) for r > ! . We next prove the second result. It suffices to verify that for r ;::: 1 , algebra.

w(x)

Indeed, for any integer r

lI a; (v w ) I ",

:$

r -l

;:::

1

w(x) ;:::

1,

I: C�IIVIl I'+ 1,w I l a; - l' w I L + l I a;v ll ", I l w l h.", :$ cllvll r,w I l W l r,w .

1' =0

Spectral MetHods' 'and Their Appli"cations

184

Then the space interpolation extends this result to any real numbers r

;,:::

1.

•

Now, let

aw(v, w) = L o",v(x)o",(w(x)w(x)) dx and define the linear mapping A (HJ.w ( A ) ) ' HJ. w (A) by aO/ ( Av , w) = (v, w), w E HJ.w ( A), ( 4.124) where (- , . ) denotes the duality pairing between (HJ.w ( A ) ) ' and HJ.O/ (A) . Lemma 4 . 2 5 . For any ;,::: 0, A is a linear, bounded and continuous mapping from H� (A) into H�+2 ( A ) n HJ.O/ ( A). For any - 1 :5 r < 0, A is continuous from (Ho.� ( A) ) ' into H�+ 2 ( A ) n HJ.O/ ( A ). :

-+

'if

r

Proof. If r is a non-negative integer, then the integration by parts in (4. 124) yields

- o�(Av) = v

I Av l r+ 2 .w cllvll r.O/ .

For real number r > 0, the same result follows from the and so :5 space interpolation. Finally for -1 :5 r < 0, the result comes as in Corollary 4.5.2 of Bergh and Lofstrom (1976) . •

A A

HJ.w ( A ) . Moreover, Lemmas L�(A) into HJ.O/ ( A ). Let HJ.w ( A) L!(A) G(A, v) = A (vo", v - J) . ( 4.125)

Now by Lemma 2.8, is a bounded mapping on 4.22 and 4:25 imply that is a compact mapping from -+ by A = .; and define the mapping G : IRI x

Clearly G is infinitely differentiable. Let DG be the Frechet derivative of G. Then D 2 G is bounded on any bounded subset of IRI x Moreover, since C we get

HJ.O/ (A).

LOO(A),

HJ.w ( A)

(4. 126)

HJ.w ( A ) HJ.O/ (A), by F(A, v) = v + AG(A, v).

Finally, define the mapping F : IRI

x

-+

Then problem (4. 123) can be written equivalently as follows: find that F A, U = 0.

( )

v E HJ.w (A ) such (4. 127)


185

Let > o}. Using the techniques in be any compact interval in = Lions (1969) , we can prove that there exists a branch of isolated solutions of (4. 123) , denoted by According to the description in Section 3.2, it means that there exists a positive constant such that

A*

IR! {A I A

(A, U(A)).

6, ( 4.128) 1 (1 + AD G(A, U(A)))v l kw � 6 1 v l h.w, V A A*, v H�,w ( A ). We begin to approximate the isolated solutions of (4. 127) . Let PN be the L! orthogonal projection and Pi/o be the HJ,w-orthogonal projection, i.e. , ( 4.129) aw (v - pi/ov , E IP�'o .

If v E H' (O) n H.;"(O) and 0 :5

p,

:5 m

:5

r,

then (5.12)

Let xU ) = ( xiii ) , . . . , 0 :5 jq :5 N, x�iq ) being the Legendre-Gauss type inter polation points. O N is the set of all xU ) . The Legendre interpolant INV(X ) E IP N of a function v E C(O) is defined by

, x !!n » )

If v E Hr (o), r > '2 and 0 :5 p, :5 r, then n

II v - IN VII ,. :5 cN2 ,.+� - r ll v llr .

(5.13)

In some special cases, we can improve the above result. For Legendre-Gauss interpo lation and r >

i,

For Legendre-Gauss-Lobatto interpolation and 0 :5 P, :5 min( 1 , 2r - n ) ,

Spectral Methods ' and Their Applications

192

For the Chebyshev approximations in several dimensions, let 1 n w(x) = II 1 - X�) - 2 , The Chebyshev polynomial of degree I is q =l

(

1j(x)

n = ( - 1 , l )n and

n

II TI • (x q ) .

=

q=l

v

The Chebyshev transformation of a function E L� (n) is as

Sv(x) =

vm(x) L: 1 =0 00

11

with the Chebyshev coefficients 2 n n 1 VI = - II - f v(xm(x)w(x) dx, I II = 0, 1 , . . . , 7r q =l CI . in where for 1 � q � n, C/ . = 2 for lq = 0 and CI . = 1 for lq � 1 . For any ifo E IP N and 1 � p � q � 00 . (5. 14) lI ifollL2; � cN � - Z; lI ifoIlL� ' Also for any ifo E IPN, 2 � P � 00 and I k l = m,

()

(5. 15) and for any r

�

0,

v vm(x). L: 1 =0

(5.16)

The L�-orthogonal projection of a function E L�(n) is If r � 0 and I-' �

PNv(x) =

r,

N

11 then for any E H� (n) ,

v

0'(1-', r ) is given in (2.53) . Furthermore, we define the inner product of H:: (n) by

(5. 1 7)

vw

( , ) m ,w

=

L: (a:v, a:wL ·

The H::-orthogonal projection P'N : H::(n) -+ IPN is such a mapping that for any E H:: (n) ,

v

O$ I " I $ m

Chapter 5. Spectral Methods for Multidimensional and High Order Problems

193

v

If E H�(f!) and 0 $ P, $ 1 $ r, then

(5. 18)

Moreover, let

am,w (v, w) = kL:m (a;v, a;wL I l=

and

(5.19)

(5.20) am,w (v, w) = kL:m (a;v, a;(ww)) . I l= In particular, a w(v, w) = a l,w (v,W) and a w(v, w) = a l,w (v, W). The HJ,w-orthogonal projection PIt : HJ,w(f!) lP� is such a mapping that for any v E HJ,w ( n ), -+

If E H� (f!) n HJ,w(f!) and

v

r ;:::

1 , then

The other HJ,w-orthogonal projection for any E H , (f! ,

v Jw )

pit : HJ,w(f!)

-+

lP� is such a mapping that (5 . 21)

If E H�(f!) n HJ,w(f!) and 0 $

v

p,

$ 1 $ r, then (5.22)

x U) (x iM , . . . , x !tn» ) v

x �j.)

Let be the Chebyshev-Gauss type interpo = , 0 $ jq $ N, lation points. f!N is the set of all The Chebyshev interpolant of E a function E C(O) is defined by If E H�(f!) , r > "2 and 0 $

v

n

x U) .

p,

INv(x) lPN

$ r, then

(5.23)

194

Spectral Methods

n

w(x)

(-� xq) .

an d

Their Applications

We now turn to the orthogonal approximations on unbounded domains. First of all, let

= (O, oot

and

=

exp

The Laguerre polynomial of degree

LI(x) qII=l LI, (xq). The Laguerre transformation of a function v L� ( n ) is Sv (x) 1L:11=0 vILI (x) I is

n

=

E

=

00

with the Laguerre coefficients

VI in v (x)LI(x)w(x) dx, I I I =

For any
0

and

with the norm

For any

H� (n, ) { v I l v l r,w , a a

v H::, (n, ) and a

E

=

0 :::;

Jl

E

=

:::; r ,

of degree I is

n

= ( - 00 , oot

E

H�(n) I v tl xl H� (n) } Il v tJ + Xq ) t Ilr,w E

(1

where a is the largest integer such that Finally let

N

( 5.25 )

a

< r

( 5.26 ) + 1.

w(x) exp (- qt=l x;) . The Hermite polynomial HI(x) qII=l HI, (xq) .

and

=

=

n

Chapter

5.

Spectral Methods for Multidimensional and High Order Problems

The Hermite transformation of a function v

E

195

L! ({l) is as

v(x) = L v,H,(x) 1 1 1=0 00

with the Hermite coefficients

For any t(x) , 1#0

�

Obviously I ud ::; lid which implies that for any

(5.42)

1 ::; m ::; 3.

r � 0,

Hence by imbedding theory, (5.42) reads

+

I b(v, w , z) 1 ::; cl v ll l 1 w llL' ( lI z I l L' lI u Il L' ) cl v h ll w l l � lI z ll � ::; cl v h ll w lh ll z ll � ·

On the other hand, if w

E

Ht (n),

for v

then

Proof. It is easy to derive the above estimates.

E

Ht (n) .

For instance, the last one comes

immediately from (5.44). Theorem 5 . 1 .

(5.41)

Proof.

•

The representation (5. 31) is equivalent to (5.37) with (5. 34), or to

with (5. 34) ·

It is clear that a solution of (5.31) is also a solution of (5.37) with (5.34) . Now, suppose that is a solution of (5.37) with (5.34) . We have from Lemma 5 . 1 , Lemma 5.2 and (5.37) that for any E V(n),

(U, P)

v

E H- � (n) and so E H� (n) n Ht (n) . By Moreover by Green ' s formula, imbedding theory, E L12 (n) n W1,¥ (n) . By virtue of (5.37) and Lemma 5.4,

U

/).U

U

I fLa ( u, v) l :5 c ( ll U llw1'lf ll U l / o 2 + Il f ll ) Il v ll· By Green ' s formula again, /).U L2(n) and so U H� (O) n H2 (O) . The integration by parts in (5.37) leads to (5.31). A standard argument tells us that the normal derivative of U is also periodic and so U H; (O). Furthermore by (5.34), L: 1 ((U . V)U(j) - f(i), ¢l) 1 2 . L: 1 1 I� I (P, ¢l W :5 L: j=l Since (U . V)U - f L2(f!), the above series converges and so P H; (f!) . Using E

E

E

1

E

(5.34) again, we have

1

3

E


203

and so P E, Finally it follows from (5.35) - (5.37) that On the other hand, by integrating by parts, we obtain that for 1 f 0,

l1-a (U, VI/JI ) = O.

H�(S1)nL�(S1).

a (U, V (I>t)

=

1 {)mU(j)(X){)m{)j �I(X) dx j,m=l 3

E

0

=-

3

E Im 1j

j,m=l

i l l l � E Ij Ulj l ,;" i l l l � (V · u, M . j=l (V . u, I/Jo) = 0 and so V . U = o . 3

1 {)mU(jl (X)�I(X) dx 0

(5.45)

In addition, The above statements lead to that P) is a solution of (5.3 1 ) . We now turn t o the equivalence of (5.37) with (5.34) and (5.41) with (5.34). If and so it U is a solution of (5.37) , then for any is also a solution of (5.41 ) . Conversely, if is a solution of (5.41) , then by taking VI/JI , 1 f 0 in (5.41) and arguing as in the previous paragraph, we can deduce that V · = o . Henc� V and for any E u, This means that is a solution of (5.37) . This completes the proof. _

(U,

v=

U

v E H�{S1 ), b(U, U, v) = b° (U, U, v) U v H�{S1), b(U, U, v) = b° (U, v).

UE

U

We are going to construct the Fourier-spectral scheme. For this purpose, we put (5.37 ) in an abstract framework. Let >. ..!:. and = Since

= E {S1 ) (HJ ( S1 ) ) ' . H! {S1 ) Hp! ( S1 ), E {S1 ) H- ! ( S1 ) and also E( S1 ) (H� ( S1 )) ' . We define an inner product on H� ( S1 ) by (5.46) (v, wh = a(v, w) + (v, w), V v , W E H;(S1), and a linear mapping A : (H� ( S1 ) ) ' -+ H� ( S1 ) by (5.47) (Av, wh = (v, w ) , V w E H;( S1 ). Moreover, by the theory of linear elliptic equations, A is a continuous mapping from W ( S1 ) with � - into Hr+2 ( S1) n H;( S1 ). Since E( S1 ) H-! ( S1 ), A is continuous from E( S1 ) into H�(S1)nH�(S1). But the latter space is comp&.ctly contained in H� ( S1 ), whence A is a compact mapping from E( S1 ) into H; ( S1 ) . Define G : rn.1 H; ( S1 ) -+ E( S1 ) by (5.48) (G( >' , v), w) = >'( b(v, v, w) - F(w)) - (v, w). From Lemmas 5.1 and 5.2, we know that for any >. E IR1 and v E H;( S1 ), I I G(>., v) II E :5 cl>'1 (II v ll � + Ilfll) . c

c

r

L

.

11-

C

C

x


204

Moreover G is a Coo mapping. Let D be the Frechet derivative of G. Then . for any positive integer m , DmG is bounded over any bounded subset of lR,l x According to (5 . 46) and (5.48) , (5.37) is equivalent to

(U, V ) l + ( G( .\, U) , v) = 0, \:I v E

Hi(f!).

H�(f!).

F(A, v) = v + AG(.\, v) . By (5.46)- (5.48) , we get that

an

(5.49)

equivalent form of (5 . 37) . It is to find U E

F('\, U) = O.

Hi(f!) such (5.50)

Let A* be a compact interval in IR\ and assume that (.\, U( .\ )) is a branch of isolated solutions of (5.50) . So there exists 8 > 0 such that for all .\ E A* and v E

11 (1 + ADG ( .\, U( .\ ))) vlll

H�(f!),

� 811vlll .

(5.51)

Now let VN be the same as in Section 5.1, and PN be the L 2 -orthogonal projection upon VN . It can be checked that for any v E

H�(n),

H;(f!) with

(5.4) asserts that PN is a continuous mapping in By (5.47) and (5.52) , AN E .c VN and

((H�(f!))' , )

(5.52) r

� o . Let AN = PNA. (5.53)

Maday and Quarteroni (19 82b) provided a Fourier-spectral approximation to (5.50) . It is to find UN E VN such that (5.54)

By (5.53) , (5.54) is equivalent to finding UN E VN such that By arguing as in the proof of Theorem 5.1 , it can be shown that \7 . UN = O.


205

E

Let { (A, U(A» , A A" } be a branch of isolated solutions of (5.50). There exists a neighborhood @ of the origin in H;(n), and for N sufficiently large, a unique Coo mapping A A" -+ UN( A) VN such that for any A E A", FN( A, UN( A» = o and U(A) - UN(A) E @. Moreover, if the mapping A E A* U(A) E H; (n) is continuous for some r 2:: 1, then for any A, there is a positive constant b1 depending on I U(A) l I r such that Theorem .5 . 2 .

E

E

-+

Proof. We first use Lemma 4.26, with B = E(n), W = H;(n), nN = PN, A = A and g (A,v) = G(A,v ) . Clearly condition (i ) of Lemma 4.26 is satisfied. Further, by ( 5.4) ( 5.55 )

and a density argument,

Since

A is a c?mpact mapping from E(n) into H;(n), we derive that Nlim �oo II (PN I)A llc(E,Hl ) = O. -+oo II AN - A llc('E H') Nlim

Therefore both ( 4.132 ) and (4.133 ) are also fulfilled. So there exists a unique branch E A*}. The combination of ( 4. 134 ) , ( 5.4 ) and ( 5.50 ) yields that '

{ (A, UN(A» , A II U(A) - UN(A) l h

� � �

P

=

-

(5.56)

p

c ( I U(A) - PNU(A) l h + I ( A - AN)G(A, U(A» 1 1) c ( I U(A) - PNU(A) l h + 1 (1 - PN)AG(A, U(A » lh ) ( 5.57 ) c Il U(A) - PNU( A) 1I1 � cN1 -r Il U(A) lI r .

For the L 2 -error estimate, we put Lemmas 5.3 and 5.4 ensure the first part of cQndition ( ii ) in Lemma 4.27. By C condition ( iii ) of Lemma 4.27 is also satisfied. Since ( 5.4 ) ensures condition ( 4.135 ) . Finally by ( 5.3 ) , ( 5.4 ) and ( 5.57 ) ,

((H;)' ,H;) ,

(I - PN)A, I U(A) uN(A) II K < I U(A) - PNU(A) I K + I PNU(A) - uN(A) I K < I U(A) - PNU(A) I K + Nl ( I U(A) - PNU(A) l h + I U(A) - UN(A) l d < cN- lII U(A)1 I 2 . �

E

A A - AN =

Spectral Methods B{ld Their .�pplications

206

Hence condition (iv ) of Lemma 4.27 is also fulfilled. Now, using (4. 136) and ( 5.4 ) , ' we obtain that

II U(,x) - uN(,x) 1 I .

� =

cII FN (,x , U(,x)) ll = II(A - AN)G(,x , U(,x» II II (PN - I)AG(,x , U(,x» II = II( PN - I)U(,x) II � cN-r I l U(,x ) ll r

which completes the proof.

•

In actual computations, we can use the Newton procedure to evaluate UN (,x) . For instance, let uWl(,x) be the m-th iterated value of UN (,x). Then the iteration is as follows

(I + ANDG ( ,x , uW) (,x»)) (uW+Il(,x) - uWl(,x») = - (u W ) (,x ) + ANG ( ,x , uW ) (,x»)) , m � O. On the other hand, the approximate pressure PN (,x) = PN( X, ,x ) could be determined

by

with

PN (,x) =

PN,O(,x ) = 0, PN,I(,x ) = I I� ( 1 . « UN (,x) . V) UN (,x) - f) , 1/>1) ,

:

1 � O.

E

( 5.58 )

n

Assume that the hypotheses of Theorem 5.2 hold, P(,x) H;(n) and r � 1 . Then there is a positive constant b2 depending on IIU(,x) ll r such

Theorem 5 . 3 .

L5(n),

that

L PN,I(,x ) Ip, ( ,x )

I I ISN

Proof. Let

IIP(,x) - PN (,x) I + N- 1 IIP(,x) - PN (,x) lh � b2N -r . PN(,x) = PNP(,x) and p(,x) = PN (,x) - PN(,x). By (5.34 )

(P (,x) , rpo ) = 0

and for I � 0,

�

and ( 5.58 ) ,

, · I l � (l · [(UN ( .\) V) UN(.\) (U(,x ) · V )U(,x) ] M . I l l � ( I . [« UN (,x) - U(,x» V) UN (,x ) + (U(,x) V) ( UN(,x) - U(,x»] , rpl ) . By integrating by parts and using V · U(,x) = V . UN (,x ) = 0, we obtain that for 1 � 0,

(P(,x), rpl)

=

-

�

.

I (P (,x ), rpl ) I

=

.

Further, define

rn'!

Vk as (GN( A, V), W ) = A ( b� (v , v , w) - F(w » ) - ( V, W)N,

GN

:

x VN -+

(5.60)

and let

Let

AN

be the same as in (5.53) and set FN

:

]Rl VN x

-+

VN,

A Fourier-pseudospectral approximation of (5.41 ) is to find

UN E VN

such that (5.62)

By (5.53) , (5.60) and (5.61) , (5.62) is equivalent to finding

UN E VN

such that

By an argument as in the proof of Theorem 5 . 1 , it can be checked that We have the following result. E

'\7

.

UN = o.

Let { ( A, U( A», A A * } be a branch of isolated solutions of (5 ./.1), and rl > � , r2 > 2 . If f E H;' (O) and the mapping A E A* -+ U( A ) E H;2 ( O ) is continuous, then for N :?: No large enough, there exists a positive constant b3 depending on IIU( A ) II '2 and IIfll ." and a unique Coo mapping A E A* -+ UN ( A ) VN such that for any A E A *, FN ( A, UN( A» = 0 and

Theorem 5 .4.

.

E

N

In order to obtain an improved L 2 -error estimate, let M > be a suitably chosen integer. The corresponding set of Fourier interpolation points, the discrete inner product and the Fourier interpolation are denoted by OM, ( . , ·)M and 1M. If then M>

Nr':' ,

Define the mapping Giv( A,

v)

as


where :FN

: YN -+ C(a) is given by :F;' (v) = ( f, V)M - L 1 /11 2 ( I · VI ) (1 . f, (Pt)M · III$N,I#O 2

209

Let F;' (A, V ) = v + ANGiv(A, v) . A new Fourier pseudospectr al approximation is to find uiv E VN such that (5.63) F;' (A, Uiv(A)) = O. We have the following result.

Let { (A, U (A)), A E A*} be a branch of isolated solutions of (5. ..{.1) and iv be the solution of (5.63). If f E Hr -1 ( n) with r > 2 and the mapping A E A* -+ U (A) E H: ( n) H;(n) is continuous, then there exists a positive constant b4 depending on II U ( A) lIr and IIf llr - 1 such that for any A E A*,

Theorem 5 . 5 . u

n

In actual computations, we can use the Newton procedure to solve (5.62) or (5.63) . For example,

(I + ANDG'N (A, U�m) ( A) ) ) (U�m+1 ) (A) - U�m) ( A) ) - (U�m) (A) + ANGiv (A, U�m) (A))) , m � O. =

On the other hand, the approximate pressure iN(A) is given by

with Piv,O ( A) = 0 and for

1 "# 0,

�

Piv,I ( A) = �I « UiV (A) · l ) (UiV (A ) . 1) + if · 1 , (Pt) M · If M > N

.:1 , then

(5.64)

where bs is a positive constant depending on I I U(A) l I r and I I f IlT- 1 . Theorem 5.4, Theorem 5.5 and (5.64) were proved in Maday and Quarteroni (1982b ) .


210

For the Fourier approximations of unsteady N avier-Stokes equation, we refer to the early work in Orszag 1971a, 1971b ) , Fox and Orszag 1973a, 1973b ) , Fronberg 1977 ) , and Moin and Kim 1980 ) . Hald 1981 ) , Kuo 1983 ) , Guo ( 1985 ) , Guo and Ma 1987 ) , Ma and Guo 1987 ) , Cao and Guo 1991 ) , and Rushid, Cao and Guo 1994 ) provided various schemes with numerical analysis for compuational fluid dynamics. The applications of Fourier approximations to other multi-dimensional equations can be found in numerous articles, such as Wengle and Seifeld 1978 ) , Feit, Flack and Steiger 1982 ) , Guo, Cao and Tahira 1993 ) , and Cao and Guo 1993b ) .

( (

(

(

5.3.

