Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations

2.

Now (2.14) becomes < 2Pop3Am 1+i" -dtd - iql 2 + 9(1 ~ \ ~ - - ' *~t m-i/2~lai/2,~l + i ] Ixi ' . i l 2 -Using (2.11) we get d 1 3 -1 1. -dt- Iql2 + ~Am+llql 2 _< 2poPlAm+ By the standard Gronwall inequality we obtain (2.12).

We adapt the method used for the NavierStokes equations by Titi [58, 59] to introduce a similar approximate inertial manifold for the Kuramoto-Sivashinsky equation. Let

= { p ~ P H : Ilpll ~ 2p1}, ~.L = [q ~ QH: Ilqll ~ 2 p l ] . As for the Navier-Stokes equations we follow Foias and T~mam [21] and Titi [58, 59] to show that there exists a mapping ~s: ~ __+Q H satisfy-

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashinsky equation

ing

tiating (3.3) one obtains

OT,,(q),7 a------q--

+ ¢ ' ( v ) ) = o,

+Qn(p + p~.

Theorem 3.1. l e t m be large enough so that Am +1 -> max( 4r~, rl2/4p 12),

(3.2)

where +

~J~-r~2,~ -1/8,~"*rn-1/4 ~ v ~ P'I'*I + 1

,

r 2 = 1 + 8PlAi-1/SAml+/14, with P0, Pl as in (1.1), (1.2). Then there exists a unique mapping ¢~: ~ - - , Q H satisfying (3.1), whose graph defines a C-analytic manifold. Moreover,

IlCS(p)ll~A~+lr 1, for all p ~ ~ .

- A - 1 Q [ B( 77' P + q)

+B(p +q,n)].

(3.1)

The graph of Cs, denoted ..~/s,contains all the steady states of (1.3). For this reason we will call the manifold .~s the steady approximate inertial manifold.

r I = 2Pl

45

Using again (1.4) we have that

aq

_ Am1/1211mll÷ 8P1A~-1/SAm3+/141bTII A,L/12r211II ½16711.

It follows that Tv is a contraction mapping. Thus for any p ~ ~ there exists a unique fixed point of Tp we denote by ¢~(p), which defines ¢~: ~---> ± satisfying (3.1). As in ref. [21] it follows that ¢ s is a C-analytic manifold. The mapping ¢s, which is given by the implicit relation (3.1), is our candidate for an AIM. In the next theorem we introduce a sequence of simple explicit approximating functions for Cs. These approximating functions will also be associated with AIMs, which are the ones used in part of our actual computations in section 5.

Theorem 3.2. l e t m be large enough so that (3.2) holds. Define

Proof. For each p ~ ~ we define ¢0(p)=0

Tp(q) -..~A-1/2q - A - I Q B ( p

for a l l p ~ ,

+ q , , + q),

for all q ~ ~ _L

and (3.3) ¢i+1(P) = Tp(¢i(P))

We begin by showing that T,: ~ ± --* ~ 1. From (3.3) we have

for all p ~ ~i~, and for i = 1,2,3 . . . . . Then

IITp(q) II -< IA -

~/4ql

+

~

- 3 / 4 Q B ( p + q, p + q ) I.

I1¢111---~ " Ic ~-3/4 J-x 3,'~.m + 1

(3.4)

,

Using (1.4), it follows that

and

IITp(q)ll < 2plAto 1+/12+ 8~/2-plzA1 l/8~t'~m+l-3/4

lieS(p) - ¢ , ( P ) II ~ 2K3a'A~,3+/14,

= A,:1+/12rl < 2Pl. We now show that Tp is a contraction. Differen-

where r 3

=

4~-li-'/Sp~,

a

=

r2Xml/12.

(3.5)

M.S. Jolly et al. /Inertial manifolds for the Kuramoto-Sivashinsky equation

46

Proof. Let p ~ ~ be fixed. Using (1.4) we find

and

that ][q(t) - ~i(p(t))ll <--2K2Am+ -7/4x + 2K3a ih ,-3/4 ,+l,

[l~l(p)l[- [A3/4QB(p,P)I

for a l l t > T * *

and

i=1,2 .....

(3.8)

<_A~3+/141B(p, p) I where T** and K 2 are as in theorem 2.2, and K 3 and ot are as in theorem 3.2.

_<4~-A ~-X/Sp12/~m3+/14 proving (3.4). Since Tp is a contraction mapping, with contraction constant a (see proof of theorem 3.1) we have

lu(t)l-
and

Ilu(/)ll-
for all t > T**(p).

II~k+ X(P) -- ~ k ( P ) II ~ akll~x(P) -- 4 0 ( P ) II = akll~l(p)l[

Proof. Because of (1.1) and (1.2) we have

In particular qbS(p(t)) is well defined and satisfies 114,S(p(t))ll_< 2pl (see theorem 3.1). Let A ( t ) = q ( t ) - ~ S ( p ( t ) ) , for t > T**. From (1.3q) and (3.1) we have

for k = 0,1,2 . . . . .

