the spectrum of a chebyshev-fourier approximation for the

Journal of Scientific Computing, Vol. 5, No. I, 1990

The Spectrum of a Chebyshev-Fourier Approximation for the Stokes Equations

U. Ehrenstein 1

Received March 9, 1990

A Chebyshev-Fourier approximation to the solution of the two-dimensional Stokes equations in the vorticity-streamfunction formulation is considered. The expansion in a Fourier series in the direction of periodicity leads to a family of one-dimensional Stokes-type problems being approximated by a Chebyshev- collocation method. First some results about the spectrum of the corresponding operators are derived. Then we consider a discretization in time by means of a class of semi-implicit finite differences schemes and we describe the influence matrix technique used to solve the resulting system at every time step. The properties of the spectrum of the Chebyshev-Stokes operator are used to derive some results about the stability of the resulting time marching algorithm.

KEY WORDS: Stokes equation; Chebyshev-Foufier approximation; initial value problem.

1. I N T R O D U C T I O N

Spectral methods for the solution of the incompressible Navier-Stokes equations have largely developed during the last ten years. Generally these methods consist in an approximation of the solution by a Fourier series (for periodic problems) or Chebyshev-polynomial series (for nonperiodic problems) in the space variables (see, for example, Canuto et al., 1987; Gottlieb and Orszag, 1977; Peyret, 1988). The discretization in time, however, is generally obtained by finite differences where the nonlinear terms are treated explicitly.

One major problem in the numerical solution of the incompressible Navier-Stokes equations is to satisfy the incompressibility condition. To

1 Laboratoire de Math6matiques, Universit6 de Nice, Parc Valrose, F-06034 Nice Cedex, France. Present address: Institut Frangais du P6trole, 1 et 4, avenue de Bois Pr6au, BP 311, F-92506 Rueil Malmaison Cedex, France.

55

0885-7474/90/0300-0055506.00/0 �9 1990 Plenum Ptlblishing Corporation

56 Ehrenstein

overcome this difficulty for the spectral approximation of the equations in the velocity-pressure variables Kleiser and Schumann (1980) proposed the so-called influence matrix technique; a theoretical analysis of this method can be found in Canuto and Sacchi Landriani (1986). This approach has been generalized by Le Quere and Alziary de Roquefort (1985) to the two-dimensional, nonperiodic Navier-Stokes equations.

An alternative formulation of the two-dimensional Navier-Stokes equations consists in the use of the vorticity co and streamfunction ~ as dependant variables, the drawback of this formulation being the lack of boundary conditions for the vorticity, while there are two conditions for the streamfunction. Again it is possible to use an influence matrix method to obtain a Dirichlet boundary condition for e) being equivalent to the Neumann boundary condition for ~; this approach has been applied by Tuckermann (1983) to the Chebyshev-Tau approximation of the equations. This technique has been generalized by Vanel et aL (1986) for a hybrid Chebyshev-Tau collocation method for the two-dimensional, nonperiodic Navier-Stokes problem. In Ehrenstein and Peyret (1986, 1989) an analysis of this method applied to the Chebyshev-collocation approximation of the Navier-Stokes equations is given [-see also Ehrenstein (1986) for a comparison between the collocation and the Tau approximation of the equations].

In the present paper we consider only two-dimensional Stokes problems whose solutions are periodic in one space direction. An approximation by Fourier series leads to a family of one-dimensional Stokes-type problems which are approximated by a Chebyshev collocation method in space. First of all we investigate the spectrum of the resulting discrete Stokes operators. We show that the eigensolutions are even or odd polynomials and that the eigenvalues associated to the even eigensolutions are real, negative, and distinct as well as those associated to the odd eigensolutions. To obtain this result in Section 3 we recall first of all in Section 2 some results about the location of zeros of polynomials. Note that the above result is analogous to that obtained by Gottlieb and Lustman (1983) for the spectrum of the Chebyshev-collocation operator for the heat equation (for general boundary conditions). This result about the discrete Stokes operator in the vorticity-streamfunction formulation contrasts with the fact shown in Gottlieb and Orszag (1977), that a Chebyshev-Tau method applied to the fourth-order streamfunction equation leads to unstable eigenvalues.

In Section 4 we present a class of semi-implicit finite differences schemes for the time discretization and we describe the influence matrix technique used to solve the resulting system at every time step. We show that the corresponding time marching algorithm is unconditionally

Chebyshev-Fourier Approximation 57

unstable for a certain range of parameters of the schemes. Finally we give some results about the spectral radius of the amplification matrix of the time stepping algorithm by use of the properties of the spectrum of the Chebyshev-Stokes-type operators.

2. SUMMARY OF RESULTS FOR THE LOCATION OF ZEROS OF POLYNOMIALS

This section provides some technical tools to assure that a polynomial has negative, real, and distinct roots; the following results can be found in Gantmacher (1960). First of all we introduce the notion of a positive pair which has been used by Gottlieb and Lustman (1983) to evaluate the spectrum of the Chebyshev collocation operator for the heat equation.

Definition 1. We say that two polynomials h(u) and g(u) of degree m (or the first of degree m and the second of degree m - 1) form a positive

t ! t U ! pair if the zeros Ul,..., um of h(u) and the zeros ul ..... Um (or ul . . . . . . 1) of g(u) are all distinct, real, and negative, if they satisfy

t t t U l < b / 1 < b / 2 < b / 2 < . . . < H m < b l r n < O

(or u ~ < u ' ~ < u 2 < . . . <U~_~<Um<0)

and if the highest coefficients of g and h are of like sign.

Lemma 1. The polynomial f ( z ) = h ( z 2) +zg(z 2) is a Hurwitz polynomial (i.e., all of its roots have negative real parts) if and only if h(u) and g(u) form a positive pair.

For the proof see Gantmacher (1960), p. 228.

Definition 2. If f ( z )=aozn +bo zn l +alzn 2 q-blZn-3 + ... (ao-r is a real polynomial, its Routh scheme is a table of numbers

ao, a l , a2,...

bo, bl, b2,...

C0~ CI~ C2~...

do, dl, d2,... . , .

where each row is obtained by the two preceding rows as follows: From each element of the upper row one subtracts the corresponding element of

58 Ehrenstein

the lower row multipled by a number such that the first difference vanishes�9 By omitting the zero value the elements of the third row are

~o (b0#0)" co=al-~obl,

those of the fourth row become

ao c 1 = a 2 - ~ o b 2 , . . .

(Co#0) do=b1 bo �9 - - - cl, dl : b2 - b~ c2 . . . .