( (

(

(

(

(

(

(

(

(

Spectral Methods for Nonlinear High Order Equations

Many nonlinear partial differential equations of high order occur in science and en gineering, such as the Korteweg-de Vries equation possessing solitons, the Laudau Ginzburg equation in the dynamics of solid-state phase transitions in shape mem ory alloys, etc. . In this section, we consider the spectral methods for such kind of equations. We shall take the initial-boundary value problem of two-dimensional Navier-Stokes equation in stream function representation as an example to describe the Legendre spectral method for solving initial-boundary value problems of nonlin ear revolutionary equations of fourth order. The corresponding numerical solution converges to the weak solution of original problem. The spectral accuracy is obtained for smooth solution. The prediction-correction method is also used for raising the total accuracy of numerical solution. The numerical experiments coincide with the theoretical analysis. 1 1 2 and 80 be a non-slip wall. The initial-boundary value problem Let 0 of two-dimensional Navier-Stokes equation is of the form

{

=( , ) -

E

8tU + (U . V)U - p.!:iU + 'VP = /, x 0, 0 < t ::; T, 'V . U = 0, x 0, 0 ::; t ::; T, ( 5.65 ) U = O, x 80 , 0 < t ::; T, x n. U(x, O) = Uo(x) , Let V = { v I v V(O), 'V . v = OJ. The closures of V in L2(0) and H l ( O ) are E E

E

E

denoted by H and V respectively. Let

a(v, w) = 10 'Vv(x) · 'Vw(x) dx, b(v, w, z) = 1o (w(x) . 'V)v(x) · z(x) dx.


211

The weak f�rm of (5.65) is as follows ·

(Ot U, v) + b(U, U, v) + p a(U, v) { U(O) Uo,

It is shown in Lions (1969) that if a unique solution in Tj V) n conservation =

L2 (O,

=

(f, v),

v

v E V, 0 < t :5 T, (5.66) x E n. and I E L 2 (O, Tj V'), then (5.66) has

Uo E H LOO(O, Tj H) . The solution of (5.66) possesses the

For numerical simulations, we can approximate (5.66) directly. But in this case, we should construct a trial function space whose elements satisfy the incompressibility. It is not convenient in actual computations. To avoid it, several alternative expressions are concerned. Firstly, we can take the divergences of both sides of (5.65) , and get

Ot eV' . U) + V' . ((U . V')U) - p l:l.(V' . U) + l:l.P = V' . f. By the incompressibility, we obtain

(U . V')U - pl:l.U {. Ol:l.Pt U ++ �(U) V' . I, =

+ 'V' =

P I, x E n, 0 < t :5 T, x n, 0 < t :5 T E

(5.68)

where

�(U) = 2 ( 02 U(1 )fhU(2) - fhU(1 )02 U(2 ») . Conversely if U and P satisfy (5.68) and . U(O) = 0, then for all t � 0, V' . U = 0. Thus (5.68) is equivalent to (5.65) provided that U and P are suitably smooth. The second formula of (5.68) is a Poisson equation for P at each time t. If we use it to 'V'

evaluate the pressure, then we need certain boundary conditions for the pressure. As pointed out in Gresho and Sani (1987) , we have that

O"P(x, t) = v . ( pl:l.U(x, t) + I(x, t)), x E On, 0 < t :5 T where O"P is the outward normal derivative of P . A simplified condition is O"P(x, t) = N(x, t), x E On, 0 < t :5 T. But the first condition is too complicated in actual calculations, while the simplified one is not physical.


212

The second way is to introduce the vorticity H(x, the stream function w(x, and J(v, w) = 82 w iAv - 81 w82 v, V V, w E Hl (n) .

t) ,

Then

{

(5. 6 5) i s equivalent to

atH + J(H, w) - p,/:).H = '\l -/:).w = H, H(x, O) = Ho (x) ,

X

x E n, x E n, x E n.

J,

0 < t ::5 T, 0 ::5 t ::5 T,

t)

(5. 69)

Since the incompressibility is already included, we do not need to construct the trial function space with free-divergence. Clearly w(x, = 0 on the boundary. But it is not easy to deal with the boundary values of the vorticity. In some literature, it is assumed approximately that H (x, = on an. But it is not physical. The third representation of Navier-Stokes equation is the stream function form. Let

t)

t) 0

{

Then the stream function form is as follows

8t /:).w + G(w, w) - p,/:).2 W = g , aw w = - = o, an w(x, O) = w o (x) ,

x E n, x E n, xEn

0 < t ::5 T, 0 < t T, ::5

(5. 70)

where 9 = - '\l x J. The main merits of this expression are remedying the problems mentioned in the above and keeping the physical boundary values naturally. But the appearance of the terms G(w, w) and /:). 2 W brings some difficulties in theoretical analysis. We shall derive the weak form of For any v, z E w 1 •4 (n) and w E H 2 (n), let

(5. 70).

Obviously

J(v, w, z) + J(z, w, v) = O.

(5. 7 1)

J(v, w, z) = -(G(v, w), z).

(5.72)

It can be verified that We can estimate IJ(v, w, z} 1 in several ways.

Chapter 5. Spectral Methods for Multidimensional and High Order Problems Lemma 5 . 7. For

' an y

213

v E HJ (fl),

Proof. By a density argument, it suffices to prove the conclusion for such a function, we observe that for x = (xt , X 2 ) E fl,

1:1 :

(8lv(X» 2 :5 2 18lv (Xl, x 2 ) 1 1 8�v (Xl, X 2 ) 1 dXl , (�v(X» 2 :5 2 1 ' 18lV (Xl! x 2 ) 1 1 �82 V (Xl! x 2 ) 1 dX 2 , Thus

v E CO" (fl) .

For

etc.

Similarly This completes the proof. Lemma 5 . 8 . For

any

•

v, w E HJ (fl), I J(v, v, w ) l :5 2 11'\7 v ll ll � v llll � w ll ·

Proof. By integrating by parts, we obtain that

J (v, v, w )

=

- 10 (8lV82 V8�w - (81 v? 8l82 w + (82 v) 2 �82 W - 8lV82 V8�w) dx + 10 (81 V82 W8�V - 81 v8l w8l 82 v + 82 v82 w81 82 v - 82 v8l w8�v) dx.

Integrating the last integral by parts again, we find that it vanishes. So by Lemma

5.7, I J (v, v, w ) 1 :5 118l v l b 1 182 v ll L4 " 8� w" + 1 1 8lV I l � . 11 8l82 W ll + 11 82 V Il�. 11 8l82 w ll + 118lV Ib 11 82 VI lL' " 8�w " :5 (2 11 8lV I l�. 1 I 82 V I l�. + 1 1 8lV lli. + 11 82 V lli. ) . 2 I 81 82 w II 2 + " 8�W ,, 2 + " 8�wl :5 2 11'\7 v ll ll� v llll� w ll ·

(

n

1

1

2"

2"

•

Spectral Metnod� and Their Applications

2 14

If v, z E HJ ( f! ) and w E H2 ( f! ) , then

Lemma 5 . 9 .

I J (v , w , z) l :5 2 11� w ll (IIVvll ll� v Il IlVzlI lI� zll) ! .

Proof. By Lemma 5.7, I J (v , w , z) 1

v'211� w Il IlV v IIL' IIV zII L' :5 2 11� w ll ( IIV v ll ll� v Il IIV z ll lI�zll) ! . :5

•

Lemma 5 . 1 0 .

For any v , w , z E HJ ( f! ) ,

Proof. By integrating by parts, we have that J (v , w , z) = 10 (OlW02Z0:V ..... �W�ZOl02V + 02W02Z01 02V - 02WOIZO�V) dx + 10 (OlVOIW��Z - 02VOIWO:Z + OlV02WO�Z - 02V02WOI02Z) dx. (5.73) Then the desired result follows from an argument as before. Lemma 5 . 1 1 .

Let v , w , z E HJ ( f! ) . If in addition w E Hr ( f! ) with r > 2, then

I J (v, w, z)1 :5 cll� vlll l w ll r IlV zlI · Proof· We first let 2 r 25 ' Set r = I-' + 1 , 2 _2 I-' Hl ( f! ) '-+ LP ( f! ) and Hr - 2 ( f! ) '-+ La ( f! ) Thus

<
2, then

I J(v, w , z) 1 � c ll v llrll.6. wI I IIV z ll , I J(v, w , z) 1 � c ll v llr IIV w ll ll.6. z ll .

Proof. Since W- 1 (!1)

'-+

I J(v, w , z) 1

L<Xl ( !1), we get that �

:S

IlolV IlLoo II.6. w ll llo2Z I1 + II 02V llLoo II.6. w ll llolZ II c ll v llrll.6. w I I IIVzll·

On the other hand, we have that

\In 02VOlwoiz dX \ � I I 02V IILoo IIVw ll l l.6. z ll ·

Moreover for

and for

2 < r < '25 '

r 2: % ,

The remaining terms i n (5.73) can b e estimated similarly. Finally, the second con clusion follows from imbedding theory as in the proof of the previous lemma. _ The weak solution of ( 5 . 7 0) is a function that

{

111

E L 2 ( 0, Tj Ha) n L<Xl ( 0, Tj H1 )

such

(OtV1l1 (t), V v) + J (1l1(t), 1l1(t), v ) +p( .6. 1l1(t), .6. v) + (g(t ) , V )C(H-2(n),H�(n» = 0, v v E Ha ( !1), ° < t � T, x E n. 1l1 (X, 0) = 1l1o(x),

(5.74) It is shown in Guo, He and Mao (1997) that if 1l1o E HJ ( !1) and 9 E L 2 ( 0, Tj H- 2 ), then (5.74) has a unique solution. The solution satisfies the conservation

Spectral MetJ; ogs .and Their Applications

216

We can adopt Legendre approxi � ations or Chebyshev approximations to solve (5.74) numerically. For the sake of simplicity, we focus on the Legendre spectral method. Let lP�f = lPN n H� (O) and

(v, wh = (Vv, Vw), (v, wh = ( b. v, b. w),

v

v, W E HJ (O) , V v, w E H� (O) .

It can be checked that in the space H� (O), the inner product (v , wh here is equal to the inner product a 2 (v, w) in Section 5 . 1 , and I v l 2 = lI b.v l l . The orthogonal projections P;:;'o : HO' (O) --+ lP�o , m = are two such mappings that for any v

E

1 , 2,

HO' (O),

(v - p;:;,o v , ¢>L = 0,

V ¢> E lP�o , m = 1 , 2. It is not difficult to sh�w that for any v E HJ (O) , I l v - Pk o v l 1 1

--+ ° as N 00 . Clearly the projection p 2 ,0 is the same as the projection p;/ in Section 5 . 1 . Now let tPN be the approximation to W. A Legendre spectral scheme for (5.74) is to find tPN(X, t ) E lP�o for all t E f4 (T) such that

{

--+

(Dr VtPN(t), V¢» + J (tPN(t) + orDr tPN ( t ) , tPN(t) , ¢» +/1- (b. (tPN(t) + O' rDr tPN(t)) , b.¢» + (g(t), ¢» = 0,

tP(O ) = tPN,O(X),

where 0' , (j are parameters, ° ::; 0', 0 ::;

(5.76)

1 and

I I W o - tPN, o l l l -+ 0, as N --+ 00 . (5.77) 0 1 For W o E HMO), we take tPN,O = PJII WO. While for W o E H� (O) , we take tPN, O = Pi/W o o Both of them satisfy (5.77). Thus the approximations on the initial level are well defined. Now, assume that tPN(t - r ) is known. Let r be suitably small and A(v, w) = (Vv, Vw) + /1-O'r ( b. v, b. w) + or J (v, tPN(t - r) , w) , V v, w E lP�o . By Lemma 5.9, A(v, w) is a continuous bilinear elliptic form on lP�o x lP�o . Hence by Lemma the numerical solution at the time t is determined uniquely. So (5. 76) has a unique solution as long as W o E HJ (O) and 9 E L2 (0, T; L2(0) ) . Especially, if 0' = 0 = then we deduce from ( 5 . 7 ) and ( 5 7 ) that

2.1 ,

�,

�

1

.6

IIVtPN(t ) 11 2 + /1-r L I Ib.V'N(s) + b.tPN(S + r ) 1 I 2 +r

seRr(t - r )

L (g(S), tPN(S) + tPN(S + r) ) ds = IIVtPN,0 11 2

seRr(t - r )


217

which is a reasonable simulation of (5.75). In practical computations, a good choice of basis functions is essentially important. Shen proposed a nice set of basis functions. Let Lm (Y) be the Legendre polynomial of degree m. Set

(1994)

Then

lP�P = {(Pi(x) 1 0 :S It , 12 :S N - 4}.

B y the orthogonality of Legendre polynomials, both the matrix with the elements ((Pz, � ' 7 = (N-'\ ) , A � max (�, 7 - 2ro ) , (b) 0' = � ' 7 (N-4 ) , 2 I ) 0' < "21 ' 7 N � ' being a positive constant such that for any J.! c�(1 _ 2 0' ) C(O, Tj L2)

=

0

=

( C

4

CJ

Cp

v E IP�o , II� v ll � cpN2 11V' v ll .

J.!,

Then there exists a positive constant bI depending only on n and the norms of and g in the mentioned spaces, such that for all t E RT (T), If 0' > � and 6 is suitably large, then the above result is vaild for any 7 --+ O .

I}i


2 19

Proof. LeOliN = P't;° ifl . We derive from (5.74) that

{

(D,. V'WN(t), V'ifJ) + J(WN(t) + OrD,. W N(t), WN(t), ifJ) + j.£(�(WN(t) + urD,.wN(t)), �ifJ) o = -(g (t), v) + B(w(t), WN(t), ifJ), V ifJ E IP� , 0 < t ::; T, W N( O ) = p;/ Wo, x E fl

(5.79 )

where

(V (D,.iflN - 8tifl ) , V ¢1) + J(iflN + 8r D,. iflN, iflN, ¢1) - J(iJ!, ifl, ¢1) + par (D,. � ifl, � ¢1) . Let W = .,pN - iJ!N. We obtain from (5.76) and ( 5 7 9 ) that (D,. VW (t) , V ¢1) + J (W (t) + 8r D,. W (t) , iflN(t) + W et), ¢1) + J (iflN (t) + 8r D,.iflN (t ), W et), ¢1) +o p (� (W (t) + ar D,. W(t») , � ¢1) = - B (ifl(t) , ifl N(t), ¢1) , 'V ¢1 E IP� . By taking ¢1 = 2 W (t) + e r D,. W (t ) and using an argument as before, we obtain from (5.71) that D,.I I VW (t)1 1 2 + r (e - 1) li D,. VW( t) 1 2 + 2pl l �W(t)1 I 2 +pr (a + n D,.II �W (t)1 I 2 (5.80 ) +pr2 (ea - a - �) I D,. � W(t)1 I 2 = � Gj (t) .

where

G1 (t) = -r(e - 28)J (W(t), W(t),D,.W(t») , G2 (t ) = -r(e - 28)J (W (t), iJ!N (t ), D,. W (t») , G3(t ) = -2J (iflN(t) + 8r D,. iflN(t), W et), W (t») , G4 (t ) = -fod (iflN (t) + 8rD,.iflN(t), W(t) ,D,. W(t») , Gs(t ) = 2 ( D,.iflN(t) - 8tifl (t), �W (t») - for (V ( iflN (t) - 8tifl (t» ,D,. VW (t») , G6 (t) = -2par (D,.� ifl(t ) , � W(t») - fopur 2 (D,.� ifl (t) , D,. � W(t») , G7 (t) = -2J (iflN (t) + 8rD,.iJ!N (t) , iflN(t), W et») + 2J (ifl(t), ifl(t ), W(t») , Gs (t) = -erJ (ifl N (t) + 8r D,. ifl N(t ) , iflN(t), D,. W (t») + er J (ifl (t), ifl(t), D,.W(t») .

220

Spectral Methods aDd Their Applications

I Gj (t) l .

c

D

Let > ° and be an undetermined positive We are going to bound Lemmas and 5.8-5.10, we have that constant. By (5.71)

I G1(t)1 S 2TI � - 281 \ VW (t)\J I �W(t) I I DT� W(t ) 1I S coT I DTVW(t)1I 2 + CITN4 I1 VW(t)1I 2 1 � W(t) 1I 2 , I G2 (t)1 S 2TI � - 2ol l � wN(t)1 1 ( I VW (t) I I �W (t) I I DT VW (t) I I DT�W(t)\1 ) � 2 (Vrl � - 20II l � WN (t)I I I \ � W(t)l \ t I DT VW (t) l I t) (T I VW(t)I I I D T�W (t) l I ) t S T I V W (t) I I DT�W(t) 1I + T(� - 20)21 1 � WN (t)1 1 2 1 � W(t) I I DT VW (t) 1I S cO IlT2 I1 DT�W(t)r + 4coll1 I V�(t)1I 2 + cO T I DT VW(t)\r + C2 T I �W(t)\r , IG3(t )1 S 2 \J (W (t) , W (t) , WN(t) + 8TDT WN(t)) \ S 2D I V W (t) I I � W (t)ll l I � (WN(t) + OTDT WN(t ) )11 S D I � W(t)11 2 + c3 1I VW (t ) 1I 2 , 1

2

_

I G4 (t)1 S 2�T I VW (t) l I ! I � W(t)lI ! (I I DTVW(t) lI ! I DT�W(t) lI t ' I � ( WN(t) + OTDT WN (t))1 1 + I DT�W(t)II I V (WN(t) + OTDTWN(t))l l t I � (WN(t) + OTDT WN(t)) I ! ) S 2�TI I WI I C(o,T;H2) (II DT VW (t) lI t I DT�W(t) lI t I V � (t) I ! I � � (t) lI ! + I DT�W (t) I I VW(t) lI ! I �W (t) l I t) S 2�TI I Wl l c(o,T;H2) I DT VW (t) 1I 2 I 1 DT�W (t) 1 I 2 I1 VW (t) 1 I 2 1 � W(t) 1I 2 + cOIlT2 I1 DT�W(t) 1I 2 + c:: I l wl l �(O.T;H2) I VW (t) I I �W (t) 1I S 2�TI I WI I C(o.T;H2) I DT V � (t) l I ! I DT � � (t) l I ! I V � (t) l I t 1I � � (t) lI ! + cOIlT2 I1 DT�W(t) 1I 2 + D I �W(t) 1 I 2 + c4 1I VW (t) 1 I 2 S 2coIlT2 I1 DT�� (t) 1I 2 + 4C�Il Il VW (t) 1I 2 + cO T I DT V � (t) 1I 2 + D I � W(t)1I 2 + c4 I1 VW (t) 1I 2 + C4T I �W(t ) 1I 2 1

1

1

1


where

221

Cl = c1o ; ( e - 2D) 2 , C3 = IIwll � (o,T ;If2 ) ' -c

�

I Fs (t) 1 and I F6(t) l . It can be verified as before that IID1'WN(t) - otw(t) 1I 2 ::; IID1'WN ( t) - D1'W (t) 11 2 + IID1'w (t) - otw(t) 1I 2 +1' 21' r+1' w ( s) 2 ds . (5.81) ::; -;;:2 itr 11 0. (w (s) - wN(s))1I 2 ds + "3 it I o� 11 A similar estimate is valid for 11 '\7 (D1'wN(t) - Otw (t» 1I 2 . Thus 1' I Gs ( t) 1 ::; D 11L).� ( t) 11 2 + cOT IID1' '\7� ( t) 11 2 + � + 110. ( W (S) - wN(s» 11 2 ds + CS * 110. (W(S) - wN ( s» l I� ds + CsT * II O�W ( s) 11 2 ds 1' + CST 2 + II O� W ( s) l l : ds

We next estimate

[

1

where

Cs =

[ max ( e, � ) . Clearly 2�0

1

1

t I G6( t) 1 ::; D 1 1 L).� ( t) 11 2 + CO/1- T 2 1ID1'L).� ( t) 1 1 2 + l1;T +1' lIo.w(s) lI� ds where Ca = /1- (1' 2 of Lemma 5.10,

(� + 4�0 e T 2 ) . Finally, we estimate I G7(t) 1 and I G8 (t) l . By virtue

2 1 J (WN(t) + DTD1'WN(t) , W (t) - WN (t), � ( t») 1 ::; 8cllwl l c(0,T ;H2 ) IIW (t) - WN(t) lI f IIW (t) - WN(t) lIl llL).� (t) 11 ::; D 11L).�(t) 11 2 + c7 ,1 Ilw (t) - wN(t)ll l ll w(t) - wN (t) 1I 2 16 ' . t Lemma where C7, 1 = c 11 W 11 2C T 2 . Accordmg 5.9, D

(0, ;H )

( 5 . 82 )

�

2 \ J ( wN(t) + DT D1' W N ( t) - W (t), W (t), � (t») I ::; 4c llwll c(0,T ;H2) 11L).� (t) 11 IIW (t) - WN(t) ll f IIW (t) - WN(t) ll l + DT IID1'WN (t) 1I 2 ::; D 1IL).� ( t) 1 1 2 + C7,1 1IW (t) - WN (t) III IIW (t) - wN(t) 11 2 t + C7,2 T itf +1' II 0.W ( s) 11 22 ds (5.83 )

(

)

Spectral Methods ahd Their Applications

222

� 52cI I 1l1 11�(o,T;H2)' Let C7

where C7, = 2 to (5.83) leads

=

2C7,1 + C7, . 2

The combination of (5.82) and

t

r IG7 (t) 1 ::; 2D I tJ.W (t) 11 2 + c7 11 1l1(t) - 1l1N(t)1I1 1l1l1(t) - 1l1N(t) 11 + C7T + 118.w(s)ll� ds. 2

I Gs (t) 1
2 ' then the values of { and all do not depend on 7 N4 • By Lemma 3.2, we only require A 2 (t) � 0, i.e., Cj

(5.88)

Spectral Methoids and Their Applications

224

being a positive constant independent of N and T. Since � 2, a sufficient is that T = o(N->' ) , ).. � max 7 - 2 . Finally, the above statements together with. the triangle inequality lead to the first part of this theorem. then we can take e = 28 and so C = = O. If in addition 0' and 8 � Therefore the condition ( a) can be weakened to any T -> O. •

b

2 condition for (5. 8 8)

(�,

�,

>�

ro)

ro

l C2

Theorem does not provide an optimal estimate. But if \[f is a little smoother, then the optimal estimate exists, stated as follows.