Since oc

qbS(P)

-- Cibi(P) =

E

[qbk+l(P)

-- ¢ J ~ k ( P ) ] ,

AA -Ax/2a + Q[ B( A , p + ~S(p) ) + B(u,A)]

k=i

dq

we have

+-d-T = 0 . co

II~S(p)

-

Using (1.4) we get

~i(p)ll-< E akll~x(P)ll k=i

IAAI < IAX/2al

2a'll~,(p)ll.

+ v~-Ia 11/211al11/211p + ~S(p)II We now substitute (3.4) to get (3.5).

+ Following Titi [58, 59] we show in the next theorem that all trajectories of (1.3) are attracted exponentially fast to a layer about graph ( ~ 0 , which is significantly thinner than that about PmH given in theorem 2.2.

Theorem 3.3. Let m be large enough so that (2.11), (3.2) and A--X/2m+I"+ A,~ ~t--7/8 -[- J~-,~l/2,~l/2Jt - 3 / 4 < 1 -r~X"m+X v~YO /"X " * m + l --

"~1('~'~

i

ll,211u:ll

ll +

From the above and (2.13) we obtain IAAI -< (~-1/2 v-,,+x + 4,°xX,7,,7/~8 + ,/~-~1/2,,~1/21 - 3/4 ~] IA zi I + K 2 A m l+ xv'~'P'0 V'l "~m+l

We now substitute (3.6) to reach (3.7). The estimate (3.8) follows directly from (3.7) and (3.5).

hold. Then for every p > 0 and for every solution u(t) = p ( t ) + q(t) of (1.3) with lu(0)l _
From estimates (3.7) and (3.8) we infer that the graph of qb2 is as close to the global attractor as the steady AIM, graph(qb0. We will refer to the manifold given by the graph of qO2 as the

[A(q(t) - ~ ( p ( t ) ) )

pseudo-steady approximate Menial manifold.

I < 2g2Arn+X ,-1

(3.7)

M.S. Jolly et al. / Inertial manifolds for the Kuramoto -Sivashinsky equation

Remark 3.4. The formal derivation of Tp from the implicit relation

Aq-A1/Zq + QB(p + q,p + q) = 0 amounts to simply bringing both the destabilizing linear t e r m al/2q, along with the bilinear term, to the right-hand side before operating on both sides by A-1. An alternative approach would be to bring over only the bilinear term, and then operate by ((A-AX/2)Q) -1, leading to the mapping

Tp( q) = - ( ( A - A ! / 2 ) Q ) - I Q B ( p + q, p + q). We remark that, for large enough m, one can show that Tp is also a contraction mapping defined on ,fi~.t. Accordingly, one can approximate ~s by the successive iterates of Tp, namely ^

4. Analytic comparisons of approximate inertial manifolds In this section we compare both the motivation and error estimates for three AIMs in the literature with three new ones introduced here as variants on the pseudo-steady AIM. For the purpose of this discussion it is convenient to consider the general evolution equation du

d----[+ Au + F(u) -- 0.

o(p) -= o, f o r p ~ ~ , and i = 1,2 . . . . . In addition, one can restate the analogue of theorems 3.2, 3.3, with ~i replacing ~,.. In doing this, however, there is no gain in the rate of approximation of ~i over that of ~;. Moreover, ((A-A1/2)Qm)-I cannot be inverted for the parameter L -- 2.rr(m + 1), which is precisely the value at which the trivial solution undergoes a pitchfork bifurcation making the dimension of the unstable manifold of the origin m + 1 dimensional. While beyond this parameter value there could be no true inertial manifold of dimension m, we will see in section 5 one instance in which the m-dimensional version of ~2 continues to capture important qualitative behavior. Nevertheless, ~1 does have the advantage of involving a major instability term which is not present in ~1. We will not address this issue in detail in this work, but we will comment on it later in section 4.1.

(4.1)

In the case of the KSE we lump -AX/2u and B(u, u) in (1.3) together as the nonlinear term F(u). We will often refer to the approximate inertial form (AIF)

dp + a p + e F ( P + ~ a ( p ) ) = O , dt

p~eH, (4.2)

^

=

47

where the AIM is the graph of ~a: P H ~ QH. The error in all the methods is described in terms of the maximum vertical distance in the H-norm between the global attractor ~¢ and the graph(~a). More precisely, for u - - p + q a~', we have Error(~a) = max Iq~a(p) - q l.

^

^

Since all the error estimates in this paper are of the form kAm~+X for some positive constants k and y, both independent of m, we will simply speak of the order of the error as am~+l.