C0 CO

and so on.

Lemma 2. [Routh criterion; for a proof see Grantmacher (1960), p. 180]. A real polynomial f(z) is a Hurwitz polynomial if and only if the first column of its Routh scheme consists of elements which are all nonzero and have the same sign.

These results will enable us to examine in the next section the spectrum of the Chebyshev collocation operator for the Stokes problem.

3. SPECTRUM OF THE CHEBYCHEV-COLLOCAION OPERATOR FOR STOKES-TYPE PROBLEMS

We consider the two-dimensional Stokes equations in the vorticity co and streamfunction ~ formulation whose dimensionless form is given by

where

&o 1 0--~- = R----~ dco + f, At)=o~ (3.1a)

9 2 0 2

A = ~-~x2 ~ 0y2

in the domain 0 < x < 2~, - 1 < y < 1, l > 0, with conditions of periodicity in x = 0 and x = 2~. The boundary values are

6 = ~ - ~ = 0 in y = +_1 (3.1b)

and the initial condition is given by

~o = co ~ for t = 0 (3.1c)


The function f in (3.1a) is a forcing term being periodic in x of period 2~, and Re is the Reynolds number. We expand the solution (co, 0) in Fourier series

o0= ~ ~ok(y,t)e ikx, O = ~ Ok(y, t )e ikx k = --oo k= --o~

The Fourier coefficients (o)k, 0k) are solution of the following one- dimensional Stokes-type problems:

&ok 1 [?2~ok(y, t) k2~ok(y ' t)] + A ( Y , t) 8t (y' t)-- Ke l_w-- 3y2 A

y ~ ] - l , l [ , t>O

?20k (y, t)--k2Ok(y, t )=oh (y , t) ?y2 (3.2)

80h 0~(_+1, t)=O -~-y (+1, t )=0, t > o

~ ( y , 0) = ~o~ y e ] - 1, ll-

where f~(y, t) and o)~ are the Fourier coefficients of the forcing term and the initial condition. In practice the Fourier expansion is truncated at some finite value K such that -K~< k ~< K, and by the condition that (co, 0) has to be a real solution the system (3.2) is solved only for 0 ~< k ~< K.

The space discretization of (3.2) consists in the approximation of the solution as polynomials of degree N. This is achieved by use of the Chebyshev-collocation method (see Canuto et al., 1987; Gottlieb et al., 1984) and we obtain the following system for the polynomials C%N, 0k, N of degree N in y :

?O)k,N(Yi, t) 1 ~02cok,:v(yi, t) ~t Re [_ ~y2

8 2 0 k , N ( Y i , l ) k 2 ?7 2 " Ok, N(Yi, t)=cOk, u(Yi, t)

t ) = ~ ( _ + l , t )=0, t > 0 Ok, N(+I ,

cox, N(ys, 0) = o)~ 1 ~< i ~< N - 1

q k2gOk, N(yi, t)[ + fk(Y,, t)

I~<i~<N--1, t > 0

(3.3)

where

Y i = C O S

are the Chebyshev-collocation points.

O <<. i <~ N

60 Ehrenstein

To examine the spectrum of the system (3.3) we put f k = 0 and we are looking for solutions of the form (we drop the subscript k)

CON(y, t) = e~t&N(y, 2), ON(Y, t) = eX'(kN(y, 2) (3.4)

Clearly d~N(y, )~) and (ON(y, 2) are solutions of

r [- 2caN ] 2oSN(y i, 2) = Ree L - - ~ y 2 (yi, 2) - ctd)N(y,, 2)

l~ i<<.N-1

a2q;U (y,, 2 ) - ~ ( y , , 2 )= &N(Y,, 2) (3.5) ay 2

~N(-- 1, 2) :~yy (+-- 1, 2) = 0

with c~ = k% Note that we can set Re = 1 in (3.5), for if ,~ is a solution of (3.5) with Re = 1 then )~/Re is a solution of the system with Re # 1.

First of all we want to write the system (3.5) as equalities between polynomials.

To do this we observe that the collocation points y~ = cos(i~/N) for 1 <~ i<~N-1 are the zeros of the derivative TN(y) of the Nth Chebyshev polynomial T j r (y)= co s(N arccos y). It follows that a polynomial E N (y ) of degree N being zero in the points Yi, 1 ~< i~< N - 1 is of the form E~v(y)= (Ay + B) TN(y) for some constant coefficients A and B. Consequently the system (3.5) can be written as equalities between polynomials (Re--1)

02(DN 2 ( ~ ) N - - O~O-)N-~ (ay + b) T'N(y) @2

02(kN O~N=ChN+(Cy+d) r'u(y) (3.6) Oy 2

1, x)= (+_ 1, = 0

for some coefficients a, b, c, and d. For simplicity we suppose from now on that N = 2M; the results of the section will not change if N = 2 M + 1.

To obtain results about the solutions of (3.6) we need the following equality:

( D _ 7 l ) _ 1 = . . . . 1 / + I D + . . + (3.7) 7 7

where D is the operator 02/@ 2 applied to the set of polynomials of degree N = 2M which is nilpotent, and l is the identity operator (7 # 0).


Furthermore we need the following preliminary result:

Lemma 3. Suppose that 2 is solution of (3.6); then # = 2 + c ~ is nonzero.

Proof We suppose that # = 0; consequently the cb N solution of the first equation of (3.6) has to be of the form

(bN= Ay + B

for some coefficients A, B. If c~ = 0 the solution ~N of the second equation of (3.6) with (f)U of the form above has to be a polynomial of degree at most three. We conclude that ~N (and hence &u) has to be identically zero by the boundary conditions ~[]U = (~IN/~Y = 0 for y = _+_1. Now we suppose that ~ > 0 and we define

7 (y )= ~ ~ j -1 [yT'N(y)] j= o ~y2j

M-1 02j+1

j = 0 ~y2j+ l

By the equality (3.7) the solution ~N with ~)N as above becomes

~ N = --a- '(Ay + B) -- cy(y) -- da(y)

(3.8a)

(3.8b)

The boundary conditions for ~N lead to the system

- ~ - I A -- c~-iB-- c~(1) - da(1) = 0

~ - ~ A - c ~ - l B - c T ( - - 1 ) - d a ( - 1 ) = 0

- -~- tA --c~'(1) - d a ' ( 1 ) = 0

--c~-~A -- c7'(-- 1)-- do- '(- 1) = 0

By use of the fact (N being even) that 7(Y) defined above is an even and a(y) an odd polynomial, a straightforward computation shows that the determinant of the above system is

-47 ' (1) a -2[a ' (1 ) - a(1)]

It can be shown (cf., for example, Gottlieb and Orszag, 1977) that

- - p--1 dPTN ( + I ) = ( _ I ) N+p I~ (N2-k2) / (2k+ l) dYP k=O

62 Ehrenstein

therefore 7'(1) consists (ct being positive) of a sum of positive numbers, hence 7 ' (1)~ 0. Note that

dNTN (1 } = t iN- 1TN dy N ! dyN_ I (1)

therefore

~ - ~ 1- a2j+2 ~2j+, ] a'(1)- a(1)=/--oZ "-J-i LOy2S+2-- T~v(1)- 0y2a+~ TN(1) 3

and this expression is again a sum of positive numbers. We conclude that the determinant of the above system is nonzero, consequently A = B = c = d = 0 and therefore @N and &N are zero.