5.7

Theorem 5 . 8 . Assume that (i) condition (i) of Theorem 5.7 holds; (ii) ro � rl > 2, or ro > rl = 2 and E LOO (O, T; Hr) for some r E ( 2 , ro); (iii) one of the following conditions is fulfilled (a) 0' > �, T = (N->') , ).. � max (�, 6 - 2ro) , (b) 0' � 'T 0 (N-4 ) , (c) 0' "21 ' TN4 :::; Jl C� (12 20' ) Then for all t E k,.(T), \[f

0

=

=

then E LOO(O, Tj for E We have

r1 2,

w

Hr ) r (2, r1 ) ' 4 G7 (t) =. jL=l Gr,j (t)

225

I R2 (t)1 more

where

G7,1 (t) = 2J (WN(t) - W (t), W(t) - WN(t), q,(t)) , G7,2 (t) = 2J (w(t) + t5-r DTw(t), W(t) - WN(t), q,(t)) , Gr,3 (t) = 2J (w(t) - WN (t) - t5-r DT WN(t ), W(t), q, (t)) , G7At) = 2t5-rJ (DT WN(t) - DTW(t) , W(t) - WN(t), q,(t)) . By Lemmas 5.8 and 5.12, we obtain that

I G7,1 (t)1 ::; 4 1 � q,(t) l l I V' (W(t) - wN(t)) l l � (W(t) - WN(t))1 I ::; D 1 � q, (t) 1 2 + C7,3 1 I W (t) - WN(t)l I � and

I G7,2 (t)1 ::; 2c Il w (t) + t5-rDT W(t)l l r I W (t) - WN(t)1 1 1 1 � q,(t) 1 ::; D 1 � q,(t) r + C7,4 I 1 W (t) - WN (t)l I � where C7, 3 = � I W llioo (O , T ; H2 ) and C7,4 = � I W l i oo (O , T ; H r) ' Moreover by ( 5.71 ) and Lemma 5 . 1 1 ,

Since

I DT WN(t)1 I 1 ::; -r- � (1* I O.WN(S)I I � dS) � ::; -r- � ([+T I O.WN(S)I I � dS) � and I l otWN(t)1 I ::; I l otw( t)1 I ' we see that 2 2 I G7,3 (t) 1 ::; D I � W- (t) 1 2 + C'i,4 I 1 W (t) - WN (t)1 I 21 + Cr,s-r itr+T l I o.w(s)1 1 22 ds

226 where

Spectral Methods

and'

Their Applications

C7,S = 282C7,4 ' Using Lemma 5.9, we deduce that

. l i D,. (llT(t) - IlTN(t)) ll i � D 1 � q, (t)1 1 2 + C7, S T 2 1 1 � (llT(t) - IlTN(t))1 I 2 1 I D,. (llT(t) - IlTN(t))l l l . l i D,. (llT (t) - IlTN(t))1 I 2 +,. � D I � q, (t) r + C7, S T I � (llT(t) - IlT N(t)) 1 1 2 ([ 118. (llT(s) - IlT N(S )) I � dS ) � . (1* 118. (llT(s) - IlTN(s)) I � dS) < D I � W (t) 1 2 + C7,S [+,. 11 8. (llT(s) - IlTN(s)) ll ; ds + 41 c7,sT 2 1 � (llT(t) - IlTN(t)) 1 I 4 i(t +,. 118. (llT(t) - IlTN(t)) ll � ds 82 where C7 ,6 = c ' The above statements lead to D I G7 (t)1 � 4D 1 � q, (t) 11 2 + c7 111lT(t) - IlTN(t) l I � + C7 It t+,. 118. (llT(s) - IlTN(s)) I � ds + C7T It t+,. 1 8. IlT(s) 1 I 22 ds + C7T2 11� (llT(t) - IlTN(t))1 1 4 1t t+,. 118. (llT(s) - IlTN(s))1 1 22 ds where C7 = max (C7 ,3 + 2 C7 , 4 , C7, S , C7 ,S ) ' Similarly, 1

1

2

where

Cs

=

(C7, 7 + 2C7,S , C7,9 , C7,10) and 4e ce 2 T',Hr) , C7,7 = -l ' coil l llT llioo(O'T,'H2) , C7,S = -4coil-I l IlT I Loo(o

max

Now, by taking = obtain from ( 5.89 ) that

D i6

C7,1 0 = c482oe . ll

and using an argument as in the derivation of ( 5.84 ) , we c

D,. 1 \7q, (t) 1 2 + T(e - 1 - 4co) I D,. \7q, (t) 1 2 + � 1 � q, (t) 1 2 + IlT (0- + �) D,. I � q, (t) I 2 + Il T 2 (e o- - 0- - � - 8co) I D,. tlq, (t) I 2 � Al I \7q, (t) I 2 + A2 (t) I � W (t) I 2 + A3 (t), ( 5.90 )


227

where

Let

qo ;::: 0

and choose

�

as follows. For

(1' > 21 ' we take

)

� ;::: 6 = max 1 + 4co + qo , 2(1'2(1'+-16co 1 . Then

�(1' - (1' - 21 � - 8co ;::: 0

(

and so

(

�

7"(1-�-4co ) li D.,. V � ( t)1 1 2 +1'7" 2 �(1' - (1' - � - 8co 7" = 0 ( N-4 ) , then we take

If (1' =

�

and so

(5.91) also holds.

and

) I D.,.6�(t)11 2 ;::: qo7" li D.,. V �(t)11 2

.

(5.91)

Finally, for (1' < � and

2

2 4 . 7" N < Jt cr>2 ( 1 - 2(1' ) ' we put

and then (5.91) also follows. Now, by the same procedure as in the proof of Theorem 5.7, we obtain from (5.90) an estimate similar to (5.86). In particular, we have

p (t ) = 7"

L: A3(S) sERr(t-.,.)

=

0

(7" 2 + N2 - 2ro + N- 2rl + 7" N2 - 2rl )

•

and

Spectral Methods Their Applications

228

T = ( N-4 ) , then for all t E R... (T) , p(t) ::; � 2�TN . On the other hand, for > �, we require that A2(t) ::; 0 , i.e. , ) ::; b4 , T ::; b4 , T N4 (T + +T b4 being a positive constant independent of and T. The proof is complete. We now turn to numerical experiments. For describing the errors, let x �jq ) and w{jq) be the Legendre-Gauss-Lobatto interpolation points and weights. The error If

0"

::;

�2

and

0

N 2 - 2 To + N- 2 T,

2

0"

N2 - 2 T,

N

_

E( tPN, t) is defined as

1 eN (1 + cos ) (1 + cos 2) (5. 92) 10 The numerical results of scheme (5.76) are presented in Tables 5.1-5.3. The first table shows the high accuracy of (5. 7 6) even for small and the good stability in long time calculation even for large T. Table 5. 2 indicates that when -+ and 0 proportionally, the numerical solution converges to the exact solution. Table 5. 3 shows that the computations are also stable for very small even if = O. Table 5.1. The errors E(tPN, t) , fl = 0. 5 , 0" = 0. 5 ,b = O , = 10. T = 0. 5 T = 0.1 T = 0. 0 5 T = 0. 0 1 T = 0.005 T = 0. 0 01 t = 10 2. 497E - 2 4. 9 95E - 3 2. 4 97E - 3 4. 9 95E - 4 2. 4 97E - 4 5.000E - 5 t = 20 2 . 4 85E - 2 4.969E - 3 2. 4 84E - 3 4. 9 70E - 4 2 . 4 85E - 4 4. 9 83E - 5 t = 30 2. 438E - 2 4.880E - 3 2. 440E - 3 4. 8 79E - 4 2. 4 39E - 4 4. 8 76E - 5 In the previous part, we considered the two-level spectral scheme (5.76). Since we I

We take the test function

W(x, t)

t

=

7r X l

7rX

•

N,

T

N

-+

fl,

00

fl

N

adopt partially implicit approximation for the nonlinear term, the convergence rate in time t is of first order. It limits the advantages of spectral approximations in the space. A modified version is the following

{

(D ... \11/;N(t) , \1 0 for 1 $ k $ Po and ),,. inner product of Vk with (5.94), we obtain

(5.95)
'I , I + ¢ >'I, I) TI ( Y (x)) . 1=0

Since IT1 (y) 1 ::; 1 for IY I ::; 1, we know from the above two equalities and the last formula of (5.120) that for x E [-1 , a] ,

In a similar way, we see from the proof of Lemma 5.16 that for

x E [a, 1),

The above estimations lead to the conclusion.

•

Theorem 5 . 9 . The iteration applied to (5. 1 1 6}- (5. 1 1 8) converges, as m

---t 00 .

Proof. The result follows from Lemmas 5 . 1 7 and 5.18. Actually, since E i s nonsingular, the convergence of ej, m implies that of U,,/, N, m - U,,/,N , 'Y = 1 , 2. •

In the final part of this section, we present some numerical results. The time discretization considered is the second-order Crank-Nicolson scheme with the mesh size T = 0.01. The test function is

U ( 1 ) (X, t)

=

et arctg

(ax + i) ,

U ( 2 ) (X, t)

=

e t arctg

(ax i) . -

We use the Chebyshev pseudospectral domain decomposition method to solve this problem. For comparison, we also use single domain method for this problem. In Tables 5.8 and 5.9, we list the maximum norms of the errors between the exact solution and the corresponding numerical ones at t = 1 , -with different values of a and

.


247

N. It can be seen that t �e domain decomposition procedure provides better numerical results, in particular, for the solutions possessing oscillations. Furthermore, if the matrices associated with the pseudospectral method are full, then the use of several sub domains allows the reduction of the overall complexity. It is noted that similar results are also valid for multi-sub domain method and the Legendre pseudospectral domain decomposition method.

Table 5.8. The errors of single domain method. a = 10 a=1 a=4 N = 4 4.767E - 4 1 . 288E - 2 2.812E - 2 N = 8 1 . 094E - 4 5.079E - 3 2.345E - 2 N = 20 2.572E - 7 2.860E - 4 1 . 005E - 2

Table 5.9. The errors a=1 N = 4 1.937E - 4 N = 8 7.377E - 7 N = 20 2.577E - 7

of two domains method. a = 10 a=4 6.1 19E - 3 3.668E - 2 1 .024E - 3 1.893E - 2 1.943E - 4 9.231E - 3

Another early work of spectral domain decomposition methods for hyperbolic sys tems was due to Kopriva (1986, 1987). We also refer to Quarteroni (1987b), Macaraeg and Streett (1988) , and Macaraeg, Streett and Hussaini (1989) for this technique and its applications. Carlenzoli, Quarteroni and Valli (1991) also used it for com pressible fluid flows. Recently Quarteroni, Pasquarelli and Valli ( 1991) developed heterogeneous domain decomposition method arising in the approximation of certain physical phenomena governed by different kinds of equations in several disjoint subre gions. Various applications of domain decomposition methods can be found in Chan, GlowinBki, Periaux and Widlund ( 1988) , and Canuto, HUBsaini, Quarteroni and Zang (1988). For the applications of Chebyshev pseudospectral methods to other nonlinear problems, we also refer to Hitidvogel, Robinson and Schulman (1980) , and Maday and Metivet ( 1986).

248 5.5.

Spectral Methods and Their Applications Spectral Multigrid Methods

Multigrid methods have been used for spectral approximations since the beginning of the last decade. Zang, Wong and Hussaini (1982, 1984 ) described this approach and its applications to periodic problems and non-periodic problems. It makes the spectral methods more efficient. In this section, we present some basic results in this topic. We first let A = (0, 211') and consider the model problem

{

x < 00 , - 00 < x < 00 . 2 and /j = J (xU» ) . Let N be an even positive integer. Set xU ) = approximation to the value U (xU» ) , 0 :5 i :5 N - 1 . Then -o�U = J, U (x) = U (x + 2 11') ,

Uj =

if - I

E

/ l12

-00
E VM.N and 2 � Proof. Set

4>1 CX I )

Then 4>1 E

114>II Loo

.•

.•

dI L oo(At} cMN i ll 4> ll . �

��

�

00,

= -2111" lA. 4> (Xl , X2 ) e-i1"2 dX2 .

lPM•1 and by Theorem 2.7, �

p

N. VM.N = lPM. I VN.2

f

� cM �

��

C6 . 1)

II4>dI L2 (At} � cMN i ( � 1 4>l I i'(At} )

i

��

Consequently, which leads to the conclusion.

•

Chapter 6. Mixed Spectral Methods

259

As a consequence of Theorem 6. 1, we have that for any , 'IjJ E

VM.N, (6.2)

Theorem 6.2. For any E VM•N , Proof. Let

that

, (X I)

1101 11 2

be the same as in

(6.1). We know from the proof of Theorem 2.8

= 27l' IILI $N II Ol d/;" (Al) � 1 !7l' M4 IILI$N lI d/i,'(A,) = � M4 I11I 2 .

On the other hand, 1102 11 2

=

27l' IILI $N 12 II 11 i,. (A,) � N2 11 1I 2 .

Then the desired result follows.

•

v v(x) mL= L vm .I Lm (xd ei1x, O 1 ' 1 =0

The mixed Fourier-Legendre expansion of a function E L 2 (!1) 00

IS

00

=

where

Vm. 1 is the Fourier-Legendre coefficient, Vm.l = 217l' (m + �) 10 v(x)Lm (X I) e -i1x• dx.

The L 2 (!1)-orthogonal projection PM.N : L; (!1) VM.N is such a mapping that for any v E L; (!1), - PM.N V, N :::; cM- 2r 2::: II V d11r(At l cN- 2 • 2::: 1W' l I v d l i'(Al ) 111> N 111$ N :::; cM- 2r l l v l l i'(A2 ,w( Al » CN- 2 B l v I 1' (A2 P (At l ) · 27r

+ +

It is noted that for any v

E

�

H�:;(!l) and r, 8

•

0, (6.4)

When we use the mixed Fourier-Legendre approximations for semi-periodic prob lems of partial differential equations, we need different kinds of orthogonal projections to obtain the optimal error estimates. For instance, the H l (!l)-orthogonal projection Pk,N : H� (!l) -+ VM, N is such a mapping that for any v E H� (!l), (V ( v - PiI,N V) , V¢»

+ ( v, ¢»

=

While the HJ (!l)-orthogonal projection plf?N (!l) any v E HJ,p (!l) ,

Theorem 6.4. If v E M;'B (!l) with r, I I v - piI,N v l l " :::; c (Ml - r

8

0, V ¢> E VM, N .

-+ V,,'},N is such a mapping that for

� 1 , then for J.L = 0, 1 ,

+ Nl-B) (M,,-l + N,,-l ) II V I l Mr,

• .

If, in addition, for certain positive constants Cl and C2 ,

(6.5) then

Chapter 6. Mixed Spectral Methods

261

Proof. Let VI ( Xl ) be as before, and Pk be the Hl (l\.t )-orthogonal projection. Set V. (X) 1 L1 :5N (PiIVI (Xl )) ei1x, . =

By Theorem 2. 10,

I l v -w l � :::; I v - v. l I � < 211' I!LI $N ( l i vi - PiIvdl k1 (A') + [2 1 vl - PiIvdl i' (A, » ) + 211' L ( I vdl k1 (A' ) + 12 I 1 vdl i2 (A, » ) :::; cM2-2IlrI>lLN N ( I vdl kr (A' ) + 12 I 1 vdl kr-1 (A1» ) I :5 + cN2 - 28 L 128 - 2 ( I vdl k1 (A, ) + 12 1 I vdl i2 (A' » ) :::; c (M1 -r +I!NI>Nl -S) 2 1 I v l �r" . inf

wE VM.N

Then the first conclusion with J.L = 1 follows. By means of the duality, we deduce that

I v - PiI,NVI l :::; c (M1 -r + N1 -S) (M-1 + N-1) I V I l Mr" .

The last result comes from the above inequality and (6.5) .

v

Theorem 6 . 5 . If E HJ,p(n)

n

M;,8 (n)

with r, s 2: 1, then for J.L = 0, 1,

If, i n addition, (6. 5) holds, then

pj,l

Proof. Let Vl (Xl ) be as before, and be the HJ (Al )-orthogonal projection. Put v. (X ) 1!LI:5N (pltvl(xl )) ei1x2• =

By virtue of Theorem 2 . 1 1 ,

•

262

Spectral Methods and Their Applications �

+

+

�

�

+

c 1LN (i vi - pl/ vd1-'(A') [2 1 v - pli�NvdI 12 (A, ) ) 1 I$ c I ILI >N (I vd1-'(A' ) [2 1 I vl lli2(A, ) ) cM2-2r ILN ( I vdl 1-r(A' ) [2 I1 vdl 1-r-l (A, ) ) I I$ CN28-2 ILN [28-2 (Ivd1-'(A, ) 12 1 I vdl i'(Al) ) I I> c (M-r N-8 ) 2 I 1 VIl �r, +

+

+

+

• .

The remaining results come from an argument as i n the last part of the proof of Theorem 6.4. • In numerial analysis of mixed Fourier-Legendre spectral schemes for nonlinear problems, we also need to bound the norm One of them is stated in the following theorem.

I Pli�NVllwr,p .

Theorem 6.6. Let (6. 5) hold and 8

n

> !. IIv E HJ.p ( O ) H; ( A2 , H1 ( A1 )) , then

I Pli�NVI I Loo � c ll v I H'(A2 .H1 (AtllIf, i n addition, v E X;· 8 ( O ) with r � 1 , then Proof. We first let VI ( Xl) be as before, and let Obviously

VI E HJ(A1 ) and vi E lP�. l ' Define

We assert from the orthogonality of the Fourier system and the property of that Obviously

PlioN

Chapter

6. Mixed Spectral Methods

263

By Schwarz inequality and Poincare inequality, we deduce that

I VI -v71 I iT' (A, ) + [2 1 I vl - v7I1 i' (A, ) :::; c ( l i vi -

!,

l L V LOO(A, I I I ::;N 1181 iI l ) :::; cM I ILI ::;N I VI - v7I H (Al) + c IlLi ::;N I vdl w+1(A, ) < (M1 -r + 1 ) II2:N I vdl w+1(A, ) I ::; < c (II2:N (1 + [2 fS ) ( I2:N (1 + [2) " I VdI W+1(A, ») I I ::; I ::; C

1

"2


264 Similarly, we have that

L I l v7 1 I LOO (A, ) s: cli v Ii H'(A2 ,H' (A, ) ,

III� N

L I l i li vi I i L OO (A, ) s: cli v I IH'+'(A2 ,W(A, ) '

III� N

The proof is complete.

•

Finally we consider the mixed Fourier-Legendre interpolation. Let 1M be the same as in the previous paragraphs and IN be the Fourier interpolation on the interval respectively. Set IM,N = 1MIN = IN IM.

A2

Theorem 6.7. If v E H: (A2 , W(A 1 )) n H; (A2 , H"'(A1 )) n H; (A2 , H>'(Ad) , O < (

( )

O

a s: min r, A ) , S: j3 S: mi n s , , and r, s , A, , > � , then I I v - IM,Nv I I H,B(A2 ,H"(A, )

s: cM2 ",-r + t Ii V I i H,B(A2 ,W(A, ) + cN,B-s l l v I I H'(A2 ,H" (AI » + cq ( j3) M2 01 - >. + t N,B --y Il v Il m ( A2 ,H� (AI » )

where q (j3) = ° for j3 > � and q(j3) = 1 for j3 s: � . Proof.

Let I be the identity operator. We have that

where

Dl

=

I I v - IM v II H,B(A2 ,H"(Ad )

+ I l v - INv Il H,B(A2 ,H"(A, ) ,

D2 = II (IM - I) ( IN - I) v II H,B(A2 ,H"(A, )

.

From Theorems 2.4 and 2.12,

If j3 > � , then by Theorems 2.4 and 2. 12,

If j3 s: � , then


•

Chapter


265

It is noted that if 0 a :::; 1 and a < min(2r - 1 , 2>' - 1 ) , then we can improve Theorem 6.7 by using (2.63) . In this case, we have that

Il v -

:::;

IM,NVl l wJ(A2 ,HQ(A, )) :::; cM"- r l v I l H�(A2 ,W(A, » + cNfj- sll vI I H'(A2 ,HQ(Al » + cq ( j3 )M"->'Nfj-,,! I l v II H� ( A2 , H A (A, ») '

q( j3 ) being the same as in Theorem 6.7. Theorems 6.3 and 6.5 can be found in Bernardi, Maday and Metivet (1987) .

6.2.

Mixed Fourier- Chebyshev Approximations

This section is devoted to the mixed Fourier-Chebyshev approximations. Let A I , A 2 1 and n be the same as in the previous section. Set w(x) = w(x I ) = (1 - XD - 2 , and

(v, w) w = Define

10

1

v(x )w(x )w(x) dx, I l v ll w = (v, v) � .

{v I v is measurable, I l v ll w < We define the spaces L�(n), H� (n), W� ,p (n) and their norms in a similar way. We L� (n)

oo}.

=

also introduce some non-isotropic spaces as follows

H�, S(n) = L 2 (A 2 ' H� (A I )) n HS (A 2 ' L�(A I ) ) ' r, S 2: 0, l �, 8(n) = H�, 8(n) n H I ( A 2 , H:- I (A I ) ) n W - ( A 2 , H�(Ad ) , r, s X�, S(n) = H8 ( A 2 ' H:+ I (Ad ) n W+ 1 (A 2 , H:(A I ) ) , r , S 2: 0

M

2:

1,

equipped with the n9rms

The meaning of Vp(n) is the same as in the previous section. The closures of Vp (S1) in H�, s(n), M�, S (n), X�, 8(n) are denoted by H�:; ,w(n) , M�:; ,w(n), X�:; , w(n) respectively. Their norms are denoted also by I . IIH:" , I . for simplicity. The corre I . The space H;:�(n) is the sponding semi-norms are denoted by I . I ':; and I .