4.1. The traditional Galerkin AIM The simplest of all AIMs is a flat one. In theorem 2.2 we have seen that for m large enough there is a layer about PmH, with thickness of -1 order Am+l, which attracts every trajectory exponentially. To employ this linear manifold PmH, amounts to simply setting ~ a - 0 in (4.2), and gives the traditional Galerkin approximation of (4.1) associated with the eigenfunctions of the

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashinsky equation

48

operator A. We use this as the standard to compare against any other AIF. More sophisticated AIMs have some curvature due to nonlinear terms in q~a. For this reason the following AIFs are often referred to as "nonlinear" Galerkin methods of approximating (4.1).

4.2. An earlier approach to

(1) 1

The following method was introduced by Foias, Manley and T~mam for the 2D Navier-Stokes equations in bounded domains. Projecting onto the space of divergence free functions, one can show that the Navier-Stokes equations with no slip or with periodic boundary conditions reduce to a special evolution equation (cf. refs. [10, 37, 54]), du

d--'-t+Au + B ( u , u ) =f.

(4.3)

Splitting (4.3) as was done for (1.3), and expanding the bilinear term, we have for the higher modes dq

4.3. The Euler-Galerkin AIM We now turn to a method which approximates a true inertial manifold, rather than just the global attractor. The key to the Hadamard approach to prove existence of inertial manifolds is the evolution forward in time starting with appropriate initial data [10, 11, 39]. For example, the semi-flow induced by the solution of (4.1) takes an initial manifold ~/0, under appropriate assumptions, to a true inertial manifold ~ as t ~ oo. Since .Z/is exponentially attracting, after a relatively short time interval ~', . ~ , the image under the semi-flow of . d 0, can be shown to be already very close to .~. This is the motivation behind the AIM in ref. [20]. The key in that approach is to approximate the true evolution with implicit Euler integration for small time r. This yields, for ~" small enough, an AIM q = ~ ( p ) satisfying

q = -~'(~'A + I ) - ' Q F ( p +q), which is the fixed point of the contraction mapping

+Aq + Q [ B ( p , p ) + B ( q , p ) + B ( p , q ) ~ff-: q~, - ~ ' ( I + r A Q ) - I Q F ( p + q ) .

+ B ( q , q)] = Qf. It is shown in ref. [17] that for large enough m and uffip+q on z~' one has that Idq/dtl, IQB(q,p)I, IQn(p,q)l, and Ian(q,q)l are relatively small. Neglecting these terms, they obtain the AIM given simply by the graph of

(Pl( P) = - A - 1 Q [ f _ B( p, p ) ] ,

(4.4)

which has an error of order A~3+/12. Here, however, the eigenvalues of A, the Stokes operator, only grow linearly [44]. Notice the similarity between qfil in (4.4) and ~1 in theorem 3.2. There we also estimated an error on the order of Am-3/2 +l , except that for the KSE the eigenvalues grow like m 4 (see section 1).

(4.5)

The first iterate of the mapping 2V- with initial guess .~o=Pm H, i.e. q = 0, yields an explicit function qtl,~: PmH ~ QmH given by

qtl,~(p ) = - r ( I + ~ ' A Q ) - X Q F ( p ) .

(4.6)

It is shown in ref. [20] that the maximal vertical distance between graph(qtl,~) and graph(g'~) is comparable to that between g r a p h ( ~ ) and .£~, which is, for appropriately small r, comparable to that between d(~ and .~. In the case of the KSE this error is of order A~7/14. We refer to the manifold defined by (4.6) as the Euler AIM and the AIF (4.2) with qb, __ ~1,, as the Euler-Galerk/n method.

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashimky equation

49

Observe that as z---, % so that the Euler integration is no longer rigorously justified, we have that ~ --->~s and ~1,~ ~ ~1, where ~1 is as in theorem 3.2.

For (ii) we start by observing that again by (3.4)

4.4. New methods: variations of the pseudo-steady AIM

Estimating Ilpxll~, we have by Agmon's inequality

<~C3p~/2 Ipl 1/4 IApl 1/4

= A-1/2CI)l( p )

< c3P~/2p 1/4 lAp[ 1/4

-A-1Q[ B ( p , p ) + B( qOl,p) +B(p,

~1) + B(~I,

~1) ] .

(4.7)

In the proposition below we estimate certain terms in q~z which are later dropped to produce three new AIMs.

Proposition 4.1. Let m be large enough to satisfy (2.11). Then for every u = p + q ~ ~ ing estimates hold:

the follow-

[A-1QmB(+I(p),~I(p))[

(i)

9 J ~ - / ( 2 ) , - 21/8 -~< ~ v ~ 3 Z t m + 1 ,

(ii)

[A-1QmB(~l(p),p)l ~.. ,.~1/2~1/4,.~ 1/41 -- 2 -~<~36,1 6,0 6,2 "*m+l,

(iii)

[A-IQmB(P,~I(p))I ,/rg-~ 1/2~ 1/2~ - 7/4/,,-~< v~6,0 6"1 "*m+1"~3,

where/9 2 is as in theorem 2.1, Po and Pl are as in (1.1) and (1.2), respectively.