Therefore (# being nonzero) we can write by use of (3.7)

M 02j M--I 02j+a

d)K=--a ~,, #-S-'Syzy[yT'N(y)]--b ~ #-J , j=o j=o 0y2i+1 TN(y) (3.9)

Note that in the first equation of (3.6) (a, b) r (0, 0) because a polynomial cannot be a solution of the equation

~2(~ N #&N = Oy 2 with # = 2 + a ~ O

We define

M g ~2(i4-j) p(y, #) = ~ #-i-1 ~ o~--j--10Y 2(i+j~---~ [yT'N(y)] (3.10a)

i=0 j = 0

and

M-- 1 M-- 1 (~ 2(i+j) + 1

Q(y, I.t)= ~ #-i-~ • ~-:-, 1 TN(y) (3.10b) i=O j = 0 8y2(i+j)+

M--1 ~2j

P(Y, U)= ~, #-j-2 8Y 2j [yT'N(y)] (3.11a) j=O

M--2 6q2j+1

0 ( Y ' # ) = E # - - J - - 2 0 - ~ - + - I TN(y) (3 .11b) j = o

The solutions of (3.6) are characterized by the following lemma.


Lemma 4. If ~ > 0 in (3.6) the solution ~N is given by

(IN(Y, #) = aP(y, #) + bO(y, #) - cT(y) - riG(y) (3.12)

with P, Q defined by (3.10a), (3.10b) and 7, a defined by (3.8a), (3.8b). If = 0, ~U becomes

~N(Y, #) = -aP(y, #) - bQ(y, #) + a'y + b' (3.13)

where a', b r are some constant coefficients and P, Q are defined by (3.11a) and (3.1 lb).

Proof By applying (3.7) to the second equation of (3.6) with (J)N given by (3.9) one obtains the equality (3.12) for 7>0. If ~=0, the polynomial C2~sfi3y 2 of degree at most N - 2 equals the polynomial ChN+ (cy + d) T'N(y). For the latter one to be of degree at most N - 2 the following equality has to hold:

( - a # - l +c)a lyN + [ a 2 ( c - a # - l ) + a l ( d - b # 1)]yN-l:---0

with

T~v(y ) = a lyN-l+a2yN-2q_ . . .

['(J)N is given by (3.9)]. We conclude that c= a# -1 and d= b# -~ therefore

~2~N M O2j M--1 ~2j+l Oy2 - - - a J•= 1 #-J-J Oy2j [yT 'N(y)] -b j•= 1 # - j ' CY 2j + 1 TN(y) (3.14)

By integrating (3.14) twice we obtain (3.13) for some constant coefficients a',b r.

The solutions ~lN(y,# ) given by (3.12) for a > 0 and by (3.13) for a = 0 have to satisfy the boundary conditions ~N(_I,#)=(C~N/Cy) ( + 1, #) = 0. This leads to the system

aP(1, #) + bQ(1,/~) -c7(1) - da(1) =0 aP(-1,/~) +bQ(-1,#) -cT( -1) -da( -1)=0

~y~t' ~yQ a-z-(l,# ) +b ( 1 , # ) - c 7 ' ( 1 ) - d a ' ( 1 ) =0 (3.15)

a~y (-1,#)+b (-1,#)-c7'(-1)-da'(-1)=O

64 Ehrenstein

for a > O, and to

-aP(1,/~) -bQ(1, #) + a ' + b ' = 0

- a P ( - 1 , # ) - b Q ( - 1 , p) - a ' + b ' = 0

- a - z - ( l , # ) - b (1,#) +a ' =0 (3.16)

# ) - b =0

for 7---0. The functions P, ~,/~ are even polynomials in y while Q, a, and Q are

odd polynomials, and a straightforward computation shows that the determinant of the system (3.15) is

DI(B) = 4 P(1, p) 7 ' (1 ) - ~y (1, #) 7(1) ( 1 , # ) a ( 1 ) - a ' ( 1 ) Q ( 1 , u)

(3.17)

while the determinant of (3.16) becomes

D2(kt) = 4 ffy-y (1, ~t) -fffy (1, #) - ~)(1, p) (3.18)

For the system (3.15) [respectively, (3.16)] to have a nontrivial solution a, b, e, d (respectively, a, b, a', b') for some value of/z, the above determinants have to vanish. The functions D~(#), D2(#) are polynomials in v = 1//~. We introduce the coefficients

M O 2 ( i + j )

pi= ~ a - J - l - - ( y T u ) ( 1 ) , O<~i<~M (3.19a) j = 0 ~ y 2 ( i + j )

as well as

M-- 1 .~ 2(i+ j) + i

p;= 0y2(~+j)+~ (yT;v)(1), O<~i<~M- 1 (3.19b) j = 0

and by (3.10a)

P 1, = v ~ pi vi, ~y 1, = v ~ p~v ~ (3.20) i = 0 i = 0


Similarly we define

M--1 qi = E ~--j--1

j=o M--2

j=o

and by (3.lOb)

- ,

Q1, i=O

(~2(i+j) + 1 ~y2(i+jl+t TN(I), O<~i<~M--1 (3.21a)

~2( i+ j )+2

(3y2(i+j)+2TN(1), O ~ i ~ M - 1 (3.21b)

qS, 1, =v • q;v' (3.22) i=0

We observe that by (3.8a) 7(1)= Po and 7'(1)= p~ while by (3.8b) a(1)= qo and #(1)=q~. Combining the above definitions and equalities we can rewrite the equation (3.17)

D~(!)=-4v4p~(v)Q~(v) (3.23a)

with

and

M M--1 P~(v)= ~ pip'ov ' - 1 - ~ p;po vi-~ (3.23b)

i=1 i = l

M--1 M--1 Ql(v)= ~ , i - l 1 qiqov - ~ q~qo v~- (3.23c)

i=1 i=1

By the definitions (3.11a), (3.11b) we can write for (3.18)

D2 (~)=4v4p2(v) Q2(v) (3.24a)

with

M•I (~2i+ 1

P 2 ( v ) = i=O

(3.24b)

and

M--2 Q2(v) =

i=0

I (~2i+2 (~2i+ 1 1 TN(1)--Oy2i'----- ~ T~v(1) v' (3.24c)

We still need the following preliminary result:

66 Ehrenstein

Lemma 5. The polynomial

M M - - 1

R(v)= ~ p~v2'+ ~ p;v 2i+1 (3.25) i = 0 i = 0

with Pi, P~ defined by (3.19a), (3.19b) as well as

M - - 1 M 1

S(v)= E qi v2i+ E q~ v2i+l (3.26) i = O i = 0

with qi, q~ defined by (3.21a), (3.21b) are of Hurwitz type.