1 M:' " Il x.:;' · 1 M':;' " H "

Spectral Metbods and Tbeir Applications

266

closure in H:;;" ( n) of the set containing all infinitely differentiable functions with the period 211" in the x 2 -direction, etc. Let M and N be any positive integers. The subspaces lP M, ! , lP1t, l ' have the same meanings as in Section 6 . 1 . We first give some inverse in and equalities.

VN, 2 , VN,2 , VM,N

VA},N

Theorem 6.8. For any 1> E

VM,N and 2 ::; p ::;

00 ,

Proof. Let 1>1 ( Xl) be the same as in (6. 1 ) . According to Theorem 2.13, we deduce

that

1 1> I Loo whence

::; IILI �N I 1>dI LOO(A, ) (IILI�N ::; c ( MN) !

::; cMt

1 I ILI �N l 1>dI L�(A, ) l 1 1>d l h(A ) = c (MN ) ! II 1> ilw ')

1

2'

As a consequence of Theorem 6.8, we have that for any 1> , 'IjJ E

This completes the proof.

Theorem 6.9. For any 1> E

11 1> 'ljJ I I � ::; cMNII 1> II � II'ljJII � ·

•

VM,N ,

(6.12)

VM,N ,

Proof. Let 1>1 be the same as in (6. 1 ) . We see from the proof of Theorem 2.14 that

The mixed Fourier-Chebyshev expansion of a function v E L�(n) is

The rest of the proof is clear.

v(x)

= mL=O ILII=O vm,ITm( Xl)eilx2 00

00

•

Chapter


267

with the Fourier-Chebyshev coefficients

(

1 v(X)Tm(x l )e -ilx2 dx . Vm , l = 1r 2 Cl in The £ 2 (O)-orthogonal projection PM, N : L; ,,,, ( O) -+ VM, N is such a mapping that for any v E L; ,,,, ( O ) ,

or equivalently,

PM,NV ( X )

=

M E E vm ,l Tm ( Xl ) eilx• . m =O I I I:5 N

Theorem 6 . 1 0 . For any v E H;:�(O) and r, II v

�

s

;:::

0,

PM, NV Il", � C (M-r -+- N-' ) II v I l H':;'· .

Proof. Let vl (xd b e the same as in (6.3) , and PM : L�(A l )

orthogonal projection. Then

PM,N V (X)

=

-+ IPM,l b e the L�(Al )

E PMvl (x l )ei lx2 . Il l:5 N

Using Lemma 2.5, we find that

IIv - PM,NV I I

=

2 1r E I I Vl - PMvdl ii!,(At l

Il l :5 N

+ 21r EN II vdlii!, (A, ) l' I >

+ cN-2• EN W· ll vdli� (A, ) I l l> I l l:5 N cM-2r l l v l l p(A. ,H':;(A,)) + C N- 2 0 l l v llh'(A2 ,Li!,(A, ) ) .

� cM - 2 r E II vdlh':; (A' ) �

It is noted that for any v E H�:; ,,,, ( O) and r, s

;:::

•

0,

(6 . 1 3) When we apply the mixed Fourier-Chebyshev approximations to different semi periodic problems of partial differential equations, we adopt different orthogonal pro jections to obtain the optimal error estimations. For instance, the H� (O)-orthogonal projection Plt,N : H!,,,, ( O) VM,N is such a mapping that for any v E H!,,,, ( O) ,

-+


268

While the HJ.w e!1)-orthogonal projection PJ.,;.�N that for any v E HJ.p .j!1),

Theorem 6 . 1 1 . If v E M;:� (n) with r,

8

:

HJ.p .we!1)

--+

Vi} . N is such a mapping

� 1 , then for f1- = 0 , 1 ,

If in addition (6. 5) holds, then

Proof. Let VI (X l ) be as before, and Pit be the H!(A l )-orthogonal projection. Put

v. (X) = L: (PIt vl (X l ) ) ei lx2 . I I I� N

By Theorem 2.16,

Il v - pk. NVII i.w �
N

N cM2-2T. ILN ( I vdl �':;(A, ) + 12 1 I vd l�':;-1 (A, » ) I I$ + cN2-2 8 L: 12 8 -.2 ( I vd��(A' ) + 12 1 I vdl h(AJ ) ) II I >N l (M -T + Nl - ·r I v l t-.:;' · · c

The rest of the proof i s clear.

•

In numerical analysis of mixed Fourier-Chebyshev spectral schemes for nonlinear One of them is stated in the problems, we have to estimate the norm following theorem.

IIpif�NVllw.:;'p .

Theorem 6 . 1 3 . Let

If in addition

S

>

!.

v E x;:�(n), then

Proof. Firstly let

Clearly

(6.5) hold and r,

VI ( X l) be as before, and set

VI E HJ,w (A1 ) and vi E lP�,l . Define

v

If E HJ,p,w (n) n H; (A 2 ' H:(A l ) ) , then

270

Spectral Methods and Xheir Applications

w E HJ,w (A t }, then by Lemma 2.8, al,w (w, w) ;::: � l w l ��(A, ) + [2 I1 w ll h (A, ) . Moreover, for any w, z E HJ,w(A 1 ), If

al ,w (vi - vi, ¢i) ° for all ¢i E lP�,I ' we see that � l I v, - vill ��(A, ) + 12 11 vI - vil l i�(A, ) ::; al,w (VI - V; , VI - vi ) al,w (lil - vi , VI - ¢i) ,

Since

=

=

V

¢i E lP�,I '

Consequently,

� l I v, - vill ��(A, ) + [2 1 vI - vil l h (A, ) ::; ..j� G ll v, - ¢i 1 l ��(A, ) + [2 1 vI - ¢i ll h (A' » ) .

Now, let pJ/ be the same as in the proof of Theorem 6.12. By taking and using Theorem 2. 18, we deduce that If'

M,l

¢i pllv, =

� l I v, - vill ��(A, ) + 12 1 vI - vil l i�(A, ) ::; cM2-2r ll vdl �':;(A, ) .

(6.14) II vl - viIlH�(A' ) ::; cM!'-r llvdIH.:;(A') , J.I 0, 1 . We now estimate IIpli�NV I w.. . oo , J.I = 0, 1 . Obviously, I l Pli�NVllw.., oo ::; IILI $N ( 11 8r viIlLOO (A, ) + ( 1 + I J.l I !' ) I vi i Loo (A' » ) , J.I = 0, 1 . (6.15) We can bound the terms on the right-hand side of (6.15) . Let 1M be the Chebyshev

By means of the duality,

=

Gauss-Lobatto interpolation on the interval A I . Then

Using Theorems 2.13 and 2.19, we obtain that for r >

I � vi - IM8I VdI LOO(A, )

�,

cMtl l 8lvi - IM8I VdI L�(A, ) ::; cMt ( l vi - VdHMA, ) + 1 8lVI - IM8I VdIL�(A' » ) < cMt I vi - vdH�(A, ) + cMt -r l l vdI H':;+ ' (A') ' (6. 1 7)
!, then Il v - IM,NV IlHl'(A2 ,H�(A, )) ::; cM2or -r ll v Il Hf3(A. ,H':;(A, )) + cNi3- s llvI I H '(A2 ,H�(A, )) + cq( j3 )Mzor - .\Ni3 -'Y llvI I H� ( A2 ,H; (A, )) where q(j3) = 0 for j3 > ! and q(j3) = 1 for j3 ::; ! ,

n

n

.


272

Proof. We have with

D1 = D2 =

I l v - IMV I H!l(A2 ,H�(Al)) + I l v - INV I Hil(A2 ,H�(Al)) ' II (IM - I)(IN - I)V Il H!l(A2 ,H�(Al)) '

From Theorems 2.4 and 2.19,

For (3 >

!, we have

while for (3

:::;

!,


•

Theorems 6.10, 6.12 and 6 . 14 can be found in Guo, Ma, Cao and Huang (1992) , and Guo and Li (1995a) . Quarteroni (1987a) also considered this kind of mixed approximation. The filterings are also available for the mixed spectral methods. For example, let

¢(x )

=

M N

L: L: �m" Tm (X1)ei'x2 . m=O 1 ' 1=0

Following the same line as in the proofs of Theorems 2.26 and 2.27, we can prove that if 0 :::; r :::; 01 and 0 :::; S :::; 02, then In order to keep the spectral accuracy, we can take OJ = oj ( N ) and oj ( N ) N � oo, j = 1 , 2, see Guo and Li (1995b).

� 00

as

Chapter

6.3.


273

Applications

In the final section of this chapter, we take two-dimensional unsteady Navier-Stokes equation as an example to describe the mixed spectral methods. For simplicity, we focus on the mixed Fourier-Legendre spectral schemes. Let and be the same as in the previous sections, and - 1 , 1 , ° :s; :s; 21l'} . I and > ° are the speed, the pressure and the kinetic viscosity, respectively, and and t) are given functions with the period 21l' for the variable We look for the solution with the period 21l' for the variable such that

At , A 2' X2

00 = {x X l = Il U = ( U( l ), U(2 ) ) . Uo(x), Po(x) f(x, X2 . (U, P) X2 , Ot U (U . V)U - Ill::. U V P = f, E 0, ° t :s; T, E 0, ° :s; t :s; T, V · U = O, E 00, ° t :s; T, U = O, x E n, U(x, O) = Uo(x), X E n. P(x, O) = Po(x), +

+

X

X

X

0 U(x, t), P(x, t)

!, r � bI ll in for some tt E R.,. (T) , then for all t E RT ( tI ) ,

Theorem 6 . 1 b35 . Let 0 p(tI) e b2 t, �

2 rM N

(6.26),

= a =

(6.33) hold.

If

where bI , b2 and b3 are some positive constants depending on Il and II UM,N II C(0,T;W1 , OO )

•

We next deal with the convergence of (6.26) with 0 = a = 0 and {} > !. Let PM1 0,N U, U = UM,N - UM,N and P = PM,N - PM,N P , We get from (6.20) and (6.26) that

[

UM,N =

( DT U(t) +_ d(U(t) , UM,N (t) + U(t)) + d( UM,N(t), U(t)) , v) + J1,a(U(t), v) - b(v, P(t) +(hDT P(t))

=

( Gt (t) + G2 (t) + G3(t), v) - b(v, G4 (t)),

(3(DT P(t) , w ) + b(U(t) + (hDT U(t), w ) U( O )

=

p( O )

=

0

=

't v E VM,N , t E RT(T) ,

- ,8 ( DT P(t) , w) + (Gs(t) , w ) , 't

w

E WM ,N , t E RT (T) ,


278

where

Gt(t) = 8tU(t) - DTUM,N (t), G2 (t) = (U(t) . V) U(t) - (UM,N (t) . V) UM,N (t), G3 (t) = "21 (V · UM,N(t)) UM,N(t), G4 (t) = p et) - PM, N P(t) - ()rDTPM,N P(t), Gs(t) = V (U(t) - UM,N (t) - ()rDTUM,N (t)) . -

•

By Theorem 6.5 and (4.24),

I Gt (t) 1 I 2 � cr l U ll k' (t,t+T;L' ) + � (M-2r + N- 28 ) I U l k1 (t,t+T;Mr, , ) .

By Theorems 6.5 and 6.6, we deduce that for

I G2 (t) 1 I 2

r, s > 1 ,

1 (U(t) . V) U(t) - (UM,N (t) . V) U(t) 1 I 2 + I (UM, N (t) . V) U(t) - (UM, N (t) . V) UM,N (t) 1 I 2 c ( II U(t) lI � l,oo Il U(t) - UM,N(t) 11 2 + I UM,N (t) 1I100I l U(t) - UM,N (t)II� ) ( M - 2r + N-2. ) ( I U(t) lI �r+ l" + l + I U(t) I tvl ,00 )

� 2

� �

C

U = 0, we derive in a similar manner that I G3 (t) 11 2 = I G3 (t) - � (V . U)U f � (M-2r + N-2. ) ( l 1 U(t) ll �r+ l" + l + I U(t) l I tv" OO ) .

Since V .

Obviously,

•

C

C

I G4 (t) 1 I 2 � (M-2r + N -2B ) I P(t) lI kr" + c()2 r ll P l k1(t,t+T;£2 ) , I Gs(t) 11 2 � (M -2r + N -2S ) II U(t) lI�r+ l , ' +l + c()2 r l U l k'(t,t+T;M" ' ) . C

Finally, by an argument as in the proof of Theorem 6.15, we get the following result.

Let 8 = q = O, () > �, r � b4J.t in (6.26), and (6.33) hold. If for

Theorem 6.16.

r, s � 1 ,

n

n

n

n

U E C (0 " T · H0t,1' Mr+1,s+ 1 Wt,oo) Ht (0 T · Mr,s) H2 (0 T · L2 ) P E e (0 T · Hr," ) n Ht (0 T· L2 ) then for all t E Rr (T) , E ( U M,N UM,N , PM,N - PM,NP, t) � bs (� ( r 2 + M-2r + N- 28 ) +(3) where b4 and bs are some positive constants depending on J.t and the norms of U and ,

,

p

P

-

P

in the spaces mentioned above.

,

,

p ,

"

p

"

p

'

Chapter

6.

279

Mixed Spectral Methods

We find from Theorem 6.16 that the best choice is f3 = OCr) = 0 (M- 2 r + N-28 ) , and the corresponding accuracy is of the order f3! . It seems that the artificial com pression brings the convenience in calculations, but lowers the accuracy. Indeed, if we compare the numerical solution to other projection of the exact solution, called the Stokes projection, we can remove this problem, and get a better error estimation for any f3 2': O. We shall introduce this technique in the next chapter. Obviously, since we use the approximation (6.24) for the nonlinear term ( U · V)U, . a reasonable simulation of conservation (6.27) follows, even the trial functions do not satisfy the incompressibility. Also because of this reason, the effects of the main nonlinear error terms are cancelled, i.e. ,

(d (u(t), u(t)) , u(t)) = (d ([r(t), [r (t) ) , [r(t)) = O. Therefore the nice error estimates follow. · Conversely, if the trial function space VM, N fulfills the incompressibility, then it is better to approximate the convective term by

d( UM,N, UM,N) ,

d( v , w)

=

81 (w(I)V) + 82 (W( 2 )V) .

In this case, the conservation can also be simulated properly. For the error analysis, we have

�

(J (u(t), u(t)) , u(t)) = (lu(tW, V · u(t)) .

As V . u(t) =I- 0 usually, this term cannot be cancelled. This fact shows that a good approximation to the nonlinear convective term plays an important role. Bernardi, Maday and Metivet (1987) applied the mixed Fourier-Legendre spectral methods first to semi-periodic problems of steady Navier-Stokes equation. The mixed Fourier-Chebyshev spectral methods have also been used for semi periodic problems of various nonlinear differential equations, such as the vorticity equation in Guo, Ma, Cao and Huang (1992) , and the Navier-Stokes equation in Guo and Li ( 1996). Recently, the Fourier-Chebyshev pseudospectral methods have also been developed, e.g. , see Guo and Li (1993, 1995a) .

CHAPTER 7 COMBINE D SP E CTRAL METHOD S

In many practical problems, the domains are not rectangular. We might use some mappings to change them to rectangular and then use pure spectral methods to solve the resulting problems numerically. But it is not easy to do so. On the other hand, such procedures usually bring in the complexity and errors. Thus it is better to adopt the finite element methods in those cases. However, sometimes the intersections of the domains may be rectangular, such as a cylindrical container. So it is reasonable to use the combined spectral-finite element methods or the combined pseudospectral-finite element methods. Since finite element methods are based on Galerkin method with interpolations of functions, the latter seem more attractive. In this chapter, we present the combined Fourier spectral-finite element methods for semi-periodic problems, and the combined Legendre spectral-finite element methods and the combined Chebyshev spectral-finite element methods for non-periodic problems. Finally, we take Navier Stoke� equation as an example to show the applications of these combined methods.

7. 1 .

Some Basic Results in Finite Element Methods

In this section, we recall some basic results of finite element methods in one dimen sion, as a preparation of the forthcoming discussions. Let A = ( - 1 , 1 ) and x E A.. Decompose A into 2 M subintervals D..j , -M $ j $ M - 1 . hj represents the length of D..j . Let h = -M<max j <M-l hj . Assume that there exists a positive constant p inde- -

pendent of the partition such that IIl,ax h $ p. Let k be non-negative integer -M$J$M-l j and nh , k be the Lagrange interpolation of degree k over each D..j . Define h

Sh , k = {v I v is a polynomial of degree $ k on D..j , -M $ j $

280

M - I}.

7.

281

Chapter Combined Spectral Methods

n

Moreover Sh,k = th,k n Hl (A) and S�,k inverse inequalities exist.

= Sh,k

Lemma ·7. 1 . For any E Sh,k and 1 ::5

p ::5 q ::5

HJ (A) . In the space Sh,k , several

00,

l1j

Proof. is a polynomial of degree at most k on each with the left extreme point = xU) . Let y x - xU») - 1, A = {y I l y l < I } and �(y) = (x). By virtue of

:( .

Theorem 2.7,

J

On the other hand,

Consequently, 1 I I I Lq(llj) ::5 c

from which the desired result follows.

Lemma 7.2. For any E Sh,k and 0

(��)

::5 r ::5

1

1

. - r;

1 I I I LP(llj ) •

1' ,

( h ) r -/I-

1 1 1 1 /1- ::5 c k2

I I l I r .

Proof. By Theorem 2.8 and an argument as in the proof of the previous theorem, we

deduce that Hence

1 1 1 1 /1- ::5

( h ) r-/I-

C k2

1 1 1 1 . ,

I'

=

0, 1.

The general result comes from induction and space interpolation. The L 2 (A)-orthogonal projection Ph ,k : L 2 (A) any v E L 2 (A),

--+

•

Sh,k is such a mapping that for

(7. 1 )

282


For simplicity, let c(k) be a generic positive constant depending on k. We have from Ciarlet ( 1 978 ) that for any v E � < r :5 k + 1 and 0 :5 I-' :5 min(r, 1 ) ,

W(A),

IJ v - n h,k V I I :5 c (k)hr-I-' Iv l r , I-' Il v - nh,kv lJwl',oo :5 c( k) hr-I-' Iv l wr,oo . Lemma 7.3. For

any v E Hr ( A ) , �

I I : , p, = 0, 1 . . I I I� N _ Finally the desired result comes from induction and space interpolation. The L 2 (51)-orthogonal projection Ph,k,N : L; (51) Vh,k,N is such a mapping that for any v E L; (51) , (v - Ph,k,NV, ¢» = 0, V ¢> E Vh,k,N. Set f = mi n ( r , k + 1 ) . For simplicity, Ph,k denotes the L 2 (A 1 )-orthogonal projection on Sh,k, 1 , and PN denotes the L2 (A 2 )-orthogonal projection on VN, 2 ' Similarly, Pl.,k stands for the H 1 (A 1 )-orthogonal projection on 8h,k, 1 and P�:� stands for the HJ (A 1 ) orthogonal projection 'on 8£,10, 1 ' Theorem 7.3. If v E H; , S (51), r > ! and s � 0, th en 1 1 &� ¢> 1 I 2 :::; C

-+

IIv

-

Ph,k,NVIl :::; c(k) ( hr + N - ' ) I l v l l w,

• .

Proof. Let

Then Ph,k,NV(X) = Using Lemma 7.3, we obtain that I l v - Ph , k,NV I 1 2

E

I I I� N

E Ph, vI ( 1 ) e il", x

k

I'I� N

I l v l - Ph,k vt l l i' (A ' )

< c( k ) h 2 r

E

I I I� N

+E

•.

I I I> N

I l v t l l i' (Al )

I l v t l l �' (A, ) + CN-2B

E

I'I> N

I Ws l l v t l l i' (A, )

:::; c( k ) h 2 r l l v l l i' ( A. ,w (AI l ) + CN-2B l v l �' (A2 , L2 (A, ) ) '

In particular, for any v E H�:;(51) , r >

!

-

and s � 0,

The H 1 (n)-orthogonal projection Pl.,k,N : H; (n) for any v E H; (n) ,

( V (v - Pl,k,N V ) , V ¢» + (v , ¢»

-+

= 0,

Vh,k,N is such a mapping that V ¢> E Vh,k,N '

286


The H6 (n)-orthogonal projection for any E H6,p (n) ,

v

P�:�,N

:

( v (v - P�:�,N V) , V¢»

H6 ,p (n)

-+

V�k,N is such a mapping that v,.�k,N '

If v E M;, O ( n ) and r, s � 1, then Il v - Pl,k,N vt :s; c( k ) (hr- I + NI - . ) (hl -1' + NI'-I ) I v Il M" " If, in addition, for certain positive constants Cl and C2 , = 0,

V ¢>

E

Theorem 1.4.

p. =

0, 1 .

clN :s; h1 :s; C2 N,

then

(7. 10)

Proof. Let VI (Xl) be as before, and

v. (x) = LN (pl, k VI (Xl)) eil"' I) I �

By Lemma 7.4,

Il v - Pl, k ,N V I :

:s;

:s;

:s;

:s;

•.

I v - v. l I � v -P V L v -P V I I I �N ( 1 l l,k II I :'(A ) + 12 1 I l,k III :' (A1 ) ) + LN (I I vdlk1(A, ) + 12 1 I vdl i' (A, ) ) II I > c( k) h 2r- 2 ILN (i l vdl k' (A') + 12 1 I vd l k.- 1 (A, ) ) I I� + cN2 - 2 • LN 12 0- 2 ( 1 I vdl k1(A') + 12 1 I vdli2 (A1 ) ) III> 1 k) c( (hr + NI -O r Il v l �.. . . 1

S o the first conclusion i s true for

If v E M;" (O)

n

p. =

1 . The rest of the proof i s clear.

and r, s � 1 , then I v - P�:�,N vt :s; c( k ) (hr- 1 + NI -o ) (h l -1' + NI'-I ) I v I M" " If, in addition, (7.10) holds, then

Theorem 1 . 5 .