Proof. By Agmon's inequality and (3.4) we have IA-1Qmn(fI)l( p),

t~l(p))[

-< ,LT,'+,ll'ar'1(p) IMl'ath(p) II _< ,LT,~+i v~-ICh(p) 11/211~,(p) II3/2 -< x,7, ~+i v'2 ~,7,,~+/~8II¢,i(p) II2 _

,I - I

,]-~~

- 1/81,-

2 A - 3/2

<'*m+lV~"~*m+laX3

_<,~m2+~llPxlL.

IIp~IL ~ c3lp~l WZlp~l~/~ < cap]/2lA1/2p[1/2

Recall from theorem 3.2, the function qD2(p)

IA-1QmB(~I(p),p)[ < Am111~(P)I IlPxll®

m+l "

Since Ihpl ~P2 by theorem 2.1, we obtain estimate (ii). Finally, we have by Agmon's inequality, (3.4), (1.1) and (1.2) that [m- 1Qmn ( p, t ~ l (

P))I

A~,~+IlIplLII~I(p)II - 1 1 V/~ IPl -~
1/211PIIt/2KaA~3+/14

- 7 / 4 J'~- ,~1/2.1/2/d" -~<'tm+l v ' 6 , 0 6,1 ~x3

proving (iii). We now consider dropping some of the terms in (4.7). Since the terms in (i) and (ii) are no - 2 1, dropping either one or larger than order Am+ both will yield an AIM with the same error as for ~s. We will refer to the AIF derived by dropping only the term in (i) as "pseudo-steady II", and that obtained by dropping the terms in both (i) and (ii) as "pseudo-steady III". Dropping all three terms estimated in proposition 4.1 yields an approximation with an error dominated by the term in (iii). Thus this last variation, "pseudosteady IV", gives an error of the same order as the Euler AIM, ~-7/4 "*m+l • In fact by gathering the B(p, p) terms we find that the method formally requires just one iterate; thus it resembles the Euler-Galerkin method in form as well. We also remark that one can derive the pseudo-steady IV manifold from ~t(P), which was introduced in remark 3.4. Indeed from remark

50

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashinsky equation

Table 1 Error estimates of order A,Sn~.1 for AIMs applied to the KSE. Approximation ~(p) flat (linear)

¢~1 pseudo-steady(4,2) pseudo-steadyII pseudo-steadyIII pseudo-steadyIV Euler (~1,~,)

~,

0 -A-1QB(p, p) A- I/2~1(p) - A - IQB(p + qBl(p), p + ~ I(P))

1 3/2 2

4 2 - A - t QB( ~ l( p ), q~l( P )) cb2 - A - 1QB( ~ I( p ), p + ~ I( P )) - ( A - 3/2 + A - 1)QB(p, p ) - ' r ( I + c A Q ) - 1QB(p, p)

2 2 7/4

3.4 we have

7/4

between the m-dimensional first iterate approximations (~1, Euler, pseudo-steady IV) and P2m H .

4,,(v) = - ( A -A'/2) 'QB(v,v), 5. Computational results and therefore

¢~l(P) ~'=-

AQ) -'-k/2 B ( p , p ) .

By keeping just the first two terms in this infinite geometric series, we obtain the pseudo-steady IV manifold, with a negligible remainder. In table 1 we summarize the forms of all the approximations discussed in this section as well their error estimates. It should be emphasized that these estimates in general are not sharp. An exception is the estimate for the fiat manifold, where examples are provided in refs. [20, 58, 59]. At first glance it appears that to achieve the accuracy of an m-dimensional pseudo-steady AIM, one would have to use roughly m 2 modes in a fiat (traditional Galerkin) approximation. However, this conclusion is easily disqualified in the case of the KSE. Due to the choice of basis, and the fact that the nonlinearity is quadratic, we have that Q 4 m ~ 2 ( p ) -- 0 for all p ~ PmH (a similar observation was made in the context of a center manifold approximation in ref. [2]). While this makes the method easy to implement, as there is no truncation error, it also implies that the mdimensional pseudo-steady (II, III) AIM is actually contained in P4m H, and hence could be no closer to the global attractor than the 4mdimensional linear space. The same relation holds

We now evaluate how well the various AIFs capture the dynamics of the Kuramoto-Sivashinsky equation. Rather than examine the accuracy of individual trajectories at particular parameter values, we compare entire bifurcation diagrams of stationary and periodic solutions. To ensure that no bifurcations are missed in the continuation, we also calculate the stability of the solution branches. In this comparison we concentrate on a parameter range over which the structure of elements of the global attractor suggests the existence of an inertial manifold of dimension three. It should be emphasized that as of yet, this claim is far from being verified analytically. After rescaling time and space (KS) becomes au

o-7+

4 04u

[ i~2u

0 < x < 2rr,

0U )

=0, (KS~)

where a = L 2 / r r 2. All of our computations were performed for (KS~). The bifurcation diagrams were obtained using AUTO, a bifurcation package [13] while the phase portraits were produced with an interactive code, SCIGMA [34]. 5.1. S t e a d y states