Proof To characterize the roots of R(V)=c%vN+flovN--I+ ~vN--2+ '" we construct its Routh scheme. We define

o N - - 2 k ~N- -1 2k

ak - (yT~v)(1), bk= (yT~)(1) ~yN - 2k OyN -- 1 -- 2k

consequently

i i

o~ i=pM_i= ~, a -i tai j, f i i=P~- l - i = ~ o~-J-lbi_j j = 0 j = O

according to the definitions (3.19a), (3.19b). The elements of the first row of the Routh scheme of R(v) are ai,

0 ~< i ~< M and those of the second row are fli, 0 ~< i ~< M - 1. We define

h(v)=aoVN +boVN-I +a,vN-2 + . . . . . (yT'N)(1)V i i=o OY i

and we show that the elements of the first column of the Routh scheme of R(v) are equal to those of h(v) multiplied by a-1. Indeed a o = a - l a o , f l o = a - l b o and according to Definition 2 the third row of the Routh scheme of R consists of the elements

O~ o ~ i ~ O~i+ l - - ~o ~ i + l

• ( oo = o~--J--1 a i + l - J - - ~ o b i + l - J = E O~-j 1 c i - j ,

j = O j = O

O<~i<.M-1

But Ck, 0~<k~<M--1 are the elements of the third row of the Routh scheme of h(v) and y o = a - % o . Thus we obtain e o = ~ - l a o , f l o = e - l b o , Yo = a - l c o ..... which characterize the elements of the first column of the Routh scheme of R as being those of h multiplied by ~-1


In Gottlieb and Lustman (1983) it is shown that h(v) is of Hurwitz type and by applying Lemma 2 the same holds for R(v).

The proof for S(v) is similar; instead of h(v) above one has to define

M--I ~i+1 g(v)= E (~y/+l TN(llvi (3.27t

i=0

and the elements of the first column of the Routh scheme of S are equal to those of g multiplied by c~ -~. One concludes again by Lemma 2, g(v) being of Hurwitz type as shown in Gottlieb and Lustman (1983). 0

We come now to the central results of this section:

Proposition 1. The roots of the polynomials P~(v), Q~(v) defined by (3.23b), (3.23c), as well as those of P2(v), Q2(v) defined by (3.24b), (3.24c) are real negative and distinct.

Proof The assertion above for P2(v) has already been proven by Gottlieb and Lustman (1983). To prove the result for Q2(v) we introduce

M~I ~2i+1 O.(v) = a-y+; rN(1)v'

i=0

and we show that

~(v) = Q2(v 2) + vQ2(v 2) M--1 a2i+l M--2 ~ /~2i+2 a2i+,

= Z Oy2i+l TN(1) v2i+ Z [ _ ~ TN(1)--ay2i+------5 TN(1)j v2'+1 i=0 i=0

is a Hurwitz polynomial. It is easY to check (cf. Definition 2) that if we drop the first row of the Routh scheme of the polynomial g(v) defined by (3.27) we obtain a table of numbers that is equal to the Routh scheme of ~(v). In Gottlieb and Lustman (1983) it is shown that g(v) is a Hurwitz polynomial and we conclude by Lemma 2 that ~(v) is also of Hurwitz type. It follows (cf. Lemma 1) that the M - 2 roots of Q2 are real negative and distinct.

To characterize the roots of Pl(v) we define (p~ r 0)

P~(v) P'o P1 " ~ Pi - p; v M-i (with p~t=0) i=1

M 1 Y, p;v M ' i

i=0

68 E h r e n s t e i n

and

/~(v) = ffl(V 2) + v/31(v 2)

Again it is easy to check that if we construct the Routh scheme of the polynomial /~(v), the resulting table of numbers is equal to the Routh scheme without the first row of the polynomial

~(v)=vWR = ~ p i v N - 2 i q - ~ p;v N - 1 - 2 i

i = 0 i = 0

with R(v) defined by (3.25). But (cf. Lemma 5) R(v) is of Hurwitz type and so is/~(v). Again we conclude by Lemma 2 that/~(v) is a Hurwitz polynomial. By Lemma 1 the roots of Pl(v) are real, negative, and distinct and so are those of Pl(v).

Finally to characterize the roots of Ql(v) we define (q~, ~ 0)

0 1 ( v ) = - ~ - o Q ~ = qi_ q~ v M- i = 1 q'o q; 1--i

M 1

Q,(v )= ~ q;v M - ' - i i = 0

and

= 0 (v ) + v01(v 2)

The Routh scheme of S(v) is equal to that of

i = 0 i = 0

from which we drop the first row, S(v) being defined by (3.26). But S(v) is of Hurwitz type by Lemma 5 and by the same arguments as above we conclude that the M - 2 roots of Ql(v) are real, negative, and distinct.

We can summarize the above results in the main proposition of this section:

Proposition 2. We consider the polynomials P;(v), Qi(v), i = 1, 2, characterized by Proposition 1. The polynomials in y (cbN(y, 2), (IN(Y, 2)) are solution of (3.5) (with R e = 1) if and only if v = 1/(2+ ~) is a root of Pl(v) Ql(v) if ~ > 0 , or a root of P2(v) Qdv) if ~=0 . The M - 1 ( N = 2 M ) roots vi, 1 ~<i<~ M - 1, of Px(v) [respectively, P2(v)] are real, negative, and distinct and the eigensolutions

03N(y, 2~), ~N(y, 2i), l<<.i<~M-1


with

1 2 i = - - ~ < 0

vi

are even polynomials in y. Similarly the M - 2 roots vi, M<~i<~ N ' - 3 , of Ql(v) [-respectively, Q2(v)] are real, negative, and distinct and the eigensolutions

(~N(..V, 2i), ~lN(y, Jvi), M<~i<~N--3

(2i = 1/vi - c~ < 0) are odd polynomials in y.