H6,p (n)

p.

(7. 1 1 ) _

= 0, 1 .

Chapter

7.

Combined Spectral Methods

287

Proof. Let VI(Xl) be as before and

v. (x) 1 L:1:5N (p�:�vl (xd) eil:r2 . Il v - p�:�,NV l l � c in! I v -wi t � cl v - v.l l · =

Then

W EVh,k,N

By an argument as in the der�va.tion of (7. 1 1 ) , we prove the first result for Finally by means of the duality and (7.10), we complete the proof.

Il p�:�,NV l wr,p . Let (7.10) hold, k and r, > ! . For any v H; (A 2 ' Hr (A 1 )), Il p�:�,NV I LOO � c( k) l I v I l H' (A2 ,W(Ad) · If, in addition, v then We can also bound the norm

�

Theorem 7.6.

E

1

s

E

fJ, =

1. _

HJ,,,(fl)

n

X;" (fl) ,

VI (Xl ) be as before and pl,k,NV (X) 1 L:1:5N vi (xl) eil:r2 . Obviously vi S� , k , l . By an argument as in the proof of Theorem we know that for any S� , k , l ' I VI - vil l �' (Ad 12 vI -vil l 12 (Ad � c( l I vl - l k1 (Ad 12 vI - 1 I 12 (A') ) · By taking P�:� v" we obtain from Lemma that Proof Let

E

=

E

=

6.6,

+ 11

+ 11

7.5

By means of the duality, It is clear that for fJ,

=

( 7.12) 0, 1 ,


288

We estimate the terms on the right-hand side of (7. 13). Firstly, we have

1 81 ViI I LOO(A,J :::; 1 81 Vi - nh,k 81 VdI LOO (A, ) + 1 81 v/ - nh,k 81 VdI Loo (A, ) + 1 81 VdI LOO (A, ) ' (7. 14) From Lemma 7. 1 , ( 7. 2 ) and (7.12), 1 81 vi - nh,k Bt vdI LOO (A, ) :::; c(k) h - t ( l i vi - vdI Hl (A, ) + 11 81 v/ - nh, k 81 vdI L2 (Atl ) < C(k) ( l I v dI H!(Al) + hr- t l vdlw+ l (Atl ) . By (7.3),

11 81 v/ - nh,k 81 VdI Loo (A1) :::; c(k) Il vdl w+l (A1) '

By imbedding theory,

11 81 VdILOO (A1 ) :::; I vdlw+ l (Al) '

The above statements yield that

:::; c(k) /EN Il vdl w+ l (Atl I I� 1 " " :::; c(k) ( I /EI �N ( 1 + 12 r ' ) ( I /EI �N ( 1 + [2 ) " I Vdl kr+ l (Atl ) (7.15) :::; c(k) l I v I l H'(A2 ,W+ l (Al)) ' 1

Similarly, it can be shown that

v , I /EI �N I iIl LOO (A, J :::; c(k)llvI l H'(A2 ,W (At l ) 1 E v I / I �N III I i I Loo (Atl :::; c(k) l I v I l H·+ (A2,W(At l ) ·

The combination of (7.13) and (7.15) -(7.17) completes the proof.

(7. 16) (7. 1 7) •

Finally we consider the combined Fourier pseudospectral-finite element approxi be the Fourier interpolation on the interval mation. Let associated with the set of interpolation points. Define = = Set 2� [ = = 2N 1 E

IN

A2 , h,k,N Ph,k IN INPh,k ' 1· dX N v(x)W(X) l (v, v v) ' II l i k i + "'2 EA2,N A1

(V, W)N Then for any v E e (A 2 ' L2(Al )), (v - Ih,k,Nv, ( ' , X 2 ) E lPN, 2 ' By Lemma 7. 1 and Theorem 2.7,

ck 11 ¢> II Loo :::; sup I I ¢> I I LOO (AI ) :::; . fL sup 11 ¢> lI v ( A I ) xEA2 x2 EA2

yh

� ( r sup ¢>2 (X) dX I ) t :::; ck: II ¢> I I . yh y h lA, x2 EA. c

E li,k, N and 0

Proof. Let

:::; r :::; 11 ,

N

1=0

¢>(x) = :L ¢>I(X I )L,(X 2 ) ' Then ¢>,(X I ) E Sh,k,l ' As in the proof of Theorem 7.2, we have (7.18)

On the other hand, 02 ¢>( X) =

N

:L ¢>1 (X I ) 02 L,(X 2 ) ' 1=0

By (2.49) and the same argument as in the proof of Theorem 2.8,

II02 ¢>(X I , · ) II L' (A. ) Therefore

N

:::; :L fi{l+l) I ¢>,(X I ) I :::; cN2 11 ¢>(XI , · ) ll v (A. ) · 1=0

r

II ¢> (X I ' · ) l I i' (A. ) dX I = cN4 1 1 ¢> 1 1 2 I I 02 ¢> 11 2 :::; cN4 lAl

which together with (7.18) imply the result for 11 = 1 and r = O. The proof is complete by induction and space interpolation. • The L 2 (n)-orthogonal projection Ph,k, N : L 2 (n) Vh,k,N is such a mapping that for any v E L 2 (n) , (v Ph,k,NV , ¢» = 0, 'if ¢> E Vh,k, N .

-

-+

Chapter

7.

Combined Spectrill Methods

Theorem 7.10.

If v E HT"' (!1) , r > !

291

and s :2: 0, then

Proof. Using Theorem 2.9 and Lemma 7.3, we find that Ilv

- Ph,k,NVi l

In particular, for

::; II·v - PN Ph , k Vl l ::; IIPN (v - Ph,k ) II + IIv - PNvi l ::; �( k ) ( hr llvllu(A" H'(A' ) + N- s llvllH' (A,p (A1» ) < c( k) ( hr + N -' ) IIvll w,"

v E H�"

•

(!1) ,

The HI (!1)-orthogonal projection for any v E H I (!1) ,

Pl,k,N

:

HI (!1)

-+ Vh,k,N is such a mapping that

(\7 (v -PI, k,N v) , \7 9'» + (v, 9'» 0,

The HJ (!1)-orthogonal projection any v E HJ (!1) ,

P�:�,N

=

:

HJ (!1)

V 9'> E

Vh,k,N .

-+ Vh�k,N is such a mapping that for

If v E MT, S (!1)

and r, s :2: 1, then fo r J.L = 0, 1, Il v - Pl,k,N v t ::; c( k) ( hr-I + h-I N- s + NI- S ) ( hl - 1L + NIL-I ) IIvIIM" " If in addition (7.10) holds, then

Theorem 7 . 1 1 .

Proof. Let v.

=

Ph,k P�V Vh,k,N . It is easy to see that E

Using Lemma 7.3, we obtain

IIv

-

Ph,kVl l �

0,

I!PN V I l LOO(A.)

� �

c I!PNV - vIILOO(A.) + IIv Il LOO(A.) c (l + In N)N-" IIv l lw&. OO (A.) + IIv I l L OO(A.) '

Chapter

7.


Taking Jl
O.

In the same manner, we verify that

B1 � B2 � B3 �

c(k ) ll vI I H!(A2 ,Hl (Al» ' c(k)ll v I H�(A2 ,L2 (A, ) ' c(k)ll v I Hi + 6 (A2 ,H � + ' (A, ) '

6'

> o.

The above six estimations together with (7.33) provide the bound of •

I l p�:2,N V l wl 'OO '

Finally we consider the combined Chebyshev pseudospectral-finite element ap proximation. Let IN be the Chebyshev-Gauss-Lobatto interpolation on the interval 11. 2 , associated with the interpolation points x� ) and the weights w (j ) , O � j � N. Define = Ph,kIN = INPh,k . Set N 1 (

h,k,N (V, W)N = � JfAI V (X t , X 2(j») W (X t , X 2j» ) (j) dx t , I vll N = (v, v) 1 · Then for any v E e (11. 2 , L2 (A 1 )), (v h,k,NV , ¢» N = 0, V ¢> E Vh,k,N ' Theorem 7 . 2 1 . If v E H�,S(f!) and r, s > �, then W

-

Theorems 7.17, 7.19 and 7.20 can be found in Guo, He and Ma ( 1995) .

Proof. The result follows from Lemma 7.3 and Theorem 2.19. 7.5.

•

Applications

In this section, we again take the two-dimensional unsteady Navier-Stokes equation as an example to describe the combined spectral-finite element methods. We focus on the combined Fourier pseudospectral-finite element schemes. But the main idea and the technique used can be generalized to non-periodic problems defined on a

Chapter

7.


301

three-dimensional complex domain, and to other kinds .of combined spectral-finite element methods and combined pseudospectral-finite element methods presented in the previous sections. Let if, P and I' > 0 be the speed vector, the pressure and the kinetic viscosity. We consider the semi-periodic pr.oblem and require P ( x , t) to be in the space and A ( P ( x , t)) = for 0 :::; t :::; T. The L� ,in) , i.e. , P has the period for bilinear forms a ( v , w) and b ( v, w) are given by We adopt the artificial com pression ( .2 3 ) with the parameter (:J � O. When (:J = it is reduced to the original problem and so requires the incompressibility automatically. As explained in Section 6.3, a reasonable approximation of non-linear convective term plays an important role in preserving the global property of the genuine solution and in increasing the com putational stability and the total accuracy of n�merical solution. But it is not easy to do so for pseudospectral methods, since the interpolation and the differentiation do not commute. Besides, for the improvement of stability, a filtering will be used. Now let � 0 and N > 0 be integers. We use all the notations of Section 7.2. For simplicity, let be the set of scalar be the set of vector functions and functions, defined by

(6. 22) 271" .X 2 (6. 21 ).

6

0

0,

k

Vh,N

Wh,N

Wh,N = {w I W E Sh,lc,l VN,2 ,A(w) = O } . ®

The interpolation n h , lc+ 1 is denoted by nlc+ l for simplicity. Furthermore, for v (x)

N

=

L v/ (xl ) e i/",. , /=0

the filtering with single parameter a � 1 is given by

( ) v = � ( 1 - 1 � 1 " ) v/ ( xl ) ei/"'2 . 1 1 D(v , w , z) = 2 (( w . V') v , z) - 2 ( ( W . V')z, v) .

RN V = R

N a

Let

If V' . w = 0, then D( v , w , z) = (( w . V' ) v , z) . Thus we define the following trilinear form d(v, w , z) = 2 . V')z), v) . V') v ) , z) 2

1 (IN((w ·

1 (IN((w

We shall approximate the nonlinear term by d(v , v, z) . Clearly -

d (v , w , z) + d( z, w , v) = o.

(7.34)

302


Guo and Ma (1993) proposed a scheme for (6.22) . Let 7 be the mesh size in time. Uh,N and Ph,N are the approximations to U and P, respectively. 8, and 0 are parameters, 0 ::; 8, 0 ::; 1 . The combined Fourier pseudospectral-finite element scheme for (6.22) is to find Uh,N(X, t) E Vh,N, Ph,N (X, t) E Wh,N for all t E Rr(T ) such that (DrUh,N(t) , v ) (RNUh,N(t) 87 RNDrUh,N(t), RNUh,N(t), RNV) (Uh,N(t) DrUh,N(t), v ) - b (V, Ph,N(t) 0 7 DrPh,N(t » v E Vh,N, t E J4 (T), = (IN nk+1 J(t) , v ) , 0 [1 (DrPh,N(t), w) b (Uh,N(t) 7 DrUh,N(t), w) = 0, w E Wh,N, t E J4(T),

a-

a-,

+d + a-7 +

+/La

+

+

+

Uh,N(O) = IN nk+1 Uo , Ph,N(O) = IN nk+1 Po

V

(7.35)

where Po is determined by �Po = 'V . (f - (Uo ' 'V) Uo) .

a-

If 8 = = 0 = 0, then (7.35) is an explicit scheme. Otherwise a linear iteration is needed to evaluate the values of unknown functions at the interpolation points. In particular, if 8 = = 0 = �, then we get from (7.34) and (7.35) that 2 1 2 2

a-

II Uh,N(t) 1I =

+ f3 IIPh,N(t) 1 I + 2/L 7 Bellr(t-r) 1: IUh,N(S) + Uh,N(S + 7) 1 1

l I u h,N( 0 ) 11 2 + f3 I 1 Ph,N( 0 ) 1 I 2 + 7 1: ( IN nk+ 1 J(S) , Uh,N (S) + Uh,N (S + 7 » . BeRr(t - r)

Clearly it simulates the conservation (5._6 7) properly. We analyze the generalized stability of scheme (7.35). Let constant such that for any v E Vh,N ,

CI = cI ( k ) be a positive (7.36)

Assume that

a- > -21 '

or

A = 7 ( h - 2 + N2 ) < /LCI ( 12- 2a-)

Let c be a suitably small positive constant and qo > two cases, ( i ) 20a- ;::: 0 ( ii ) 20a- ::; 0

O.

(7.37) '

We consider the following

+ a- + 2c and 20 ;::: 1 + 2c + qo, + a- + 2c and -1 20 ;::: (1 + 2c + qo + /LACI( a- + 2c » ( 1 - /LACI G - a-))

(7.38) (7.39)

Chapter

7.

303


Uh,N(O),Ph,N (O), !N nk+1 j and the right-hand side of the second uo , p, j and g , respectively, which induce the errors ( 7.35 ) , Ph, N u p for simplicity. Then from ( 7.35), it follows that Uh,N (DTU(t), v) + d (RNU(t) + 87" RNDTu(t), RNUh,N (t) + RNU(t), RNV) + d (RNUh,N (t) + 87" RNDTUh,N (�)' RNU(t), RNV) + J-Ia (u(t) +0"7" DTu(t), v) - b (v,p(t) + 97" DTP(t)) = (j(t) , v) , v E Vh,N , t E R, (T), ,8 (DTP(t), w ) + b (u(t) + 97" DTu(t), w) (g(t), w ) , V w E Wh,N , t E �( T). ( 7.40 ) Take v = u + 97" DTu in the first formula of ( 7 . 40 ). Since 1 297" (DTu(t), j (t))1 � e J-l7" 2 I D.Tu(t) 1 21 + c9e 2 1 Jet) 1 2- 1 ' J-l we get from (1.22) and ( 7 .34) that DT I l u(t) 11 2 + 7"( 29 - 1) I DTu(t) 1 I 2 + J-I(2 - e) l u (t) l � + J-I7"(0" + 9)DT l u (t) l � +J-I7" 2 (290" - - 9 - e) I DTU(t) l � - 2b (u(t) + 97" DTu(t),p + 97" DTP(t)) ( 7 .41) + t Fj (t) � C (le: 92 ) I l j(t) l � l

Suppoee that have the errors formula of denoted by and and of

{

=

0"

where

2d (RNUh,N (t) + 87"RNDTUh,N (t), RNU(t), RNU(t)) , 297" d (RNUh,N (t) + 87" RNDTUh,N (t), RNU(t), RNDTu(t)) + 27"( 9 - 8)d (RNU(t), RNUh:N (t), RNDTu(t)) , = 2 7"(9 - 8)d(RNU(t), RNU(t), RNDTu(t)) . Next take w p + 97" DTP in the second formula of ( 7. 4 0). We obtain that ,8DT I p (t)1 1 2 + ,87"(29 - 1 - e) I DTP(t)1 I 2 +2b(u(t) + 97"DTu(t),p(t) + 97"DTP(t)) � ,8 I p(t) 11 2 + (� + :;) I g (t) 1 2 . ( 7.42 ) Putting ( 7 .41) and ( 7. 4 2) together, we derive that F1 (t) F2 (t) F3 (t)

=

( 7 . 43)

Spectral Methods

304 We are going to estimate

an d

Their Applications

I Fj (t) l · Let v, w, z E Vh,N and

By virtue of (2.23), Theorems 2.1, and Theorem 2.4,

Bq(v, w , z) � l (wOqV, Z)NI + l (wOq Z, V)NI � c(k) lIwll (IIIN(zoqv) II + Il v llLoo lzh) � c(k)lI w ll Olzoq VII + II v IlLoo lzh ) · Moreover, for any 0

< 'Y
O.

We know from Theorem 7.22 that if (j > � , then the restriction o n A disappears and so we can use larger T . If (J = Ii > 2""- 1 for (j > then the restriction on also disappears and so the index of generalized stability, This improves the stability essentially. Indeed, it is the best result. On the other hand, since we adopt the operator the effect of the main nonlinear term is cancelled, i.e.,

�,

p e t)

s = -00.

d(v, w, z),

d (u(t), u(t) , u( t)) = o. This fact again shows the importance of a reasonable treatment with the leading nonlinear terms in the approximation of nonlinear problems. We are going to analyze the convergence of scheme (7.35) . In order to obtain the uniform optimal error estimation for (3 � we first consider the corresponding Stokes problem, which is itself very important in fluid dynamics. Let be the dual space of (HJ, p( O ) and = L�,p(O) . ( ' , - stands for the duality between and Let "I and = The Stokes problem is to find and such that

0,

V. pEW

V=

E V'

r

W E W e ' W. ,

{ b(au(,uw)v)=- b(v,p) w),

)

V'

uEV

V'

( Tf, v ) , 'v' v E V, (7.48) (e, 'v' w E W Let Vh,N be the trial function space of the speed vector u. We approximate u and p by u h,N E Vh,N and ph, N E Wh,N satisfying { l-'ab (u(h,uNh,N, w, )V)=-(e,b (w),v ,ph,N) = (Tf, v) , 'v''v' wv EE Vh,Wh,N'N (7.49) ' I-'

=

(7.50) is a combined Fourier pseudospectral-finite element approximation to (7.49) . Following the same line as in Girault and Raviart (1979), it can be shown that 7 . 5 has a unique solution provided that the following BB condition ( the inf-sup condition ) is fulfilled, sup 7.5 ) �

( 0)

vE V':.N

b(v, w) cllwll , 'v' w E Wh,N ' 11 V 11 1

---

( 0

Chapter

7.


3 07

This condition is fulfilled for Vh,N = ( S�,k + ,l 181 VN, 2 , see Canuto, Fujii and Quar 2 teroni (1983) . If the components of trial functions belong to the different spaces Ah,N and Bh ,N , then it is denoted by Vh, N = Ah,N X Bh ,N. It was pointed out that ( 7.5 1 ) holds for Vh, N = S�, k + 2,l 181 VN, 2 X S�, k + l,l 181 VN, 2 . G o and Ma (1993) improved this result essentially, stated as in the following lemma.

)2

(

Lem m a 7.6.

If Vh,N

(7.51) is fulfilled.

) (

= S� , k + 1,l 181 VN, 2

(

) �

) X ( S�,l,l

181

VN, 2 )

and (7.10) holds, then

We define the linear operator P : V x W Vh,N X Wh ,N by P. ( u , p ) = ( P.u , Pop) = h (u ,N , ph ,N) where (uh,N , ph,N) is *the solution of (7.50 ) . Let f = min(r, k + 2). Using

-+

Lemma 7.6 and an abstract approximation result in Girault and Raviart (1979) , we obtain the following result. =

M; , 8 (f2) n HJ,p ( f2 ) , p E H;- l, 8- l (f2) n L5 ,p ( f2 )

Lemma 7.7.

If U J

E

Let Vh,N

Vh ,N '

(7.50) has a unique solution. Moreover if u and r, 2:: 1, then s

l, s- l (f2) n L2O (f2) Hrp , s(f2) n HQl,P (f2) ' p E Hr,P P

and r, s > 1 then -

By (7.52) and Theorem 2.26, it can be shown that if u H�" - l (f2) n L5 ,p ( f2 ) and 1 ::; ::; lX, then

s

We now deal with the convergence. Let (6.22) that

U*

E

=

P.U, p .

E

=

J

M; , S(f2) n HJ (f2) , p E

PoP. We deduce from

(D.,. U*(t), v) + d (RNU*(t) + 8r RND.,. U*(t), RN U*(t), RNV) v) - b (v, P*(t) + (hD.,. P *(t)) +/l a (U*(t) + O'rD.,.U*(t), 7 = (IN nk+ 1 f(t), v) + L Gj (v, t), V v E Vh ,N , t E R.,. (T) , j=l f3 (D.,. P *(t), w) + b (U·(t) + Or D.,.U*(t), w) = ( Gs(t), w) , V w E Wh ,N , t E R.,. (T), (7.54)


308 where

G1(v, t) = (D'TU(t) - 8tU(t), v) , G2 ( v, t) = (D'TU* (t) - D'TU(t), v) , G3 (V, t) = d (RN U*(t), RN U*(t), RNV ) - D (U(t), U(t), v) , G4(V, t ) = 07d ( RN D'TU* (t), RNU* (t), RNV ) , G5(V, t) = O'wra (D'TU*(t), v) , G6 (V, t) = -8rb (v, D'T P* (t)) , G7(V, t) = (f(t) - IN nk+ l f(t), v) , Gs( v, t) = (JD'T P* (t) . Let

U = Uh ,N - U* and F = Ph,N - P*. Then by (7.35) and (7.55) , we obtain that

(D'T U (t), v ) + d (RNU(t) + Or RN D'T U (t), RNU* (t) + RNU (t), Rv ) + d (RN U* (t) + Or RND'T U* (t), RNU (t), RNV) + p,a (U (t) + 0'7 DT U (t), v) -b (v, F(t) + 8rDTF(t) ) = - � Gj (v, t), V v E ltJ. ,N , t E R,. (T) , j= l (J (DTF(t), w) + b ( U (t) + 8r DT U (t), w ) = - (Gs(t), w) , V w E Wh,N , t E R,. ( T) , U ( O ) = IN nk+ 1 Uo - U*(O), F(O) = IN nk+ l Po - P *( O ) . 7

(7.55) We now estimate the terms on the right-hand side of (7.56) . Firstly by (4.24) and (7.53 ) ,

7 � II DTU(s) - OtU(s) 11:1 ::::; c(k)r 2 IJ UI I�(o,t;H- I)' seRr(t-T) r � IIDTU*(s) - DTU(s) 1I 2 aeAr (t-T) ::::; c(k) ( h 2f + N - 2 a ) ( IJ U II�(o,t;H" ') + II P II11(o,t;H.-I -I) ) . To estimate the nonlinear terms, we first let v, w, Z E Vh,N , ;::: I + � and I > O. Set .•

Aq = I «RNIN - I) ( RNW8qRNV ) , z ) l , Bq = I(IN ( RNW8qRNZ ) - RNW8qz , RNV ) I ,

S

q = 1 , 2, q = 1, 2.