The bifurcations of the steady states for the KSE have been examined numerically in a num-

M.S. Jolly et al. / Inertial manifoldsfor the Kuramoto-Sivashinsky equation

ber of papers, including but not limited to refs. [2, 5, 6, 27, 30, 35, 36, 45]. These studies, when restricted to the space of odd solutions, concur with the results of a twelve-mode traditional Galerkin approximation over the range a ~ [0, 70] shown in fig. la. A further motivation for using twelve modes, when eight would yield nearly the same results on the interval of interest, is that as we discussed at the end of section 4, the threedimensional pseudo-steady manifolds parametrize the fourth through twelfth modes in terms of the first three. We first introduce some terminology for the elements appearing in most of the diagrams with a brief review of the situation for twelve Galerkin modes. At a = 4 , 16,36,64, the trivial solution undergoes pitchfork bifurcations leading to solution branches referred to as unimodal, bimodal, trimodal and quadrimodal, respectively. Each solution branch is symmetric in a certain subset of modes, for which the signs can be switched. We shall distinguish the two sides of each branch as either "positive" or "negative" according to the signs of the symmetric modes near criticality. Note that the two sides of each branch will be superimposed in our bifurcation diagrams where we plot the solution norm versus parameter. These primary, non-trivial branches undergo a certain pattern of secondary bifurcations according to a predictive scaling law in ref. [50]. We follow Scovel et al.'s labeling scheme. In particular, the positive side of the bimodal branch undergoes its own pitchfork bifurcations leading to the bi-tri branch at the point R2b2, and the tri-bi branch at REb 4. At REb 3 the negative side of the bimodal branch also has a pitchfork bifurcation, leading to a symmetric giant branch of solutions which apparently grow in amplitude without bound as a increases [27]. Both the tri-bi and giant solutions play critical roles in distinguishing the performances of the various nonlinear Galerkin methods. We now compare the steady state bifurcations for the twelve-mode traditional Galerkin method with various nonlinear Galerkin methods. As in-

51

dicated in fig. lb and in ref. [16], the steady state bifurcation diagram for the Euler-Galerkin method (with 7 = 0.1) is qualitatively correct up to ot = 40. Noticeably absent however, are both the tri-bi and giant branches between a = 36 and a = 64. The same is true for the pseudo-steady II Galerkin shown in fig. lc. Only the pseudo-steady Galerkin in fig. ld successfully captures one of the branches between 36 and 64, the bi-tri, albeit at a considerably inflated value of a. While we expect the corresponding bifurcations to occur at reasonable values of a, we have also searched as high as a = 300. Henceforth, when a bifurcation is said to be missing, we mean that we have not observed it up to this value of a. We remark that we carried out similar computations using the AIM given by 41, but found the results nearly indistinguishable to those for the Euler-Galerkin manifold. We also compare the results described above with lower-dimensional traditional Galerkin approximations, to determine the minimum number of traditional modes necessary to produce the essential features of the correct diagram. In figs. le, lf, lg, and lh are the diagrams for the three-, four-, five- and six-mode traditional Galerkin methods respectively. Observe that the nth mode bifurcates vertically if the traditional Galerkin truncation is of dimension < 2n, since in that case, it is a " p u r e " mode with no nonlinear balancing, as is easily seen in the equations provided in the appendix. A quantitative comparison of the steady state bifurcation values is provided in table 2. 5.2. Periodic solutions

We now turn to the time-dependent behavior of the KSE. For comparison purposes, we start with the eight-mode traditional Galerkin approximation, which appears to share the same behavior as the true KSE, for the parameter interval of interest here. The first periodic branch we encounter while increasing a, bifurcates from the positive bimodal at HB r It initially consists of

M.S. Jollyet aL / Inertialmanifoldsfor the Kuramoto-Sivashinskyequation

52 15

15

(c)

(a) Pseudo- Steady ]I

Traditional Galerkin,12 modes

n,0 Z

3modes

o

Z

Z 0 I--

Z

3

o

0

(2 ~

70

15

O(

~ ~

15

(b) Euler-Golerkin, 3modes

70 (d)

Pseudo- Steady 3 modes

Ig Z

Z

Z

Q F-

I.D .J

£ 0

Q ___.~

OJ

70

t5

I1 -------~

70

15 (e)

Traditional Galerkin 5 modes

Traditional Galerkin 3 modes

:E nO Z "7 0 I,-

0 Z Z 0

J 0 (/)

J O

I

70

15 Traditional Galerkin :E

Traditional Galerkin 6 modes

nO z

z F2D _.1 O

Q ~

70

Q ~

70

Fig. 1. Steady state bifurcation diagrams, solution norm versus parameter a. The solution norm is

Ilull2s =

L a2, where u(x,t)= L ak(t)sin(kx)" kffil

kffil

The open squares denote steady state bifurcations, the black squares denote Hopf bifurcations. In (a) bold lines indicate where both superimposed branches are stable, while . . . . . indicates where only the "negative" side is stable.