Proof A solution &u, ~ v of (3.5) with /~=2+c~ ( R e = l ) is characterized by (3.9) and Lemma 4. The coefficients a, b, c, d for c~ > 0 (respectively, a, b, a', b' for c~=0) have to be a (nontrivial) solution of (3.15) [respectively, (3.16)]. Such a solution exists if and only if Pa(1//0 or QI(1/p) [respectively, P2(1//~) or Qz(1/p)] are zero, the characteristic equation being Dj (/t) = 0 with D~ given by (3.23a) [respectively, D2(,u) = 0 with D 2 given by (3.24a)].

If P~(1 /# )=0 for some # then by (3.17) P(1,#)7 ' (1)=(OP/@ ) (1, p) ~(1) and a solution of (3.15) can easily be seen to be

P(1,/~) b = d = 0 , c - a

~(1)

consequently the corresponding @N given by (3.12) is an even polynomial in y [P(y, i~) and 7(y) being even], and so is o5 N.

If QI(1 /#)= 0 for some # then again by (3.17)

r Q(1, #) a '(1)=-~-y (1, #) a(1)

and in this case a solution of (3.15) is

a=c=O, a = Q ( l' #----~) b ~(1)

and the corresponding ~ as well as &N are odd polynomials in y. If c~ = 0 and P2 (1 /# )=0 for some/~ then by (3.18) (~P/Oy)(1,/~) = 0 and a solution of (3.16) is b = a' = 0, while aP(1, #) = b'. By (3.13) ~v is an even polynomial and so is &u. If Q2(1/#) = 0 then again by (3.18)

0_~_Q (1, ~) = Q(1, ~) @

854/5/1-5

70 Ehrenstein

and a solution of (3.16) can be found to be bQ(1, # )=a ' , and a=b' =0. NOW @N as well as ~)N are odd polynomials.

Proposition 2 is a result about the spectrum of the Chebyshev- collocation operator of the Stokes-type problems (3.3). The spectrum consists only of real and negative eigenvalues. This seems to indicate that the Fourier-Chebyshev approximation of the Stokes equations (3.1a) (3.1c) in the vorticity co and streamfunction ~ formulation is suitable for time-dependent computations. The above result contrasts with the fact stated in Gottlieb and Orszag (1977) that a Chebyshev-Tau approximation applied to the fourth-order Stokes equations in the streamfunction formulation exhibits spurious eigenvalues--that is, those that are positive.

In the next section a family of difference schemes is applied to (3.3) for time discretization. We present the influence matrix technique used to solve the resulting system at every time step, and we study the properties of this time marching algorithm.

4. STABILITY OF THE INFLUENCE MATRIX TECHNIQUE

Again we consider the system (3.2) and we apply a time discretization by a family of finite difference schemes to the first equation (to study the stability of the time stepping schemes we can set f = 0)

(1 +e)~o ~+1-2~o ~ - (1 -~)co ~-1

2at

=01vLoon+I +O2vLco"+(1-OI-O2)vLfon-1 (4.1)

where ~om~o~(x, mAt), L=O2/Ox2-cd, (v=l/Re). This scheme is of second order for parameters such that

= 201 + 02 - 1

and of third order when (4.2)

2 e 1 02 3' 5 + ~ 201

We have the two classical schemes:

i. The Crank-Nicolson scheme for

01 = 1/2, 02 = 1/2, ~ = 1


ii. The second-order backward Euler scheme for

81=1, 82=0, e = 2

We apply the Chebyshev collocation approximation to the spatial operator and we consider semi-implicit schemes 0 < 01 ~< 1.

The total discretization leads to

0 2C~v +1 _ ( ~ § l § @2 ( Y J ) 2"O~v-AtJ r~N+'(YJ)

1--0101 --82 [_~ ~ 2(D ~'V- 1 ~ ( Y J ) - - (XC0~V-- 1 (yj ) ]

l - - e 281vAtrnu-l(yj), I<~j<~N--1

2 n+l : - - ( y j ) - - ~ U + 1 ( y j ) - - ,+ j ~ __ (.0N l (yj) , i ~< N - - 1

(4.3a)

(4.3b)

(4.3c)

(4.3d)

g,T ' (+l )=O

- - ( + 1 ) = 0 @

[oJ~ 1 ~<j~< N - 1 is given]. To start the integration (n = 0) we assume that C0N'=~0 ~ SO that the scheme gives the solution at t=2AU(l+e ) rather than at At. The solution at t=At is then defined by a linear extrapolation. In the following we present the influence matrix method used to solve the system (4.3).

4.1. Inf luence M a t r i x M e t h o d

This method makes use of a decomposition of the solution o~v and ~/~v

(2) N -- (4.4) m m m

~I N - - ~/ N § *~v l ~//1,N-~- /~2 ~/ 2, N

where (eS~, ~u), ((J)i,N, ~i,N), i = 1, 2 are polynomials of degree N satisfying 2 ~n+l 00)~v ~y2 (YJ)--(a+~)&~+l(Yj)=g~n-l(Yj), I <~j<~N-1

(4.5a) (DN(--~- 1) ---~ 0

72 Ehrenstein

and i = 1, 2

~)"-'''-2~ (yj) __ ~ t F l ( y j ) = (~)~r l ( y j ) ,

~%+'(•

I <~j<~N-1 (4.5b)

O2(Di, N Oy2 (yj) - (a + ~) co~.~v(yj) = 0, I<~j<<.N-1

(Di, N( ( - -1 ) i )=o , (Di, N( ( - - l ) i+ l ) = 1

02~ti'N (yj)--O~i,N(Yj)=(Di, N(Yj) , I < . j < . N - - 1 Oy 2

r

(4.6)

where

l + e a = ~ (4.7)

201vAt

g . . . . 1 is the right-hand side of the first equation of (4.3) [note that g ~ - I is known as a function of the initial condition e~~ In the following we consider only values of the parameters e, 0j, 02, such that a~>0.

To satisfy the boundary condition (4.3d) the coefficients 2~' and 2~ are the solutions of the system

[r - ' 111 ] (4.81 ~ ( - 1 ) ' m -

6 2 ( - 1 ) ] L ~ 2 _ 1 -~'m(-1) We denote by Mc the matrix of (4.8) and first of all we show the following:

Lemma 6. The matrix Mc above is invertible.