By Theorem 2.26 and an argument as in the proof of Lemma 10 of Guo and Ma ( 1991 ) , we know that if 0: ;::: max ( s - p" 1 ) , P, = 0, ":"' 1 and s > !, then for any

Cbapter

7.

309

Combined Spectral Metbods

II RNIN (RN V RNW) - RNvRNwII HI' (A, ) :::; c(k) NI' -· (I v I H '(A, ) l w Il H"Y+ ! (A, + I w I H '(A, ) l I vI I H"Y + ! (AJ . )

(7.56)

By virtue of imbedding theory and (7.57) , we deduce t hat

Aq :::; I (RNIN - 1) (RNWOq RNV) 1I -1 1 I zl h < c( k) N - S (i, (I Oq vl � ' -l (A2) I w ll �"Y+ !. (A ') + l w I 1'-' (A2) I Oq v l �"Y+ ! (A2» ) dX l ) t 1I ;l h :::; c(k) N -S (IIOqvIl H ' -' (A2 P(A' ) Il w Il H"Y+! (A" H"Y + ! (A ') + I Oq v Il H"Y+ ! (A, P (A, )) I w I l H' -l (A" H"Y+! (Al)) ) I l zlh · Similarly,

Bq :::; I (RNIN (RN v RNw) - RN V RNW, Oq z ) 1 " :::; c(k) N - · (i , ( I vl � '(A') II w ll �"Y+ ! (A + I w I 1 '(A, ) I H �"Y+! (A ) dX l ) Ilzl h 2» 2) :::; c( k)N-S (1 I vII H '(A,P(A l)) ll w I H"Y + ! (A , H"Y + ! (A )) 2 ' +l I w Il H' (A2 P(Al)) I I v ll H"Y+ � (A2 ,H"Y+ ! (A' ) ) Il z l k 1

The above arguments imply .that

from which and (7.54), it follows that

(RNU* (t) , RNU*(t), RNV) - D (RNU* (t), RNU*(t), v)1 (7.57) :::; c(k)N - S ( 1 I U(t) I Ml " + I P(t) I H' -' (A2 P (A, )) f I l vlll. Similar to (7.44) and (7.45), we have from Theorem 2.26 and ( 7 . 53 ) that for r � 1 , I D (RNU*(t), RNU*(t), v) - D(U(t), U(t), v)1 :::; c(k) ( II U(t) I M',"Y + l + II U*(t)II Mlo"Y+ l ) II RNU*(t) - U(t) ll ll vl h (7.58) :::; c(k) ( hi' + N - s ) ( I U(t) I Mr" + I P(t)I I Hr - " , -, ) 2 1 I vl k Id

310


The estimates (7.58) and (7.59) lead to the bound for have from Theorem 2.26 and (7.45) that

I G3 ( v , t ) I I ' Furthermore we _

Moreover, by (7.53) and (7.54) ,

11 U* (s) l I �l"'+l II DrU* (s) 1I 2 � c(k) r2 ( 1 U 1 I &(o,t ;Ml ,..,+ l ) + I P I &(o,t ;H-r (A2 P (Ad » ) ( 1 I U Il h1 (o,t;Hl ) + I P l h1 (o,t ;L2 » ) ' We can estimate I Gs(v, t) 1 and I G6(v,t)1 in the same manner. Also by Theorem 2.4 and (7.2) , I G1(v , t ) 1 I � c( k ) ( hi' + N-' ) Il f(t) l w, . l v l . r3 L

.eR. (t-r)

In addition,

1 0 ( 0 ) 1 � c( k ) ( hi' + N -B ) ( ll UoII Hf + IIPoII Hf-1 , . -d , i P (O) i � c( k) ( 1 Uo lll + I PoI I H' + 6 (A. ,H ' + 6 (Al » ) ' 6 > O. .•

Finally we apply Theorem 7.22 to (7.56) , and the convergence follows.

Assume that (i) (7.10) and condition (i) of Theorem 7.22 are fulfilled; (ii) for certain constants r � 1 and s > �, U E C (0 , T', Mpr,. Jio,l p) HI ( 0 , T ,· Hr,B ) H2 (0 ' T · H - I ) P E e (0 , T', L02 ,1') HI (0 , T' Hr - l ,.- l ) f E e (O , T H; ) (iii) for certain t l E R,. ( T ) and certain positive constant b2 , ) eb2 t1 (r2 h2i' + N-2• + (3) - r Np.hc(k 6 ( 9)2 . Then for all t E Rr(tl) , I l uh,N (t) - U* (t) 1 I 2 � ceb2 (r2 h 2i' + N- 2• + (3) where b2 is a certain positive constant depending on k, f3, and the norms of U, P and f in the mentioned spaces.

Theorem 7.23.

n

i

n

i

"

n

n

,

,

P

. w dS = - is zw dS, V w E 1)( S ) ,

,

A,

then we say that z is the derivative of v with respect to denoted by 0>. v etc. Furthermore, we can define the gradient, the Jacobi operator and the Laplacian in the sense of distributions. For example, if

is v 6.w dS = is zw dS, V w E 1)( S ) , then we say that

z = 6. v, etc. Now, let (v , w) = is vw dS, I lvl l = (v , v)� ,

and

L2( S ) = {v E 1)'( S ) I lI v ll

< oo } .

Furthermore, define

equipped with the semi-norm and the norm as

For any positive integer r, we can define the space Hr (s) with the norm I . Il r by induction. In particular, the norm I . 11 is equivalent to (11v112 II 6. v I 12)! . For any 2 real r 2: the space Hr (s) is defined by the interpolation between the spaces H[r]

0,

+

Chapter 8. Spectral Methods on the Spherical Surface

315

the space H: (S) is the duality of H - r (s) . Particularly, and H[r] + l . For r < 2 O H (S) = L (S) and Il v l l o = Il v ll . It can be verified that for any v , w H 2 (S) ,

0,

E

(8 . 1)

- (�v , w ) = (\7v , \7w ) .

. The space C(!1) i s the set of all continuous functions o n S, with the norm I I · 1 1 0 . Lemma 8 . 1 . For

C(S) .

any

0 � J.L � r , HT (S)

'-t

HI-'(S) . For

any

r >

0, W+1 (S)

'-t

Proof. The first assertion follows immediately from their definitf�ns. We now prove

the second one. Let B be the unit ball in the three-dimensional Euclidean space, and w be a function defined on B. We denote by Tw the restriction to S of w . We can take HT + 1 (S) to be the trace space of HT + � (B) with the norm II v IIW+l (S) =

By imbedding theory, W + � (B)

On the other hand, for any v and

'-t

inf

weHr + � (B) Tw=v

Il w li H "+ � (B l "

C(B) , and so for any w E Hr + � (B),

E HT+1 (S), there exists w E HT+� (B) such that Tw = v

Consequently, Il v l i o(s) = sup Iv(x) 1 = sup Iw(x) 1 xES xES

su� Iw(x) 1 � c Il w(x) I i H "+ � (B) � 2cll v ll w+I (S) . � xEB

This implies the assertion. Let

L� (S) = Lemma 8 . 2 . For

any

•

{ v E L2 (S) I A (v) = O } ,

vE

Hl (S) n L6(S),

A(v) =

is v dS.


316

Co being a positive constant independent of v .

Proof. By the Poincare inequality, we claim that and the desired result follows. Lemma 8.3. If v

E

L 2 (S), w

•

E

Hr+ 1 (8)

and

r

> 0,

then

I l v w ll :s c ll v ll ll w ll r+ 1 '

Proof. By Lemma 8 . 1 , •

L I (x)

We next turn to the L 2 (8)-orthogonal system. Let be the Legendre poly nomial of degree I. The normalized associated Legendre function is given by

{

+ l ) (m - I)! (1 X 2 ) t fi Lm ( X ) , . I,m ( X ) = (2m2(m + I) ! L I,m (X) = L _I,m (x), Moreover, the spherical harmonic function Yi, m (..\ , 0) is

L

_

x

for 1 2 0, m 2 for I < 0,

m

III,

2 I ll .

(8.2)

(8.3)

The set of these functions is the L2 (S)-orthogonal system on 8, i .e. , 21r 1r = 0) cos 0

la l� Yi,m (>', O) "f/',m'(>" 2

dOd>' 81 ,118m,ml•

The spherical harmonic expansion of a function v E L 2 (8) is 00 v= L L

v (>', O) 1= - oo m� 1 1 1 I,mYi,m

where the coefficient

VI,m is given by VI,m = l" l: v ( >. , O) "f/,m (>', 0) cos 0 dOd>'. 2


It can be checked that

- sYl,m ( \ 0) = m ( m

317

+ 1 )1I,m (A, 0) .

(8.4)

So 1I ,m ( A , O ) is the eigenfunction of the operator - � on S, corresponding to the . eigenvalue m ( m 1 ) . Thus in the space Hr ( s ) , the normJ v ll r is equivalent to

+

mr ( m

( f :E

1= - 00 m?: 111

VN = span {1I ,m 1 1 1 1

+ 1 )" I VI,m I 2 )

1

2

(8.5)

Now let N be any positive integer, and define the finite dimensional space VN as

:::; N, I I I :::; m :::; M ( l)} . Clearly, VN depends on the choice of M ( l ) . Usually we take M ( l ) N or N + I l l . For fixing the discussion, we take M ( l ) = N . Let VN b e the subset of VN containing �

all real-valued functions. In this space, we have the following inverse inequality. Theorem 8 . 1 . For any 1>

E

VN and

j.t ,

0 :::; r :::;

:E :E JI,m1l,m (A, 0) . 1> = 1=-N m=111

Proof. Let

N

N

Then by (8.5) ,

<
I I ; · •

The L 2 (S)-orthogonal projection PN : L 2 (S) any v E L 2 (S), or

equivalently,

PNV(A, 0) =

N

N

L L

1= - N m=111

-+

VN is such a mapping that for

vI,m1l,m (A, O ) .

318

Spectral Methods and Thei� Applications

Theorem 8 . 2 . For any v E Hr (s) and 0 :::;

p.

:::; r,

(8. 5), L mJ.l ( m + 1)J.l l vl ,m I 2 + L L mJ.l ( m + 1)J.l l vl ,m I 2 l i v - PNvl l ! I=LN-N m=N+ NL L 1 mJ.l(m + 1)J.l l vl,m I 2 + I LI>N m=L1 1 mJ.l(m + 1 )J.l l vl,m I 2 1=-Nm=N+l 1 1>Nm=N+l 1=-oom=N+l mr (m + 1)' I VI,m I 2 The proof of the above two theorems is due to Guo and Zheng (1994) .

Proof. By

00

:::; c

c

c

00

:::; c

00

00

:::; cN 2 J.1 - 2r :L

8.2 .

:L

00

00

:::; cN2 J.1 - 2 r l l v l l � ·

•

Pseudospectral Approximation on the Spherical Surface

(j ) w A ( - 1,1). SN {(..x ( ") 0 (; » ) l ..x ( " ) - } + 0(;) arcsin x (;) - The discrete inner point ( . , · ) N and the discrete norm I . l i N are defined by (V ,W)N + 1 E2N �N (.�(k) ,O(i») (..x(k) ,O(i» ) w(i) , I v i N The interpolation IN C(S) VN is such a mapping that for any C(S) , INv 1=N-N m=N1 1 VI,mY/,m (..x , O), (8. 6 ) where 2N N (..x(k) , OW) ,m (..x(") , O(i») w(i) . VI,m + 1 k=03=0

We now present another approximation on the spherical surface, based on the in terpolation. Let x U ) and be the Legendre-Gauss interpolation points and the weights on the interval = Define SN as a set of grid points, =�

,

=

=

2N

271"

, 0 < k < 2N, 0 < -

vE

--t

2

_71" _

2N

""

L..J �

v

J.

' " m e'., � 8",L " m ( sin B ) cos B. L.. m=O lll$m

8.

321

Chapter Spectral Methods on the Spherical Surface

I:;I,m ' denote the sum in the above formulae. Then

Let

I� 12 cos B OA
0,

(J(v, w), z) + (J(z, w), v) = O. Indeed, the integration by parts yields

(8. 13)

{" L: Z (O>.V09W - O>.W09V) d9dA " - fo 2 L: (O>.Z09W - 09Z0>.W) d9dA - l" Z (A, i) v (A, i) O>.W (A, i) dA + l" Z (A, -i) v (A, -i) O>.W (A, -i) dA. From Courant and Hilbert (1953) , the regularity of w implies that w approaches the limits independently of A, as 9 -+ ± � . This means that O>.W = 0 at 9 = ± � , and so (J(v, w), z ) =

2

2

v


323

H

(8.13) follows. Next let v = in (8.12) and integrate the resulting equation from a to t . We obtain from (8.13) that II H(t) II 2 + 2 p.

l I H (s ) l� ds

= IIHo ll 2 + 2

1o\j(s ) , H(s)) ds.

(8. 14)

Now let V� = VN n L�(S) . and lIT are approxiiQated by TiN and ""N . The spectral scheme for (8. 12) is to find TiN( ). , 0, t) E VN , ""N( ).. , 0, t) E V� for all t E R,. ( T ) such that

{

H

(DT TiN (t) + J (TiN(t) + CrDTTiN(t) , ""N(t) ) , 4» 4> E VN , t E RT ( T ) , - I' (t. (TiN (t) + UrDTTiN(t)) , 4» = (J(t), 4» , 4> E V� , t E RT ( T ) , - ( t.""N(t) , 4» = ( TiN (t) , 4» , ( ).. , O) E S TiN ( a ) = PNHo ,

V V

(8.15)

where c and u are the parameters, a � c, u � 1 . If both of them are zero, then (8.15) is an explicit scheme. Otherwise a linear iteration is needed at each time t E R.,. ( T ) . We now consider the existence and uniqueness of the solution of (8.15) with c i- a or u =F a. Let F (t) = TiN(t - r) - r ( l - c ) J (TiN(t - r ) , ""N (t - r )) +p.r(l - u )t.TiN(t - r) + rJ(t - r ) .

Then for any t E R.,. ( T ) ,

Clearly i t i s a linear algebraic system for the coefficients o f the spherical harmonic expansion of TiN (t) . Hence we only have to verify that TiN(t) is a trival solution when F (t) == a. In fact, by taking 4> = TiN (t) and using (8. 13), It implies TiN (t) == a , whence TiN (t) is determined uniquely at each time t E R,. ( T ) . In particular, if c = u = � , then we have from (8.13) and (8.15) that p.r ll TiN(t) II 2 + 2

L..J

"

= l l TiN( a ) 11 2 + r

ITiN (S) + TiN(S + r ) ll2

BERr(t-T)

�

BE Rr(t-T)

(J( ) , TiN (S) + TiN (S +. r )) .

s

(8.16)

324


This is a reasonable ana!ogy of (8. 14) . f We now analyze the generalized stability of scheme (8. 15). Suppose that and the right-hand side of (8.15) have the errors j and respectively. They and denoted by induce the errors of and Then

17N(O), �o , g, �N ¢N. 17N 1/;N, (DT�(t) + J (� (t) + 8TDT�(t) , 1/;N(t) + ¢(t)) + J (17N(t) + 8TDT17N(t), ¢ (t)) , rp) - It (� ( �(t) + (7TDT�(t)) , rp) = (l(t), rp) , V rp E VN, t E RT(T), � - ( ¢ (t), rp) = ( �(t) + g(t), rp) , V rp E V� , t E RT(T), A (¢ (t)) = 0 , t E RT(T), � ( O ) = �o , (A, 8) E S. ( 8. 1 7 ) Let e be an undetermined positive constant. By taking rp = 2 � + eT DT fJ in the first formula of (8. 1 7 ) , we get from ( 1 . 22 ) and (8.13) that DT 11�(t)1 1 2 + T(e - 1 ) I DT�(t) 1 2 + 21t 1�(t) l ; + itT (0" + �) I DT�(t) l; + ItT 2 (eO" - 0" - n I DT�(t) l ; + � Fj (t) = (l(t) , 2�(t) + eTDT�(t)) (8.18) where

Fl(t) 2 ( J (17N(t) + 8TDT17N(t), ¢(t)) , �(t)) , F2 (t) eT (J (17N(t) + 8TDT 17N(t), ¢ (t)) , DT � (t)) , F3 (t) T(e - 28) (J ( � (t), 1/;N(t)) , DT�(t)) , F4(t) T(e - 28) (J (� (t), ¢ (t)) , DT �(t)) . Further, we put rp = ¢ in the second formula of (8. 17), and then obtain that =

=

=

=

By Lemma 8.2, we find that

from which and Lemma

8 . 2,

(8 . 1 9)

Chapter

8.

325

Spectral Methods on the Spherical Surface

e

We are going to estimate I Fj (t) l . Let > O. By find that for , >

0,

I FI (t) 1

$ $ $

( 8.13 ) , ( 8.19) and Lemma 8.1, we

2 1(J (7} (t) , t,b(t)) , 71N(t) + 6TDT71iv (i)) I ep, 1 7} (t)l � + e: II 71N I I �(O,T;LOO ) 1 t,b (t) l : ep, 1 7} (t)l � + e: I171N I I �(O,T;H"Y+') ( U 7}(t) 112 + II g (t) 11 2) .

Similarly,

8.3 yields T (e - 26) 2 I I .,pN I I C(o,T;H"Y+2) 2 171 (t) II2 . 1 F3(t) 1 $ e T I I DT71 (t) II 2 + C e

Furthermore, the use of Lemma

_

_

Since

8.1 that _ T 2 (e - 26)2 II 1/N I 1 C(O,T;L2) 2 1 1/- (t ) 1 I2 · IF3(t) 1 $ e 'T I I DT71 (t) I I 2 + c N "1 e Also, by Lemma 8.3, Theorem 8.1 and ( 8. 2 0 ) , T(e 2 ) 2 1F4 (t) 1 $ e T IIDT 7}N I I 2 + C � 6 1 t,b (t) I :+ I 7} (t)l � 2 $ eT I IDT7}(t) 11 2 + C'T (e � 26) 2 ( 1 I 7}(t) II; + I Ig (t) II;) 17} (t) l � $ eT I I DT 7}(t) 11 2 + cTN2"1 (ee - 26) 2 (1 I 7} (t )1 I 2 + I l g (t)1 1 2 ) 1 7} (t) I � . Clearly, I (i(t) , 7}(t) + e� DT7}(t)) 1 $ eT II DT7} (t) 112 + C 1 7} (t)1 1 2 + C ( 1 + T;2 ) I j(t) 1 2 . By substituting the above estimates into ( 8.18 ) , we claim that DT 1 I �(t) 11 2 + T( e - 1 - 3e ) I I DT�(t) 112 + p, 1 � (t)l � + p,T (0" + � ) DT 1 � ( t) l � +p,T2 (eO" - 0" - � - e) I D �(t )l � $ Al 1 I � (t) 11 2 + A2 (t) 1 �(t )l � + A3(t) ( 8. 22)

we deduce from Theorem

T


326

l + e I1 77N ll 2c O,T;H'Y+ 1 ) , Al c ( 1 + � ( ) CTN2-y C - ( e 2 6 ) 2 ( I1 77N l c2 (O , T ; L2 ) + 1 I � (t) 1 I 2 + I l g ( t )1I 2 ) , A2 (t ) . = - p, + p, + Te ) 1 J-(t ) 1 2 + C ( 1 + e2 ) I1 77N ll c2 O,T;H"I+ 1 II g- (t )1I 2 . A3( t) = C ( 1 + e ( ) p, Let qo > 0 and c be suitably small. We choose e in three different cases. (i) u > !. We take 2 u + 2c ) e � 6 = max 1 + qo + 3c, 2 u ( 1 .

where

=

c

-

c

_

T(e - 1 3c ) IIDT�(t )1I 2 + p,T 2 (eu - u - � - ) IDT�(t) l � � qoT IIDT�(t )1I 2 . (8.23) (ii) u = !. According to Theorem 8 . 1 , there exists a positive constant c] such that I v ll � � c]N2 11 v1l 2 for all v VN. For this reason, we take e � 6 = 1 + qo + 3c + P,C] T N2 G + ) Then

c

-

E

c

and so (8.23) also holds.

( 111. . ) U .

< "2

1 and T N2

c

.

whence (8.23) is still valid. Now let

E(v , t) = II v(t )11 2 + T z: T ( l v( s ) l � + qoT IIDTv( s )1I 2 ) , ( ) p e t ) = 11� (O) 11 2 + p,T (u + �) I�( O )I� + T z: A3(S). p,

seRr t-

seRr (t-s)

.R,. ( t ) yields E (� , t ) � p (t ) + T z: (AIE ( i7 , s ) + A2 (S)E2 (� , S)) .

Then the summation (8.22) over

seRr(t-T)

Applying Lemma 3.2 to (8.24), we claim the result as follows.

(8.24)


Assume that (i) N2,,! is suitably small, being an arbitrary small positive constant; r.") > -1 or N2 CII-' ( 1 2- 2U ) ,. U 2 ·· (iii) I l g (t) 1 2 ::; T�b 2"! and p (tl) eb2 t' ::; TNbs "! for some tl E Rr (T). Then for all t E R,. (tl)

Theorem 8.4. T

{ n

T

327

I

l , ( 8.26 ) for u = l , for (j < C l-' . It leads t o the following result.

�.

If (8. 26) and condition (ii) of Theorem 8.4 are fulfilled, then for all p (t) and t E R,. (T) , the estimate (8. 25) is valid.