M.S. Jolly et aL /Inertial manifolds for the Kuramoto-Sivashinsky equation

,_o_s.oy:ct

7 Trodihonol Golerkin

i

8modes

53

(o)

o g

i11/ HB2 - " -"~ : " ~ 2 " - / I

I

I

I

I

130

g 8ct~ ,

1I

Ct ~

37

,

,

30

,

I

k'

/

CZ ~

37

(b)

od.es

Z•

IE o z

~

Pseudo-Steody

I

/ I--

I

I

I

I

I

a

~o 25

i

I

I

I

~

37

I

l

[

(e)

I

,I

a

Traditional Golerkin

Trad,tiono I Galerkin 3modes

~E

I

30

I

/

II

~

37

i

5 modes

I

(g )

I

I i

rr

o

o

Z

Z

o

....

g

.............

i---

g _ _

1

~

I

I

t

"""-'--'--~ I

I

t

CZ

40 I

Traditional Galerkin

4modes I

I

I

a

I

~

I

i

f

37 /

(f)

Traditional Golerkin

5:

6 modes

( h)l

n.o z z

nO z z o I-

F-

6

o o3

_o _J

%

i

1

i

(2 -------=-

i

I

i

J

40

30

J

t

i

I

C/ -------=--

it

|

37

Fig. 2. The solution norm is Ilvll~ as in fig. 1 for steady states, and ]lvlf2r= 1 rT n~ ~J. L

~ k=l

2 ak(7")d'r,

for periodic solutions, where T is the period and n is the dimension of the approximating dynamical system. ( - - - ) steady states; ( - - ) limit cycles.

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashinsky equation

54

Table 2 P a r a m e t e r v a l u e s a for s e c o n d a r y s t e a d y s t a t e bifurcations. Galerkin method

R2b 1

R2b 2

R2b 3

R2b 4

R3t 1

R3t 2

R4b I

R4q I

3-mode Euler-Galerkin 3 - m o d e p s e u d o - s t e a d y II 3-mode pseudo-steady 12-mode t r a d i t i o n a l 6-mode traditional 5-mode traditional 4-mode traditional 3-mode traditional

16.103 16.130 16.131 16.140 16.140 16.140 16.140 16.000

20.590 21.928 22.009 22.556 22.547 22.828 21.124 16.000

246.14 102.90 93.913 52.891 52.721 48.493 52.067 16.0??

63.737 63.278 61.856 62.355 16.0??

36.206 36.206 36.238 36.234 36.234 36.000 36.000 36.000

63.908 50.911 46.851 36.000 36.000 36.0??

64.559 64.000 64.000 64.000 -

64.275 64.000 64.000 64.000 -

stable limit cycles, which as sets are symmetric in the odd modes. At a = 32.85 the symmetric periodic orbit becomes unstable, giving birth to two stable limit cycles, both asymmetric by themselves as sets, yet also reflections of each other in the odd modes (and hence superimposed in fig. 2a). The symmetric branch becomes extremely unstable (one Floquet multiplier eventually greater than 5000), making its continuation increasingly difficult computationally. For this reason we do not attempt to continue this branch to its conclusion. A sequence of period doublings of the asymmetric branch follow, with stability inherited by the period doubled branch at each bifurcation. We have continued the original branch of the now unstable, asymmetric solutions well past the period doublings and through a "corkscrew" like behavior indicative of a Sil'nikov loop [26, 61]. The two superimposed periodic branches bifurcating from each side of the bi-tri branch at HB 2 undergo a period doubling, after which both the original and the doubled branch exhibit Sil'nikov corkscrews. For details of the periodic behavior see ref. [4]. The diagrams for the nonlinear Galerkin methods have all the features mentioned above. Two of them are illustrated in the three-dimensional phase space of the pseudo-steady AIF. In figs. 3a-3d we depict the initial stages of the period doubling sequence of the asymmetric limit cycle culminating in a "thick", apparently chaotic orbit. An example of a periodic solution near a Sil'nikov loop is shown in fig. 4. We note that the Euler-Galerkin AIF produces artificial turning

c°

(b)

(c)

Fig. 3. P e r i o d d o u b l i n g s e q u e n c e of a n a s y m m e t r i c limit cycle at (a) a = 32.975, (b) a = 32.989, (c) a = 32.993, a n d (d) a = 32.997, o b t a i n e d via t h r e e - m o d e p s e u d o - s t e a d y approximation.

M.S. Jolly et aL /Inertial manifolds for the Kuramoto-Sivashinsky equation

55

whose detection does not improve monotonically in the number of modes.