Proof Suppose that there exist (21, 22)~ Ker Me. According to the definition of Mc this would lead to the existence of

0)N = /~1 (-01,N "~/~2 ('02,N

being a nonzero solution of the system (3.5) with 2 = a and Re = 1. According to Proposition 2 this is impossible for 2 = a > 0.

Before we give some results about the stability of the time stepping algorithm (4.3) solved by the use of the influence matrix technique, we need


the following notations and definitions. To a polynomial qgN(y ) of degree N we associate the vectors

and

~N ~'~- ((PN(Yl), '", r T

~ = (~0 ~(y , ) , . . . , ~o~(yN_ ~))~

[ y i = cos( i~/N), 0 <~ i <~ N, are the collocation points]. Furthermore we define by D2, N the Chebyshev-collocation second

derivative operator as a ( N - 1 ) x ( N - 1 ) matrix such that for a polynomial ~o N of degree N with q~N( + 1 ) = 0 we have

D2,NO N = ~(~) (4.9)

[see, for example, Gottlieb et al. (1984) and Peyret (1986) for the expression of the coefficients of D2.N].

The linear forms D+ of order N - 1 are defined to satisfy

D+~bN = q~v(___ 1) (4.10)

with ON(_+ 1 )=0 (see Gottlieb et al., 1984; Peyret, 1986). Finally we define by AN the space of the polynomials q~N of degree N

satisfying

q~N ( "J- 1 ) = ~0~V( -~- 1) -~-- 0 ( 4 . 1 1 )

and by /~U CZ C N- 1 the vector space composed of vectors U such that

U= (D2, N - al)(Ju (4.12)

with q~N~ AN, ~ ~ 0 (l is the identity operator).

Now we come back to the influence matrix method used to solve the system (4.3) at each time step. This technique makes use of the decomposition (4.4), 2 7, i---1, 2 being solution of the system (4.8). We have shown by Lemma 6 that the matrix Mc of (4.8) is invertible, and we define

M~I = [mu], l<~i,j<~2 (4.13)

It follows by (4.8) and by (4.10) that

2i = ( _ m i . l D + mi, 2D_ ~m m -- ) ~ N , i = 1 , 2

where ~m m ~u N is the vector associated to ~u, solution of (4.5b). The operator O2, N - al is invertible for a ~> 0 (see Gottlieb and Lustman, 1983), and we can write

~m ~N = (Dz.N-- ~l)-~2~V (4.14)

74 Ehrenstein

( ~ is the vector associated to Ch~v); consequently

27 = ( - m i . l D + - mi,2D_ )(D2,N -- ~ l ) - 1/2~V ~m =-Mi, NY2N, i = 1, 2 (4.15)

By the definition of the vector space J*N given by (4.12) it is easy to see that

Mi, NU=O, U ~ A N , i = 1, 2 (4.16)

Furthermore we associate (2j, N to COj, M, j = 1, 2, given by (4.6) and we have

Mi, N~'2j, N = --0/j, 1 ~<0~<2 (4.17)

(6/j is the Kronecker symbol), because by (4.6) and (4.15)

M,,NOj, N = -- mi, 1 ~;(1 ) -- mi,2 C j( -- 1)

and the expression above is the (0")-th coefficient of - M ~ - I M c. These considerations allows us to prove the following lemma.

Lemma 7. According to the decomposition (4.4), the vector s associated to the solution a~v of (4.3) satisfies

m ~m ff'2 N : BN(O~)~ N (4 .18)

with

Ker[BN (~) - l] = AN

Ker BN(c 0 = C(21,N O C(22,N

that is BN(~) is a projection from C Nvl onto the subspace AN defined by (4.12).

Proof. By use of (4.15) the matrix BN(a) gives

BN(O~ ) = l'71- G N((~ )

where the ith row of G~v(~) is defined to be

Gi= Ml,uO91,N(Yi) + m2,NOgZ, u(Yi), l <<. i <~ N - 1

By (4.16) A N c K e r G N ( ~ ) and by (4.17) GN(CQE2j, N= --12j, N, j = 1, 2, which proves the lemma, the vector space ~'~N being of dimension N - 3. D


We denote by

( ' + ~ ) s (~ + a)f2~V a=2OlvAt

the left-hand side (m = n + 1) of (4.3a), and by (4.5a) we obtain the following equality:

~27v (2) (~ + ~)~27~ (D2,N-- (~ + ~) - m - - : [ ) i f 2 N

.= HN(O~, --m a)f2 N (4.19)

the right-hand sides of (4.3a) and of (4.5a) being identical. By the use of (4.18) together with (4.19) we can rewrite (4.3a) as

follows:

HN(~, ~ ) ~ ; 7 1 = _ Oll H~(~, a ) - ~1B~(~) -m

+ [ 0 1 + 0 2 - 1 ](2N (4.20) Ol HN(O~, 0.) __ z.2BN (~) - n - 1

with aO2vAt+e 2a(1 - 0 1 - 0 2 ) v A t + ( l - e )

(4.21) zl 01vAt ' T2~- 20[vAt

The influence matrix technique gives rise to the time-marching algorithm (4.20), the final solution CON + 1 at every time step is obtained by the projection

n+ 1 obtained, r +1 (4.18) together with (4.15) [note that once a solution CON is given by (4.3b) and (4.3c), (4.3d) is satisfied automatically]. Therefore the stability properties of (4.20) determine the stability of the time marching algorithm (4.3). To obtain some results about (4.20) we define OTv = s 1 and (4.20) becomes

[ ~'~ I l [~" A~ [-~r l F ~r ~vl (4.22) ~nN§ OJLoNnNJ~-~ELoN_J with

-02 F= ~ l - z l HN(a, G)- 1BN(~) (4.23)

01 + 02 - 1

A - 01 l-z2Hjv(~, ff)-lBu(Ot ) (4.24)

l is the identity operator.

76 Ehrenstein

In the following section we determine the spectral radius p(E) of the matrix E defined by (4.22) in order to obtain necessary conditions for the stability of the iterative algorithm (4.20).