Theorem 8 . 5 .

We know again from Theorem 8.5 that a suitable implicit approximation of non linear term can improve the stability essentially. Indeed in the case of Theorem 8.5, scheme ( 8. 15) has the generalized stability index = -'- 00 . In the final part of this section, we deal with the convergence of ( 8.15 ) . First of all, we derive from ( 8.4) that for v

s

PN fl ( >' , fJ) v

E H2 ( S),

m ( m l)vl , m Y/, m (>', fJ) 111:SN m �111 2: 2: VI ,m flY/,m ()., fJ) = flPN V(>', fJ). ( 8.27) I I I:SN m �111 Then we have from ( 8.12) and ( 8.27) that

- 2: 2:

+

HN = PNH and WN = PNW. (DrHN(t) + J (HN(t) + 8T DrHN(t), WN(t)) , r/J) + I-' (V (HN(t) + UT DrHN(t)) , V r/J) = (� Gj (t) + f(t), r/J) + (VG4 (t), Vr/J ) , 'V r/J E VN, t E Rr(T), (VWN(t) ; V r/J) = (HN(t), r/J ) , 'V r/J E V�, t E R,. (T ) , A (W N(t)) = 0, t E R,.(T), = HN(O) PNHo , ()., fJ) E S

Let

( 8.28 )


328 where

Gl(t) = DrHN(t) - OtHN(t), G2 (t) = J (HN(t), WN(t » - J (H(t), w (t » , G3 (t) = DTJ (DrHN(t), WN(t » , G4 (t) = J.LO'TDrHN(t). Next, set fI = 1IN - HN and q, = 'ifJN - WN. We get from (8.15) and (8.28) that (DrfI(t) + J (fI(t) + DTDrH(t), WN(t) + q, (t) ) +J (HN(t) + DTDrHN(t) , q, (t) ) , t/» - J.L ( (fI(t) + O'T DrfI(t) ) t/» = - (� Gj(t), t/» - ( V'G4 (t), V't/» , V t/> VN, t Rr( T ) - (Aq, (t), t/» = (fI(t) , t/» , V t/> V�, t Rr(T), A ( q, (t) ) = 0 , t Rr(T), fIC O) = 0, (A, 9) S. By an argument as in the proof of Theorem 8.4, we can deduce an estimate similar to (8.24) . But 1IN and ii are replaced by HN and H respectively, while p et) is replaced A

E

E

E

E

E

,

,

E

by

As before, we have

I: I Gl(s) 1 I 2 � CT 2 I 1 H I �2 (O .t ;£2 ) ' T ERT(t 8 -r)

By Lemma 8 . 1 , Theorem 8.2 and a similar estimate as (8.21) , we verify that for any 'Y > 0,

I G2 (t) 1 I 2 � c I WN I �+2 I 1 HN(t) - H(t) lI� + c Il H(t) lI� I WN(t) - w (t) I �+2 � cN-2r ( I W(t) I ;+2 I 1 H(t) I �+1 + I w (t) II�+'Y+2 I1 H(t) l I n � cN-2r ( I H(t) II ; II H(t) I �+1 + II H(t) II�+'YI I H(t) I �) .

By virtue of Lemma 8.3 and Theorem 8.2, it is easy to see that T

I: I G (s)1 I 2 8ERT(t-r) 3

�

<
O , H E C (O , Tj W+l ) Hl (O , Tj Hl ) H2 (O , Tj L2 ) . Then for all t E RT (T),

Theorem 8.6.

n

n

J-t

b4 being a positive constant depending on and the norms of H in the mentioned spaces. The above numerical analysis comes from Guo (1995) . The spectral methods on the spherical surface have also been used for the barotropic vorticity equation in Guo and Zheng (1994), and for a system of nonlinear partial dif ferential equations governing the fluid flows with low Mach number in Guo and Cao (1995) . The applications of the pseudospectral methods on the surface can be found in Cao and Guo (1997) .

References Abe, K. and Inoue, O . ( 1980 ) , Fourier expansion solution of the KdV equation, J. 34 , 202-210.

Compo Phys.,

Adams, R. A. ( 1975 ) ,

Sobolev Spaces, Academic Press, New York.

Alpert, B . K. and Rokhlin, V. ( 1991 ) , A fast algorithm for the evaluation of Legendre expansions, J. 1 2 , 158 - 1 79.

SIAM Sci. Stat. Comput., Bateman, H. ( 1953 ) , Higher Transcendental Functions II, McGraw Hill Book Com

Bergh, J. and Lofstrom, J. ( 1976 ) , Verlag, Berlin. pany, New York.

Interpolation Spaces, An Introduction, Springer

Beringen, S. ( 1984 ) , Final stages of transition to turbulence in plane channel flow, J. 148, 413-442.

Fluid Mech.,

Bernardi, C . , Canuto, C. and Maday, Y. ( 1986 ) , Generalized inf-sup condition for Chebyshev approximations to N avier-Stokes equations, 303, 971 -974.

C. R. Acad. Sci. Paris,

Series I,

Bernardi, C., Coppoletta, G. and Maday, Y. ( 1992a) , Some spectral approxima tions of two-dimensional fourth-order polynomials, J. 15, 1489- 1505.

SIAM Sci. Comput.,

Bernardi, C . , CopPbletta, G. and Maday, Y. (1992b ) , Some spectral approximations of two-dimensional fourth-order problems, 59, 63-76.

Math. Comp.,

Bernardi, C. and Maday, Y. ( 1989 ) , Properties of some weighted Sobolev spaces and applications to spectral approximation, J. 26, 769-829.

SIAM Numer. Anal., Bernardi, C. and Maday, Y. ( 1992a ) , Approximations Spectrales de Problemes aux Limites Elliptiques, Springer-Verlag, Berlin. 331

332


Bernardi, C. and Maday, Y. (1992b), Polynomial interpolation results in Sobolev spaces, J. 43, 53-80.

Compo Appl. Math.,

Bernardi, C. and Maday, Y. (1994) , Univ. Pierre et Marie Curie, Paris.

Basic Results on Spectral Methods, R94022,

Bernardi, C . , Maday, Y. and Metivet, B . (1987) , Spectral approximation of the periodic-nonperiodic Navier-Stokes equations, 5 1 , 655-700.

Numer. Math.,

Boyd, J. P. (1978a) , The choice of spectral functions on a sphere for boundary and . eigenvalue problems: a comparison of Chebyshev, Fourier and associated Legendre expansion, 106, 1 148-1191.

Mon. Weather Rev.,

Boyd, J. P. (1978b), Spectral and pseudospectral methods for eigenvalue and sepa rable boundary value problems, 106, 1 192-1203.

Mon. Weather Rev.,

Boyd, J. P. (1987) , Spectral method using rational basis functions on an infinite interval, J. 69, 1 12- 142.

Compo Phys.,

Bramble, J. H . and Pasci ak, J. E. (1985) , A boundary parametric approximation to linearized scalar potential magnetostatic field problem, 1, 493-514.

Appl. Numer. Math.,

Brezzi, F., Rappaz, J. and Raviart, P. A. (1980), Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions, 36, 1 -25.

Numer. Math.,

Butzer, P. L. and Nessel, R. J. (1971 ) , Basel.

Fourier Analysis and Approximation, Birkhiiuser,

Butzer, P. 1 . , Nessel, R. J. and Trebels, W. (1972) , Summation processes of Fourier expansion in Banach spaces II, Saturation theorems, J. , 24, 551 -569.

Tiihoku Math.

Cai, W., Gottlieb, D. and Shu, C. W. (1989) , Essentially nonoscillatory spectral Fourier methods for shock wave calculations, 52 , 389-410.

Math. Comp.,

Cai, W., Gottlieb, D. and Shu, C. W. (1992) , On one-side filters for spectral Fourier 29, 905-916. approximations of discontinuous functions, J.

SIAM Numer. Anal.,

333

References

Cai, W. and Shu, C. W. (1991 ) , Uniform high order spectral methods for one and two dimensional Euler equations, 91-26.

ICASE Report,

Cambier, L., Ghazzi, W., Veuillot, J. P. and Viviand, H. (1982) , Une approche par domaines pour Ie Calcul d ' ecoulements compressible, in ed. by Glowinski, R. and Lions, J. L . , North-Holland, Amsterdam, 423-446.

Methods in Applied

Sciences and Engineering,

Canosa, J. and Gazdag, J. (1977) , The Korteweg-de Vries-Burgers ' equation, 23, 393-403.

Compo Phys.,

Canuto, C. (1988) , Spectral methods and a maximum principle, 615-630.

J.

Math. · Comp., 5 1 ,

Canuto, C., Fujii, H. and Quarteroni, A. (1983) , Approximation of symmetric break ing bifurcation for the Rayleigh convection problem, J. 20, 873-884.

SIAM Numer. Anal.,

Canuto, C . , Hussaini, M. Y., Quarteroni, A. and Zang, T. A. (1988) , Springer-Verlag, Berlin.

in Fluid Dynamics,

Spectral Methods

Canuto, C . , Maday, Y. and Quarteroni, A. (1982) , Analysis for the combined finite element Fourier interpolation, 39, 205-220.

Numer. Math.,

Canuto, C . , Maday, Y. and Quarteroni, A. (1984) , Combined finite element and spec tral approximation of the Navier-Stokes equations, 44, 201 -217.

Numer. Math.,

Canuto, C. and Quarteroni, A. ( 1982a) , Approximation results for orthogonal poly nomials in Sobolev spaces, 38, 67-86.

Math. Comp.,

Canuto, C. and Quarteroni, A. (1982b), Erl,"Or estimates for spectral and pseudo spectral approximations of hyperbolic equations, J. 19, 629-642.

SIAM Numer. Anal.,

Canuto, C. and Quarteroni, A. (1987) , On the boundary treatment in spectral meth ods for hyperbolic systems, J. 71 , 100- 1 10.

Compo Phys.,

Cao, W. M. and Guo, B . Y. (1991 ) , A pseudospectral methods for solving Navier Stokes equations, J. 9, 278-289.

Compo Math.,


334

Cao, W. M. and Guo, B. Y. (1993a) , Spectral-finite element method for solving three dimensional unsteady Navier-Stokes equations, 9, 27-38.

Acta Math. Sinica,

Cao, W. M. and Guo, B . Y. (1993b), Fourier collocation method for solving nonlinear Klein-Gordon equation, J. 108, 296-305.

Compo Phys.,

Cao, W. M. and Guo, B. Y. ( 1997) , A pseudospectral method for solving vorticity equations on spherical surface, 13,176-187.

Acta Math. Appl. Sinica, Carlenzoli, C . , Quarteroni, A. and Valli, A. (1991) , Spectral domain decomposition methods for compressible Navier-Stokes equations, AHPCRC preprint 91-46, Univ. of Minnesota. Carleson, L. (1966) , On convergence and growth of partial sums of Fourier series, 1 16, 135-157.

Acta Math.,

Chan, F. F., Glowinski, R. , Periaux, J. and Widlund, O . (1988) , SIAM, Philadelphia.

sition Methods for P.D.E.s II,

Domain Decompo

Chen, G. Q . , Du, Q. and Tadmor, E. (1993) , Spectral viscosity approximations to multidimensional scalar conservation laws, 61, 629-643.

Math. Comp.,

Cheng, M. T. and Chen, Y. H. (1956), On the approximation of function of several variables by trigonometrical polynomials, No.4, 412-428.

Pmc. of Beijing Univ.,

Chorin (1967) , The numerical solution of the Navier-Stokes equations for an incom pressible fluid, 928-931.

Bull. Amer. Math. Soc.,

Christov, C. I. (1982) , A complete orthogonal system of functions in space, J. 42, 1337- 1344.

L2 ( - 00 , 00 )

SIAM Appl. Math., P. G. (1978), The Finite Element Method for Elliptic Problems,

Ciarlet, Holland, Amsterdam.

North

Cooley, J. W. and Tukey, J. W. (1965) , An algorithm for the machine calculation of complex Fourier series, 19, 297-301.

Math Comp.,

Cornille, P. ( 1982), A pseudospectral scheme for the numerical calculation of shocks, J. 47, 146-159.

Compo Phys.,

References

335

Courant, R. , Friedrichs, K. O. and Lewy, H. (1928) , U ber die partiellen differezenglei chungen der mathematischen physik, 100, 32-74.

Math. Ann., Courant, R. and Hilbert, D. (1953) , Methods of Mathematical Physics, Vol. 1, In terscience, New York.

Dennis, S . C. and Quartapelle, L. (1985) , Spectral algorithms for vector elliptic equations, J. 6 1 , 319-342.

Compo Phys.,

Don, W. S. and Gottlieb, D. (1994) , The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points, J. 3 1 , 1519-1534.

SIAM Numer. Anal.,

Douglas, J. and Dupont, T. (1974) , Galerkin approximations for the two-point boundary value problem using continuous piecewise polynomial sp �c� , 99- 109.

Numer.

Math., 22,

On a numerical method for integration of the hydrodynamical equations with a spectral representation of the horizontal fields, Report No. 2, Institute for Teoretisk Meteorologi, Univ. of

Eliasen, E., Machenhauer, B . and Rasmussen, E. (1970) ,

Copenhagen. Feit, M. D . , Flack, J. A. and Steiger, A. (1982) , Solution of the Schrodinger equation by a spectral method, J. 47, 412-433.

Compo Phys.,

Fox, D. J. and Orszag, S. A. (1973a) , Pseudo-spectral approximation to two dimen sional turbulence, J. 1 1 , 612-619.

Compo Phys.,

Fox, D . J. and Orszag, S. A. (1973b), Inviscid dynamics of two dimensional turbu lence, 16, 169-171.

Phys. Fluids,

Fronberg, C. B. (1977), A numerical study of two dimensional turbulence, J. 1-31.

Phys., 25,

Compo

Fronberg, C. B. and Whitham, G. B. (1978) , A numerical and theoretical study of certain nonlinear' wave phenomena, (1361 ) , 313-404.

Philos. Trans. Royal Soc. London, 289

Fulton, S. R. and Taylor, G. D. ( 1984) , On the Gottlieb-Turkel time filter for Cheby shev spectral methods, J. 37, 70-92.

Compo Phys.,


336

Funaro, D . and Gottlieb, D. ( 1988 ) , A new method of imposing boundary condi tions in pseudospectral approximations of hyperbolic equations, 5 1 , 599-613.

Math. Comp.,

Funaro, D . and Gottlieb, D. ( 1991 ) , Convergence results for pseudospectral ap proximations of hyperbolic systems by a penalty-type boundary treatment, 57, 585-596.

Math.

Comp.,

Funaro, D. and Kavian, O . ( 1990 ) , Approximation of some diffusion evolution equa tions in unbounded domains by Hermite functions, 57, 597-619.

Math Comp.,

Gazdag, J. ( 1976 ) , Time-differencing schemes and transform methods, 20, 196-207.

Phys.,

Girault, V. and Raviart, P. A. ( 1979 ) ,

J.

Compo

Finite Element Approximation of the Navier Stokes Equations, Lecture Notes in Mathematics 794, Springer-Verlag, Berlin. Gottlieb, D . ( 1981 ) , The stability of pseudospectral Chebyshev methods, Math. Comp., 36, 107- 118. Gottlieb, D . ( 1985 ) , Spectral methods for compressible flow problems, in Proc. 9th Int. Conj. Numerical Methods in Fluid Dynamics, ed. by Soubbarameyer, J. P. Gottlieb, D., Gunzburger, H. and Turkel, E. ( 1982 ) , On numerical boundary treat ment for hyperbolic systems, J. 19, 671-697. B . , Springer-Verlag, Berlin, 48-61.

SIAM Numer. Anal.,

Gottlieb, D. , Hussaini, M. Y. and Orszag, S. A. ( 1984 ) , Theory and Applications of spectral methods, in ed. by Voigt, R. G . , Gottlieb, D. and Hussaini, M. Y., SIAM-CBMS , Philadelphia, 1 -54.

Spectral Methods for Partial Differential Equations,

Gottlieb, D . , Lustman, L. and Tadmor, E. ( 1987a ) , Stability analysis of spectral methods for hyperbolic initial-boundary value problem, J. 24, 241 -256.

SIAM Numer. Anal.,

Gottlieb, D., Lustman, L. and Tadmor, E. ( 1987b ) , Convergence of spectral meth ods for hyperbolic initial-boundary value systems, J. 24, 532-537.

SIAM Numer. Anal.,

337

References

Numerical Analysis of Spectral Methods,

Gottli �b, D. and Orszag, S. A. (1977) , SIAM-CBMS, Philadelphia.

Theory and Applications,

Gottlieb, D. and Orszag, S. A. and Turkel, E. (1981 ) , Stability of pseudospectral and finite difference methods for variable coefficient problems, 37, 293-305.

Math. Comp.,

Gottlieb, D . and Shu, C. W. (1994) , On the Gibbs phenomenon II, Resolution prop erties of Fourier methods for discontinuous waves, 116, 27-37.

Compo Meth. Appl. Mech.

Eng.,

Gottlieb, D . and Shu, C. W. (1995a) , On the Gibbs phenomenon IV, Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a. piecewise 64, 1081- 1095. analytic function,

Math. Comp.,

Gottlieb, D. and Shu, C. W. (1995b) , Recovering the exponential accuracy from col- . location point values of piecewise analytic functions, 7 1 , 511-526.

Numer. Math.,

Gottlieb, D . and Shu, C. W. (1996) , On the Gibbs phenomenon III, Recovering exponential accuracy in a sub-interval from the spectral partial sum of a piecewise analytic function, J. 33. 280-290.

SIAM Numer. Anal.,

Gottlieb, D . , Shu, C. W., Solomonoff, A. and Vandeven, O . H. (1992) , On the Gibbs phenomenon I, Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. 43, 81 -98.

Compo Appl. Math.,

Gottlieb, D. and Tadmor, E. (1985) , Recovering pointwise values of discontinuous data within spectral accuracy, in ed. by Murman, E. M. and Abarbanel, S. S . , Birkhii.user, Boston, 357-375.

Fluid Dynamics,

Process and Supercomputing in Computational

Gottlieb, D. and Turkel, E. (1980), On time discretization for spectral methods, 63, 67-86.

Stud. Appl. Math.,

Gresho, P. M. and Sani, R. 1. (1987) , On pressure boundary conditions for the incompressible Navier-Stokes equations, J. 7, 1 1 1 1 - 1 145.

Inter. Numerical Methods in Fluids,


338

Griffiths, D. F. (1982) , The stability of difference approximation to nonlinear partial differential equation, 18, December, 210-215.

Bull. IMA,

Guo, B . Y. ( 1965) , A class of difference schemes of two-dimensional viscous fluid flow, Research Report of SUST, Also see 17, 242-258.

Acta Math. Sinica,

Guo, B . Y. (1981a) , Error estimations of the spectral method for solving KdV -Burgers ' equation, Lecture in Universidad Complutense de Madrid, Also see 28, 1 - 15.

Acta Math.

Sinica,

Guo, B . Y. ( 1981b) , Difference method for fluid dynamics-numerical solution of primitive equations, 24, 297-312.

Scientia Sinica,

Scientia Sinica, 25, 702-715. Spectral method for Navier-Stokes equation, Scientia Sinica,

Guo, B. Y. (1982) , On stability of discretization, Guo, B . Y. (1985) , 28A, 1139- 1153.

Guo, B . Y. (1987) , Spectral-difference method for baroclinic primitive equation and its error estimation, 3DA, 696-713.

Scientia Sinica, Guo, B . Y. . (1988a) , Differen ce Methods for Partial Differential Equations, Science Press, Beijing.

Guo, B . Y. (1988b) , A spectral-difference method for solving two-dimensional vor ticity equation, J. 6, 238-257.

Compo Math.,

Guo, B. Y. ( 1989), Weak stability of discretization for operator equation, 32A, 897-907.

Sinica,

Scientia

Guo, B. Y. ( 1994) , Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, 163, 33-54.

Contemporary Mathematics,

Guo, B. Y. ( 1995), A spectral method for the vorticity equation on the surface, 64, 1067- 1079.

Comp.,

Guo, B. Y. (1996) , Hong Kong.

Math.

The State of Art in Spectral Methods, City Univ. of Hong Kong,

339

References

Guo, B. Y. and Cao, W. M. (1988) , The Fourier pseudo-spectral method with a restrain operator for the RLW equation, J. Compo Phys., 74, 110- 126. Guo, B . Y. and Cao, W. M. (199 1 ) , Spectral-finite element method for solving two dimensional vorticity equation, Acta Math. Appl. Sinica, 7, 257-271. Guo, B . Y. and Cao, W. M . (1992a) , A combined spectral-finite element method for solving two-dimensional unsteady Navier-Stokes equations, J. Compo Phys., 1 0 1 , 375-385. Guo, B . Y. and Cao, W. M. (1992b), Spectral-finite element method for compressible fluid flows, RAIRO Math. Model. Numer. Anal. , 26, 469-491. Guo, B. Y. and Cao, W. M. (1995) , A spectral method for the fluid flow with low Mach number on the spherical surface, SIAM J. Numer. A nal. , 32, 1 764- 1777. Guo, B. Y., Cao, W. M. and Tahira, N. ( 1993), A Fourier spectral scheme for solving nonlinear Klein-Gordon equation, Numerical Mathematics, 2, 38-56. Guo, B . Y. and He, L. P. (1996) , The fully discrete Legendre spectral approximation of two-dimensional unsteady incompressible fluid flow in stream function form, to appear in SIAM J. Numer. Anal. . Guo, B . Y., He, L. P. and Mao, D. K. (1997) , On two-dimensional Navier-Stokes equation in stream function form, JMAA, 205, 1-31. Guo, B . Y., He, S . N. and Ma, H. P. (1995) , Chebyshev spectral-finite element method for two-dimensional unsteady Navier-Stokes equation, to appear in HM SCMA J.

Guo, B . Y. and Huang, W. (1993) , The spectral-difference method for compressible flow, J. Comp o Math. , 1 1 , 37-49. Guo, B. Y. and Li, J . (1993) , Fourier-Chebyshev pseudospectral method for two dimensional vorticity equation, Nume.r. Math., 66, 329-346. .