5.3. Artificial inflation of the absorbing ball

Fig. 4. An unstable limit cycle near a Sil'nikov loop at a 32.989 and the point in phase space (0.9734,- 0.5669, 1.568) on the bi-tri branch, obtained via three-mode pseudo-steady approximation.

points in the periodic branches bifurcating at H B 2 , while the pseudo-steady AIFs avoid this artifact. The fate of the periodic solutions for the lower-dimensional traditional Galerkin methods is not clear. While the three-mode traditional approximation fails to find all but a vestige of the symmetric branch bifurcating from the positive bimodal branch, both the four-mode and fivemode truncations yield the primary Hopf bifurcation at HB1, asymmetric limit cycles, and the beginnings of the period doubling sequence of the oscillatory branch bifurcating at HB 1 off the bimodal branch. In addition the four-mode traditional approximation apparently has at least one branch ending in a Sil'nikov loop, and perhaps more. The computational difficulty in tracing the periodic branches, especially around the turning points in a Sil'nikov corkscrew forces us to leave open the existence of those missing. The fivemode traditional approximation exhibits a more definite small distortion: the periodic branch bifurcates from the positive bimodal branch at HB 1, but now does so subcritically, quickly turning back in the correct direction at a spurious turning point. This artifact is interesting in that it was not present for the four-mode traditional approximation, providing a feature of the dynamic behavior

We now turn to a side effect of the AIFs employed here. As mentioned in section 5.1, the nonlinear Galerkin methods failed to find the giant branch within the interval 36 < a < 64. Considering that the bi-tri branches for the pseudosteady AIF bifurcated from the trimodal branches at an inflated value of a, it is reasonable to expect that the bifurcation of the giant branches may also be shifted. In figs. 5a, 5b and 5c we show 300

~

.

Galerkin3modes(a)

150

300 OC

o Z

15C

Z

_o I--_.J

o

CO

0 300

(c)

Pseudo-Steady 3modes 150

iO0

Of ".--,,.-

200

300

Fig. 5. Continuation of the negative bimodal branch for large a for (a) Euler-Galerkin, (h) pseudo-steady If, (c) pseudosteady methods.

56

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashinsky equation

the continuation of the negative bimodal branch for the Euler-Galerkin, pseudo-steady II, and pseudo-steady AIFs for much higher values of a. While all three methods produce branches on which the solution norm is unbounded, as in the giant branch of the full KSE, they all cause the branches to turn back, and apparently end with a vertical asymptote at a = 0. Since in each case any absorbing ball for the corresponding dynamical system must include the giant solution, the radius of the ball will grow as a decreases, in contrast to the case of the full KSE, which is dissipative. This is also unlike the traditional Galerkin approximations in which the ball suddenly exploded with a steady state bifurcation of the trivial solution. Yet one can show, following Nicolaenko et al. [48], that there exists a*(m) which is an increasing function of m, such that the traditional Galerkin approximation of order m is dissipative for all values of a which are smaller than or* (see ref. [33]). The nonlinear Galerkin case may be explained by the fact that the function representing the AIM is a polynomial of finite degree which, by its growth at infinity, allows an artificial balance of the modes to create a giant steady state at small or. One remedy for this would be to truncate in a smooth fashion the "nonlinear" term, -A1/2u + B(u, u), outside a ball of twice the radius of the absorbing ball, as is done in existence proofs of inertial manifolds (see e.g. refs. [10, 11, 19, 39]). This modification may introduce new dynamic behavior locally, but would not affect the dynamics within the absorbing ball. We investigate this remedy and others in a separate work [33].

6. Conclusions

The computational results here indicate a strong correspondence between the estimates in table 1 and the qualitative performance of the AIFs tested, even though we use such a low number of modes (m = 3) and the rigorous error estimates are asymptotic as m ~ ~. In general,

the dynamics of the nonlinear Galerkin methods seem comparable to that of the traditional Galerkin method using up to twice as many modes. In particular the pseudo-steady AIF, which is based on an AIM passing through the steady states, accurately captures time-periodic behavior, not only locally near Hopf bifurcations, but globally, to the conclusion in Sil'nikov loops. The improvement of the error estimate of the pseudo-steady AIF over the Euler-Galerkin was manifested in two ways: the capture of the tri-bi branch of steady states, and the avoidance of the artificial turning point in fig. 2b. On the other hand, the capture of the tri-bi branch also distinguished the pseudo-steady from the pseudosteady II AIF, which is obtained from the former approximation by neglecting a term estimated to be smaller than the total error. This indicates a sensitivity to certain terms that our measurement of error may not detect. While this work demonstrates the effectiveness of these nonlinear Galerkin methods for the KSE, the potential for computational savings in a large-scale application has yet to be realized. Most of the computational effort in any discretization of the KSE is concentrated on evaluating the nonlinear term. Since the nonlinear Galerkin methods involve the composition of the nonlinearity, direct implementation of the equations for these AIFs will be more costly per mode. For example, in the case of a single-iterate method such as the Euler-Galerkin method, computing the equations for three modes would require the same order of work as the first three equations of the six-mode traditional Galerkin truncation. Nevertheless, the fact that less equations are eventually used can be a distinct advantage in long semi-implicit or implicit integrations, as well as in stability and bifurcation analysis, in which calculations using a smaller Jacobian matrix become a significant factor. Moreover, in the particular case of the KSE over the parameter range of interest here, being able to investigate certain global bifurcations in a three dimensional phase space may prove to be a crucial advantage.