4.2. Spectral radius of E

Suppose that Z = (X, y ) r with X, Y~ C n- ~ is an eigenvector of E with eigenvalue #; then

X = # Y

and ( # F + A) Y= #2 y (4.25)

The operator g F + A is of the general form al l+ a2HN(e, a) I BN(~) for some constant coefficients al , a2; consequently Y is an eigenvector of HN(e, a) -a BN(~) and hence of F and A, therefore (4.25) becomes

#2 _ 7# _ 2 = 0 (4.26)

with F Y = 7 Y and A Y = 2 Y. We conclude that the eigenvalues of E are obtained by considering an

eigenvector Y of

FN = HN(C~, a) ~ BN(e) (4.27)

hence FY=~,Y, AY=)LY, and #i, i = 1 , 2 , solutions of (4.26) are eigenvalues of E with the corresponding eigenvectors

Zi = (uiX, y)r, i = 1, 2 (4.28)

Consequently we have to characterize the eigenvalues of the matrix FN given by (4.27):

Proposition 3. For all e >1 0 and cr > 0 the matrix F N defined by (4.27) is diagonalizable, it has a kernel of dimension 2, the other eigenvalues are

1 l <~i<~N-3

~.i-- ff'

where hi, 1 ~i~< N - 3 are the eigenvalues of the system (3.5) (with Re = 1) which are real and negative by Proposition 2.

Proof From Lemma 7 and (4.27) we conclude that

Ker(FN) = Ker[BN(~)] = C~e~I,N (~ CO2, N


Let us consider an eigenvector co~ of the system (3.5) with eigenvalue 2. We define a polynomial g0~ by

02~~ (yj) - (~r + a) ~o~(yj) = 0 I~j<~N--1 ~y2

(4.29) ~pA+l)=co~.(_+ 1)

and again we associate to co~ (respectively, g0~) the vector s (respectively, ~b~). We show that g'2~ - ~b~ is eigenvector of FN with eigenvalue 1/(2 -- a).

Indeed by Lemma 7 we have

B x (a)(f2~ - ~ ) = t?~

because (2)~ belongs to *~N and q~. = cox(l)~r N .

The following equality

f i N ( Q ) . - - ~,~) = H N 1 ( ~ , 0-) ~'2a = ]/(~Q,a. - - ~ ) . )

holds for some # if and only if

Q ) . = t 2 F D 2 , N - - (0~ + ~r) l](t'2). - ~ba) (4.30)

by the definition of HN(~, a). But (o~). - ~ba)( _+ 1 ) = 0 and therefore

[D2 N - - (0~ -}- ~ ) l ] ( ~ t ~ ) . - - [[~ * - - 0 ( 2 ) - - "4)(2) (0{ -'}'- 0")(~r - - 4 ) , ) , "gZ) - - ~ 2 7"). - -

[g'2(~ 2) and ~b(a 2) are the vectors associated to (82cojOy 2) and (82go~jOy2), respectively]. We conclude that

[D2, N - (a + a) l ] (O) , - ~b;) = (2 - a)O;,

because co;, is a solution of (3.5) with Re = 1 and ~0~ a solution of (4.29). Consequent ly (4.30) becomes

1 ~2~ = #(2 - er)g2)., thus # = 2 - a 0

The Propos i t ion 3 will enable us to evaluate the spectral radius of E defined by (4.22). First of all we need some definitions and results that can be found in Miller (1971).

Let f(z) be a polynomial of degree n with real or complex coefficients; we define

78 Ehrenstein

where f is the polynomial whose coefficients arc the complex conjugates of those of f and

f* (0) f(z) - f(O) f* (z) A ( z ) =

Z

We say that f is a v o n Neumann polynomial if its roots z satisfy Izl ~< 1 and f is a Schur polynomial if its roots z satisfy Iz] < 1. We have the following results (see Miller, 1971):

Theorem 1. f is a v o n Neumann polynomial if and only if one of the following two conditions is satisfied:

i. If*(0)1 > [f(0)l and f l is of von Neumann type.

ii. f l = 0 and f ' is of von Neumann type.

Theorem 2. f is a Schur polynomial if and only if [f*(0)l > If(0)l and f l is of Schur type.

We want to apply these results to the polynomial (4.26), that is to f (#) -- #2 - 7# - 2. Here f * ( # ) = 1 - 7 # - 2 # 2 and f ~ ( # ) = ( 1 - 2 2 ) # - 7(1 + 2). The first condition of Theorem 1 states

If*(0)[ > If(0)] ~ 1 > I,tl

and f l is a yon Neumann polynomial

~ 1 7_--~ ~<1

The second condition of Theorem 1 states

f l - 0 ~ = ~ 2 = - 1 or ~.=1 and 7 = 0

and f ' is of yon Neumann type

Theorem 2 states that f is of Schur type

and ~ <1

We can state the following result about the spectral radius p(E) of E:


Proposition 4. If 02 > 1/2 in the family of finite difference schemes (4.1), then p(E)> 1 with E defined by (4.22).

Proof The matrix F u = HN(~, a) -1 B:v(C~) admits zero as an eigenvalue (see Proposition 3). We denote by Y the associated eigenvector and by (4.23), (4.24)

FY=TY, A Y = 2 Y

with 7 = -02/01, 2 = (01 + 02- 1)/01. The roots of the equation (4.26) are eigenvalues of E, and by Theorem 1, a root I#l ~< 1 if and only if one of the two following conditions is satisfied:

i. J01 + 0 2 - 11 01_202 1011 <1, ~<1

-- (01 +02--1 ) ii. 01 + 02 1 _ _ 1 or =1 and 02=0 01 01

and 0_~ ~< 1

The second condition of (i) is equivalent to 02 ~< 1/2 and the first condition of (ii) becomes 02 = 1 -201, which together with ]02/2011 ~< 1 is equivalent to 02 ~< 1/2.

As a consequence we have the following corollary.

Corollary 1. The influence matrix technique applied to the solution of (4.3) is unconditionally unstable for values of the parameters such that the temporal diseretization is of third order.

Proof The temporal discretization is of third order [see (4.2)] for 02 = 2/3 and e/2 + 1/3 = 201 and we conclude by Proposition 4.

Now we examine only values of parameters, such that the temporal discretization is of second order (that is, e/2=201 + 0 2 - 1 ) and we use only values of the parameters in (4.1), such that

(H) 01~]0,1], 02=0 or 0 2 = 1 - 0 1

This allows us to prove the following proposition:

Proposition 5. By assuming the hypothesis (H) above we have

i. p(E)>l if 0<01<1)2

ii. p(E)=l if 01=1/2

iii. p(E)<l if 1/2<0 L~<I

80 Ehrenstein

Proof The result (i) is a consequence of Proposition 4 because according to (H) 0 < 01 < 1/2 ~ 02 > 1/2.