Guo, B . Y. and Li, J. (1995a) , Fourier-Chebyshev pseudospectral methods for two dimensional Navier-Stokes equations, RAIRO Math. Model. Nllmer. Anal. , 29, 303-337.


340

Guo, B . Y. and Li, J. (1995b), Fourier-Chebyshev pseudospectral approximation with mixed filtering for two-dimensional viscous incompressible flow, J. Appl. Sci. , 13, 253-271 . Guo, B . Y. and Li, J. (1996) , Fourier-Chebyshev spectral method for two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal. , 33, 1169- 1 187. Guo, B . Y., Li, X. and Vazquez, L. (1996) , A Legendre spectral method for solving the nonlinear Klein-Gordon equation, J. Compo Appl. Math. , 1 5 , 19-36. Guo, B . Y. and Ma, H. P. (1987), Strict error estimation for a spectral method of compressible fluid flow, CAL COL O, 24, 263-282. Guo, B. Y. and Ma, H. P. (1991), A pseudospectral-finite element method for solving two-dimensional vorticity equation, SIAM J. Numer. Anal. , 28, 113- 132. Guo, B. Y. and Ma, H. P. (1993), Combined finite element and pseudospectral method for the two-dimensional evolutionary Navier-Stokes equations, SIAM J. Numer. A nal. , 30, 1066- 1083. Guo, B. Y., Ma, H. P. , Cao, W. M. and Huang, H. (1992) , The Fourier-Chebyshev spectral method for solving two-dimensional unsteady vorticity equations, J. Compo Phys., 101, 207-217. Guo, B . Y., Ma, H. P. and Hou, J. Y. (1996), Chebyshev pseudospectral-hybrid finite element method for two-dimensional vorticity equation, RAIRO Math. Model. Numer. Anal . 873-905. .

Guo, B. Y. and Manoranjan, V. S. (1985), Spectral method for s.olving R. L. W. equation, IMA J. Numer. Anal. , 5 , 307-318. Guo, B . Y. and Huang, W. (1993) , The spectral-difference method for compressible flow, J. Compo Math. , 1 1 , 37-49. Guo, B. Y. and Xiong, Y. S . (1989) , A spectral-difference method for two-dimensional viscous flow, J. Compo Phys., 84, 259-278. Guo, B. Y. and Xiong, Y. S . (1994) , Fourier pseudospectral-finite difference method for two-dimensional vorticity equation, Chin. Ann. Math. , 15B, 469-488.

References

341

Guo, B. Y. and Zheng, J. D. (1992), Pseudospectral-difference method for baroclinic primitive equation and its error estimation, Science in China, 35A, 1 - 13. Guo, B . Y. and Zheng, J. D . (1994) , A spectral approximation of the barotropic vorticity equation, J. Compo Math. , 12, 173-184. Haidvogel, D. B . , Robinson, A. R. and Schulman, E. E. (1980) , The accuracy, ef ficiency and stability of three numerical models with application to open ocean problem, J. Compo Phys., 34, 1-15. Haidvogel, D . B . and Zang, T. A. (1979) , The accurate solution of Poisson ' s equation by expansion in Chebyshev polynomials, J. Compo Phys., 30, 167- 180. Hald, O . H. (1981), Convergence of Fourier methods for Navier-Stokes equations, J. Compo Phys., 40, 305-317. Haltiner, G . J. and Williams, R. T. (1980) , Numerical Prediction and Dynamical Meteorology, John Wiley & S ons, New York. Hardy, G. H . , Littlewood, J. E. and P6lya, G. ( 1952), Inequalities, 2nd ed. , Cam bridge University Press, Cambridge. Harten, A. (1989) , ENO schemes with subcell resolutions, J. Compo Phys., 83, 148- 184. Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. (1987), Uniformly high order accurate non-oscillatory schemes III, J. Compo Phys., 71, 231 -303. Harvey, J. E. (1981 ) , Fourier integral treatment yielding insight into the control of Gibbs phenomena, Amer. J. Phys., 49, 747-750. He, S. N. and Guo, B . Y. ( 1995), Chebyshev spectral-finite element method for three dimensional unsteady Navier-Stokes equation, To appear in Appl. Math. Comp . . Hille, E. and Phillips, R. S. ( 1957) , Functional Analysis and Semi-Groups, Provi dence, R.I._. Hussaini, M. Y. , Salas, M. D. and Zang, T. A. (1985), Spectral methods for inviscid compressible flows, in Advances in Computational Transonics, ed. by Habashi, W. G., Pineridge, Swansea, U. K., 875-912.

342


Ingham, D . B . ( 1984 ) , Unsteady separation, J. Compo Phys. , 53, 90-99.

Ingham, D. B. ( 1985 ) , Flow past a suddenly heated vertical plate, Proc. Royal Soc. London, 1\.402, 109-134.

Jarraud, M. and Ba.ede, A. P. M. ( 1985 ) , The use of spectral techniques in numerical weather prediction, in Large-Scale Computations in Fluid Mechanics, Lectures in Applied Mathematics, 2 2 , 1 -41 .

Kantorovich, L. V. ( 1934 ) , On a new method of approximate solution of partial differential equation, Dokl. Akad. Nauk SSSR, 2 , 532-536.

Kantorovich, L. V. ( 1948 ) , Functional analysis and applied mathematics, YHM, 3, 89-185.

Keller, H. B . ( 1975 ) , Approximation methods for nonlinear problems with application to two-point boundary value problems, Math. Comp . , 29, 464-476.

Kenneth, P. B. ( 1978 ) , c m convergence of trigonometric interpolations, SIAM J. Numer. Anal., 1 5 , 1258-1266.

Kopriva, D . A. ( 1986 ) , A spectral multidomain method for the solution of hyperbolic systems, Appl. Numer. Math., 2, 221 -241 .

Kopriva, D . A. ( 1987 ) , Domain decomposition with both spectral and finite difference methods for the accurate computation of flow with shocks, Tallahassee, FL 323064052, Florida State Univ . .

Kreiss, H. O . and Oliger, J. ( 1972 ) , Comparison of accurate methods for the inte gration of hyperbolic equations, Tellus, 24, 199-215. Kreiss, H. O . and Oliger, J. ( 1979 ) , Stability of the Fourier method, SIAM J. Numer. Anal. , 16, 421 -433.

Kuo, P. Y. ( 1983 ) , The convergence of spectral scheme for solving two-dimensional vorticity equation, J. Compo Math. , 1 , 353-362. Lanczos, C. ( 1938 ) , Trigonometric interpolation of empirical and analytical functions, J. Math. Phys. , 17, 123-199.

References

343

Lax, P. D. ( 1972) , Hyperbolic Systems of Conservation Laws and the Mathemati cal Theory of Shock Waves, ABMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.

Lax, P. D. and Richtmyer, R. D. (1956) , Survey of the stability of linear finite difference equations, CPAM, 9, 267-293. Li, X. and Guo, B. Y . ( 1997) , A Legendre pseudospectral method for solving non linear Klein-Gordon equation, J. Compo Math. , 105 - 126. Lions, J. L. ( 1967) , Problemes aux Limites dans Les Equations aux Derivee Patielles, 2 ' edition, Les Press de L ' universite de Montreal. Lions, J. 1. (1969) , Quelques Methods de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris� Lions, J. L. ( 1970), On the numerical approximation of some equations arising in hydrodynamics, in Numerical Solution of Field Problems in Continuum Physics, SIAM-AMS Proc. II, ed. by Birkhoff, G. and Varga, R. S . , AMS , Providence, Rhode Island, 1 1 -23. Lions. J. L. and Magenes, E. (1972) , Nonhomogeneous Boundary Value Problems and Applications, Vol. Springer-Verlag, Berlin.

1,

Ma, H. P. (1996a) , Chebyshev-Legendre spectral viscosity method for nonlinear con servation laws (unpublished) . Ma, H. P . (1996b ) , Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws (unpublished) . Ma, H. P. and Guo, B . Y. (1986) , The Fourier pseudo-spectral method with a re straint operator for the Korteweg-de Vries equation, J. Compo Phys., 65, 120- 137.

Ma, H. P. and Guo, B . Y. (1987) , The Fourier pseudo-spectral method for solving two-dimensional vorticity equation, IMA J. Numer. A na ., 5 , 47-60.

l

Ma, H. P. and Guo, B. Y. (1988) , The Chebyshev spectral method for Burgers'-like equations, J. Compo Math. , 6, 48-53.

344


Ma, H. P. and Guo, B. Y. (1992) , Combined finite element and pseudospectral method for the three-dimensional Navier-Stokes equations, Chin. Ann. Math. , 1 3 B , 96-1 1 1 . Ma, H. P . and Guo, B . Y . (1994) , The Chebyshev spectral methods with a restraint operator for Burgers ' equation, Appl. Math.- JCU, 9B, 213-222. Macaraeg, M. and Streett, C. L. (1988) , An analysis of artificial viscosity effects on reacting flows using a spectral multi-domain technique, in Computational Fluid Dynamics, ed. by Davis, G. de Vahl and Fletcher, C., North-Holland, Amsterdam, 503-514.

Macaraeg, M., Streett, C. 1 . and Hussaini, M. Y. (1989) , A spectral multi-domain technique applied to high speed chemically reacting flow, in Domain Decomposition Methods for P.D. E.s, ed. by Chan, T. F., Glowinski, R., Periaux, J. and Widlund, 0 . , SIAM, Philadelphia, 361 -369. Maday, Y. (1990), Analysis of spectral projectors in one-dimensional domain, Math. Comp . , 5 5 , 537-562. Maday, Y. (1991), Resultats d ' approximation optimaux pour les operateurs d ' interpolation polynomiale, C. R. Acad. Sci. Paris, 312, 705-710. Maday, Y., Kaber, S. M. O. and Tadmor, E. (1993) , Legendre pseudospectral viscos ity method for nonlinear conservation laws, SIAM J. Numer. Anal. , 30, 321-342. Maday, Y. and Metivet, B. (1986), Chebyshev spectral approximation of Navier Stokes equation in a two-dimensional domain, RAIRO Math. Model. Numer. A nal. , 2 1 , 93-123. Maday, Y., Pernaud-Thomas, B . and Vandeven, H. (1985) , Une rehabilitation des methodes spectrales de type Laguerre, La Recherche A erospatiale, 6, 353-379.

Maday, Y . and Quarteroni, A. (1981), Legendre and Chebyshev spectral approxima tions of Burgers ' equation, N'Umer. Math. , 37, 321-332.

Maday, Y. and Quarteroni, A. (1982a) , Approximation of Burgers ' equation by pseudo-spectral methods, RAIR O Numer. Anal. , 16, 375-404.

References

345

Maday, Y. and Quarteroni, A. (1982b), Spectral and pseudospectral approximations of the Navier-Stokes equations, SIAM J. Numer. A nal. , 1 9 , 76 1 -780. Maday, Y. and Tadmor, E. (1989) , Analysis of the spectral viscosity method for periodic conservation laws, SIAM J. Numer. Anal. , 26, 854-870. Majda, A., McDonough, J. and Osher, S. (1978) , The Fourier method for nonsmooth initial data, Numer. Math. , 32, 1041 - 1081 . McCrory, R . L . and Orszag, S . A. ( 1980), Spectral methods for multi-dimensional diffusion problem, J. Compo Phys., 37, 93- 1 12. McKerrell, A., Phillips, C . and Delves, L. M. (198 1 ) , Chebyshev expansion meth ods for the solution of elliptic partial differential equation, J. Compo Phys., 40, 444-452. Mercier, B. (198 1 ) , Analyse Numerique des Methodes Spectrales, Note CEA-N-2278, Commissariat a l 'Energie Atomique Centre d 'Etudes de Limeil, 94190, Villeneuve Saint Georges. Mercier, B . and Raugel, G. ( 1982), Resolution d ' un probleme aux limites dans un ouvert axisymetrique par elements finis en r, z et series de Fourier en 8, RAIRO Anal. Numer. , 16, 405-461. Milinazo, F. A. and Saffman, P. G. (1985) , Finite complitude steady waves in plane viscous shear flows, J. Fluid Mech. , 160, 281 -295. Moin, P. and Kim, J. (1980) , On the numerical solution of time-dependent viscous in compressible fluid flows involving solid boundaries, J. Compo Phys., 3 5 , 381-392. Mulholland, S. and Sloan, D. M. (1991) , The effect of filtering in the pseudospec tral solution of evolutionary partial differential equations, J. Compo Phys., 96, 369-390.

Murdock, J. W. ( 1977), A numerical study of nonlinear effects on the boundary-layer stability, AlA A J. , 1 5 , 1 167- 1 1 73. Murdock, J. W. (1986) , Three dimensional, numerical study of bo.undary-Iayer sta bility, AIAA J. , 86-0434.

346


Nessel, R. J. and Wilmes, G. (1976) , On Nikolskii-type inequalities for orthogonal expansion, in Approximation Theory II, ed. by Lorentz, G. G . , Chui, C. K. and Schumaker, L. L., Academic Press, New York, 479-484.

Von Neumann, J. and Goldstine, H. H. (1947) , Numerical inverting of matrices of high order, Bull Amer. Math. Soc., 53, 1021 -1099. Nevai, P. (1979) , Orthogonal Polynomials, Men. Amer. Math. Soc., 2 1 3 , AMS, Providence, R. 1 . . Nikolskii , S . M. (1951) , Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables, Dokl. Akad. Nauk. SSSR, 58, 244-278. Orszag, S . A. (1969) , Numerical methods for the simulation of turbulence, Phys. Fluid, Suppl. II, 1 2 , 250-257. Orszag, S. A. (1970) , Transform method for calculation of vector sums, Application to the spectral form of the vorticity equation, J. Atmosph. Sci., 27 , 890-895. Orszag, S. A. (1971a) , On the elimination of aliasing in finite-difference schemes by filtering high wave number components, J. Atmosph. Sci. , 28, 1074. Orszag, S. A. (1971b) , Numerical simulations of incompressible flows within simple boundaries: Accuracy, J. Fluid Mech. , 49, 75- 1 12. Orszag, S. A. (1972) , Comparison of pseudospectral and spectral approximations, Stud. Appl. Math. , 5 1 , 253-259.

Orszag, S. A. ( 1980), Spectral methods for problems in complex geometries, J. Compo Phys., 37, 70-92. Pasciak, J. E. ( 1980), Spectral and pseudospectral methods for advection equations, Math. Comp . , 35, 1081- 1092.

Pasciak, J. E. (1982) , Spectral methods for a nonlinear initial value problem involving pseudodifferential operators, SIAM J. Numer. Anal. , 19, 142- 154. Quarteroni, A. (1984) , Some results of Bernstein and Jackson type for polynomial approximation in Lp-space, Jpn. J. Appl. Math., I, 173- 181.

References

347

Quarteroni, A. (1987a) , Blending Fourier and Chebyshev interpolation, Theor. , 5 1 , 1 15-126.

J.

Approx.

Quarteroni, A. (1987b), Domain decomposition techniques using spectral methods, CAL COL 0, 24, 141 -177. Quarteroni, A. (1990) , Domain decomposition methods for systems of conserva tion laws, spectral collocation approximations, SIAM J. Sci. Stat. Comput. , 1 1 , 1029-1052. Quarteroni, A. ( 1991) , An introduction to spectral methods for partial differential equations, in Advance on Numerical Analysis, Vol. 1, ed. by Light, W·. , Clarendon Press, Oxford, 96- 146. .

Quarteroni, A., Pasquarelli, F. and Valli, A. (1991) , Heterogeneous- domain decom position: Principles, algorithms, applications, AHPCRC preprint 91-48, Univ. of Minnesota. Richtmeyer, R. D. and Morton, K. W. (1967) , Finite Difference Methods for Initial Value Problems, 2nd edition, Interscience, New York. Rushid, A . , Cao, W. M. and Guo, B. Y. (1994) , Three level Fourier spectral ap proximations for fluid flow with low Mach number, Appl. Math. Comput. , 63, 131- 150. Schwarz, L. (1966), Theorie des distributions, Hermann, Paris. Schwraz, L. (1969) , Nonlinear Functional Analysis, Gordon and Breach Science Pub lishers, New York. Shamel, H. and Elsasser, K. (1976) , The application of the spectral method to non linear wave propagation, J. Compo Phys., 2 2 , 501 -516. Shen, J. (1992) , On pressure stabilization method and projection method for un steady Navier-Stokes equation, .in Advances in Computer Methods for Partial Dif ferential Equation VII, ed. by Vichnevetsky, R. , Knight, D . . and Richter, G . , 658-662.

348


Shen, J. (1994) , Efficient spectral-Galerkin method I, Direct solvers of second and fourth order equations using Legendre polynomials, SIAM J. Sci. Comput. , 15, 1489- 1505. Shen, J. (1995), Efficient spectral-Galerkin method II, Direct solvers of second and fourth equations by Chebyshev polynomials, SIAM J. Sci. Comput. , 16, 74-87. Silberman, I. (1954) , Planetary waves in the atmosphere, J. Meterol. , 1 1 , 27-34. Slater, J. C. (1934) , Electronic energy bands in metal, Phys. Rev. , 45, 794-801 . Smoller, J. ( 1983), Shock Waves and Reaction Diffusion Equations, Springer-Verlag, Berlin. Stetter, H. J. (1966), Stability of nonlinear discretization algorithms, in Numerical Solutions of Partial Differential Equations, ed. by Bramble, J., Academic Press, New York, 1 1 1 - 123. Stetter, H. J. ( 1973), A nalysis of Discretization Method for Ordinary Differential Equations, Springer-Verlag, Berlin. Strang, G. (1965) , Necessary and insufficient conditions for well-posed Cauchy prob lems, J. of Diff. Equations, 2, 107- 114. Streett, C. L., Zang, T. A. and Hussaini, M. Y. (1985) , Spectral multigrid methods with application to transonic potential flow, J. Compo Phys., 57, 43-76. Szabados, J. and Vertesi, P. (1992) , A survey on mean convergence of interpolatory processes, J. Comp o App. Math. , 43, 3-18. Tadmor, E. (1989) , Convergence of spectral viscosity methods for nonlinear conser vation laws, SIAM J. Numer. Anal. , 26, 30-44. Tadmor, E. (1990), Shock capturing by the spectral viscosity method, Compo Meth ods Appl. Mech. Engrg., 80, 197-208. Tadmor, E . (1991 ) , Local error estimates for .d iscontinuous solutions of nonlinear . hyperbolic equations, SIAM J. Numer. Anal. , 28, 891 -906. Tadmor, E. (1993a) , Total variation and error estimates for spectral viscosity ap proximation, Math. Comp., 60, 245-246.

, References

349

Tadmor, E. (1993b), Super viscosity and spectral approximations of nonlinear con servation laws, in Numerical Methods for Fluid Dynamics IV, .ed. by Baines, M. J. and Morton, K. W., Clarendo Press, Oxford, 69-82: Tartar, L. (1975) , Compensated compactness and applications to par�ial differential equations, in Research Notes in Mathematics Nonlinear Analysis and Mechan ics, Heriot- Watt Symposium, ed. by Knopps, J., Vol. 4, 136-21 1 .

39.

Timan, A. F . (1963) , Theory of Approximation of Functions of a Real Variable, Pergamon, Oxford. Vandeven, H. (1991 ) , Family of spectral filters for discontinuous problems, J. Sci. Comput. , 6, 159- 192. Weideman, J. A. C. (1992) , The eigenvalues of Hermite and rational spectral differ ential matrices, Numer. Math. , 6 1 , 409-432. Weideman, J. A. C. and Trefethen, L. N. (1988) , The eigenvalues of second-order spectral differentiation matrices, SIAM J. Numer. A nal. , 2 5 , 1279-1298. Wengle, H. and Seifeld, J. H. (1978) , Pseudospectral solution of atmospheric diffusion problems, J. Compo Phys., 26, 87-106. Woodward, P. and Collela, P. (1984) , The numerical simulation of two-dimensional fluid flows with strong shock, J. Compo Phys., 54, 1 15-1 17. Zang, T. A. and Hussaini, M. Y. (1986) , On spectral multigrid methods for the time-dependent Navier-Stokes equations, Appl. Math. Comput. , 19, 359-373. Zang, T. A., Wong, Y. S. and Hussaini, M. Y. (1982) , Spectral multigrid methods for elliptic equations, J. Compo Phys., 48, 485-501 . Zang, T. A . , Wong, Y. S. and Hussaini, M. Y. (1984) , Spectral multigrid methods for elliptic equations II, J. Compo Phys . , 54, 489-507. Zen, Q. C. (1979) , Physical-Mathematical Basis of Numerical Prediction, Vol. Science Press, Beijing.

1,

Spectral Methods and Their Applications

Spectral methods and their applications

Sequential Methods and Their Applications

Sequential Methods and Their Applications

Spectral Methods. Algorithms, Analysis and Applications

Finite Element Methods and Their Applications

Finite Element Methods and Their Applications

Finite Element Methods and Their Applications

Path-integral methods and their applications

Finite element methods and their applications

Finite Element Methods And Their Applications

Mesh parameterization methods and their applications

Finite Element Methods and Their Applications

Stochastic Methods and Their Applications to Communications

Chebyshev and Fourier spectral methods

Spectral Analysis and its applications

Spectral methods in MATLAB

Spectral methods in MATLAB

Numerical Analysis of Spectral Methods : Theory and Applications

Spectral Methods in MATLAB

Spectral Methods in Matlab

Spectral Methods in Food Analysis: Instrumentation and Applications

Spectral Methods in Food Analysis: Instrumentation and Applications

Numerical analysis of spectral methods: theory and applications

Spectral Methods for Time-Dependent Problems: Analysis and Applications

Orders and their Applications

Metagraphs and Their Applications

Wavelets and their applications

Metagraphs and Their Applications

Wavelets and their applications

Cellulases and Their Applications

Spectral Methods and Their Applications