57

M.S. Jolly et aL / Inertial manifolds for the Kuramoto-Sivashinsky equation

Acknowledgements We thank one of the referees for a number of helpful suggestions, which included the addition of the appendix. This work was supported in part by NSF grants CBT-8707090, EET-8717787, DMS 881-0684, DOE Grant DE-FG02-86ER25020, the US Army Research Office through the Mathematical Sciences Institute of Cornell University, and the Packard Foundation.

We denote the complex coefficients in (A.2) by Yk, and using (A.1), rewrite them in terms of the sine coefficients as -1 ~/k

=

--

1i j

+ E

k-1 - j b - j j b k - j + E jbjbk-j m j=l

-

bjbjk,

j=k+l

Appendix

where bj = 0 for j < 1 and for j > m. The m-mode traditional Galerkin approximation is then given by

An m-mode traditional Galerkin approximation amounts to substituting the truncated Fourier expansion

db~ dt = ( - 4 k 4 + akE)bk - afl'~, 1 < k < m,

m

u(x,t)=

(A.3)

Y'~ Uk(t) e ikx where

k=-m

into (KS~) and gathering terms. The only effort comes when evaluating the bilinear term B(u, u ) = u au/ax. The restriction to the space of odd functions requires that the complex coefficients u k be purely imaginary. Setting

fl'~( b 1. . . . . bin) = 2iy k m

l ~_,jbj[bk+j+sgn(k-j)blk_il]

(A.4)

j=l

In particular for m = 3 we have

Uk = -- ½ib k for l_
for - m _
-- 0

otherwise,

dbl - -dt = ( - 4 + a ) b 1+ l a ( b 2 b 3 + b i b 2 ) , (A.1)

one has the sine representation

db3 dt = ( - 3 2 4 + 9 a ) b 3 - ~Otblb 2.

m

u ( x , t ) = Y'. bk(t ) sin(kx). k=l Yet it is easier to multiply complex polynomials to obtain

u

dt = ( - 4 k 4 + a k 2 ) b k

~_, k= --2m

We can express single iterate nonlinear Galerkin methods with the composition of the expression in (A.4) which yields db k

au(x, t) ax =

db2 1 2 - dt - = ( - 6 4 + 4 a ) b 2 + a ( - ~ b I +bib3) ,

j u j ( t ) Uk_y(t m

e ikx.

(A.2)

-- ° t f l 2 m ( b l . . . . .

1
bin,

~m+ 1. . . . . ¢~2m), (A.5)

M.S. Jolly et al. / Inertial manifolds for the Kuramoto-Sivashinsky equation

58

where for m + 1 < j < 2m, we define

is the same as that in (A.3), up to quadratic terms, with additional terms up to degree eight.

ot ta?mgb bm,O . . . . 0), 4'j=4')=- - 4 j 4 ~ J ~, 1. . . . ,

for q~a = 4)1, References

4 ' Y = ~ ' - - 1 f f ~ 4 j 4ti2m(bl . . . . ' b m ' O ' " " O )

for ~a = ~Tfl,r,

(A.6)

with ~1 and ~l,z as in section 4, only adapted for (KS~). The effect in (A.5) is to leave the quadratic terms the same as in (A.3), while adding cubic and quartic terms, which are also quadratic and cubic in the parameter a, respectively. This is most easily seen by the reconstruction of the bilinear function. Let b = (b 1. . . . . bm), 4' = (4'm+ 1. . . . ,4'2m ) and m

B~'( b;b') = 1 y , jb][bk+y + s g n ( k _ j ) b l k _ j l ] . j=l It is easy to see that B m ( b ; b ) = expanding to obtain

tim(b).

After

ti2m( b, 4' ) = B2m( b,O; b,O) -.bB2m(o,4''~b,O)

+ BZm(b,0;0,4,) -1--e2m(0, 4,; 0, 4,), we note that B2m(b,O; b,0) = tim(b, b). The new cubic terms then are introduced by B2m(0, 4,, b, 0) and B2m(b,O;0,4,), while the quartic terms are given by B2m(0, 4';0, 4'). The situation is similar for the second iterate methods. For example, the m-mode pseudo-steady A.IF is given by dbk dt

= (-4k4

+ak2)bk

- atti4m( bl . . . . . bm, 4,m+ l, . . ., 4,4m), 1 < k < m,

(A.7)

where for m + 1 < j _< 4m, we define 4,y---- ¼ a [ - - j - 2 ~ 1 - j - 4 t i ; m ( b , 4 , 1 , 0 , . . . , O ) ] ,

with 4,1 as in (A.6). Again the vector field in (A.7)

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