The condition 01 = 1/2 with 02= 1 - 0 , corresponds to the Crank Nicolcon scheme which is in fact a two time-level scheme thus (4.22) simplifies to

~"+~ = F ~ " (4.31)

with

F = - l - - v - ~ t F N

[see (4.23) with ~ given by (4.21)]. We have shown that F~v has an eigenvalue being zero, thus F has - 1

as eigenvalue. The other eigenvalues are (see Proposition 3)

4 4 ]2i= --1 1

v At(2e- a) v At 2e- 2

with 2i eigenvalues of (3.5) (with Re = 1). But 2i < 0 so

I# i [< l for all v, A t > 0

and we have p(F)= 1. Now we come to the last assertion of the proposition. (a) First we suppose that 02 = 0; then

F = - FN

A = ---~101--1l - [ 2 ~ ( 1 - 01) v A t + l - e ] 2 f f ~-l v At F N

by (4.21), (4.23), (4.24). If FNY=O, then F Y = O and A Y = 2 Y with 2 = (01-1/01), and

1/2 < 0z ~< 1 assures that the roots #i, i = 1/2 of (4.26) satisfy I/~il < 1. Now if Y is an eigenvector of FN with eigenvalue q ~ 0, then

s

F Y = ~ Y , A Y = 2 Y with 7 - O~v Airl

0 1 - 1 2~r(1--Ol)vAt+(1--e) 2 - - - - q

01 201 v At


For the roots of (4.26) to be such that I#l < 1 we have to show that

12[<1 and ~ - 2 <1

We know that t /= 1/0~- a), X<0 (Proposition 3) with a defined by (4.7), and 2 = A/B with

A = B - 2v01At )~ + 2e01

B = 01 [2v01 At )~- (1 + e)]

But e= 401- 2 > 0 because 01 > 1/2, therefore B < 0 and it is easy to see that indeed [2J < 1.

Furthermore

and clearly

7 -201e

1 - 2 2 v O 1 A t z - 2 O l e

1 ~_-~ <1

Now we come to the following case:

(b) 02 = 1 - 01. Here we have

0 2 aO2v At + ~

e - 1 A - F u

201 v At

We write

with

A = -2v01 At )~ + B

B = 20~(v At Z - -2 ) < 0

82 Ehrenstein

Clearly [A/BI < 1 if B < vO 1 At Z that is if

v A t z 01>

2(v At z - 2)

but this condition is assured for 01 > 1/2, which achieves the proof of the proposition.

4.3. Concluding Remarks

The Proposition 4 states the remarkable result that the system (4.3) leads to a time marching algorithm which is unconditionally unstable for schemes with 02 > 1/2. This result contrasts with the fact that for the heat equation approximated by finite differences Chebyshev-collocation there is always a result of stability of the form (1-201) A t < C / N a f o r schemes (4.1) with e= 1 and 02= 1 - 0 j (see Gottlieb and Orszag, 1977; Ouazzani et al., 1986). In this case even for 02 > 1/2 there is always a time step such that the time marching algorithm is stable in the sense of p(E) < 1.

We have seen in the proof of the assertions (ii) and (iii) of Propo- sition 5, how the result about the spectrum of the Chebyshev-Fourier Stokes operator stated in Proposition 2 assures that p(E) ~< 1 (for schemes of second order in time). The Crank-Nicolson scheme [Proposition 5 (ii)] is marginally stable, while schemes belonging to (iii) seem to have the best properties of stability; in particular the second-order Euler backward scheme (~ = 2, 01 = 1, 02 = 0) associated with an Adams-Bashforth evaluation of the nonlinear terms has been used in Demay et al. (1987), Ehrenstein and Peyret (1989), Pulicani (1988), and Vanel et al. (1986), for the solution of the Navier-Stokes equations. Although the property p(E)< 1 is not sufficient to conclude that the discretization is stable when At--, 0 and N ~ o e (see, for example, Gourlay and Griffiths, 1980), it assures nevertheless that the approximated solution reaches a steady state for n ~ oe (At and N fixed) whenever the exact solution of the Stokes problem reaches a steady state.

We conclude that the Chebyshev-Fourier approximation of the two- dimensional Stokes equations in the vorticity-streamfunction formulation has a spectrum consisting of eigenvalues that are real and negative and is therefore suitable for time-dependent computations. The influence matrix technique is an attractive method to solve the system (4.3) at every time step because it permits the consecutive computation of solutions of the Helmholtz equations (4.5a), (4.5b) with homogeneous Dirichlet boundary conditions. To ensure the Neumann boundary condition for the stream-


funct ion the decompos i t i on (4.4) together with the solut ion of (4.8) is used: This m e t h o d leads to the t ime march ing a lgor i thm (4.22), and P ropos i t i on 5 shows that it is a lways poss ible to choose values of the pa rame te r s in the family of schemes (4.1) such tha t the t ime discre t iza t ion is of second order and which assures that p ( E ) <~ 1, E being the ampl i f ica t ion mat r ix of (4,22).

A C K N O W L E D G M E N T S

The au tho r is very grateful to Professor R. Peyre t for helpful discus- sions and to Professor D. Go t t l i eb for his interest in this work.

R E F E R E N C E S

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Ehrenstein, U., and Peyret, R. (1989). A Chebyshev-collocation method for the Navier -Stokes equations with application to double-diffusive convection, Int. J. Numer. Methods Fluids, 9, 499-515.

Gantmacher, F. R. (1960). Matrix Theory, Vol. II, Chelsea Publishing, New York. Gottlieb, D., Hussaini, M. Y., and Orszag, S. A. (1984). Theory and application of spectral

methods, in Speetral Methods for Partial Differential Equations, Voigt, R. G., Gottlieb, D., and Hussaini, M. Y. (eds.), pp. 1-54, SIAM, Philadelphia.

Gottlieb, D., and Lustman, L. (1983). The spectrum of the Chebyshev collocation operator for the heat equation, SlAM J. Numer. Anal 20, (5), 909-921.

Gottlieb, D., and Orszag, S. A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications, BMS Regional Conference Series in Appl. Math., SIAM, Philadelphia.

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Kleiser, L., and Schumann, U. (1980). Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows, Proc. Third GAMM-Conference on Numerical Methods in Fluid Mechanics, Hirschel, E. H. (ed.), pp. 165-173, Vieweg, Braunschweig.

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84 Ehrenstein

Ouazzani, J., Peyret, R., and Zakaria, A. (1986). Stability of collocation-Chebyshev schemes with application to the Navier-Stokes equations, Proc. Sixth GAMM Conf. on Numer. Methods in Fluid Mechanics, Rues, D., and Kordulla, W. (eds.), pp. 287-294, Vieweg, Braunschweig.

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the spectrum of a chebyshev-fourier approximation for the

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