Implicit–explicit BDF methods for the Kuramoto–Sivashinsky equation

ximateethod.

des

ed andmes are

or thekwardseudo-umericald

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Applied Numerical Mathematics 51 (2004) 151–169www.elsevier.com/locate/apnum

Implicit–explicit BDF methodsfor the Kuramoto–Sivashinsky equation

Georgios Akrivisa,∗, Yiorgos-Sokratis Smyrlisb

a Computer Science Department, University of Ioannina, 451 10 Ioannina, Greeceb Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus

Available online 15 April 2004

Abstract

We consider the periodic initial value problem for the Kuramoto–Sivashinsky (KS) equation. We approthe solution by discretizing in time by implicit–explicit BDF schemes and in space by a pseudo-spectral mWe present the results of various numerical experiments. 2004 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:Kuramoto–Sivashinsky equation; Implicit–explicit BDF methods; Periodic attractors; Period doubling casca

1. Introduction

Linearly implicit methods for a class of nonlinear parabolic equations were recently constructanalyzed in [2–4] under various conditions on the nonlinearity and the schemes. These scheefficient and unconditionally stable. These schemes were not implemented in [2–4].

In this paper we focus on a concrete example, namely the periodic initial value problem fKuramoto–Sivashinsky (KS) equation, and its discretization in time by combinations of bacdifferentiation formulae (BDF) and appropriate explicit schemes. In space we discretize by a pspectral method. In this study we implement these schemes and present the results of various nexperiments. Our experiments establish thepth order of accuracy of thep-step BDF scheme anreproduce efficiently the universal attractors of the equation. Certain quantitative characteristicsuniversal attractors are calculated with the same order of accuracy.

We consider the periodic initial value problem for the KS equation: we seek a real-valued fuu = u(x, t), defined onR × R

+0 , satisfying

* Corresponding author.E-mail addresses:[email protected] (G. Akrivis), [email protected] (Y.-S. Smyrlis).

0168-9274/$30.00 2004 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2004.03.002

152 G. Akrivis, Y.-S. Smyrlis / Applied Numerical Mathematics 51 (2004) 151–169

ut + uux + uxx + νuxxxx = 0 for (x, t) ∈ R × R+0 , (1.1)

0

le

icalaklytrationin freedricalstable

mple onnamic

rees of0]. Fore.,sion

provedr the

icity of

rm

tst

u(·,0) = u , (1.2)

with u0 :R → R a sufficiently smooth, 2π -periodic function andν a positive parameter playing the roof viscosity. The solution of (1.1)–(1.2) will also be 2π -periodic in space, i.e.,u(x + 2π, t) = u(x, t) forall x ∈ R andt � 0.

KS is a simple partial differential equation (PDE) exhibiting a particularly complex dynambehaviour as the viscosity parameterν varies. It arises as an amplitude equation in long-wave, wenonlinear stability analysis, in a great variety of applications. For example, it arises in concenwaves in chemically reacting systems [19], in flame propagation and reaction combustion [24],surface film-flows of viscous liquids, and in the dynamics of interfaces in two-phase flows in cylingeometries [22]. It is one of the simplest PDEs with a convective nonlinearity and a band of unmodes, in its linearized version (around zero), and thus it has served as an appropriate exawhich the general notions of inertial manifold theory are applied. This means that the long time dybehaviour of KS is captured well by a finite dimensional dynamical system, the number of degfreedom of which is at least as large as the number of linearly unstable Fourier frequencies [12π -periodic solutions the number of linearly unstable frequencies is[ν−1/2] while the best estimatfor the dimension of the inertial manifold is O(ν−21/40) in the case of solutions of odd parity, i.eu(−x, t) = −u(x, t) [10]. This bound is useful in determining estimates of the Hausdorff dimenof the universal attractor.

The boundedness of the solutions of the KS equation, for general initial data, has beenindependently by Il’yashenko [15], Goodman [11] and Collet et al. [8]. The best estimate foHausdorff dimension of the attractor so far appears in [8]. In this work it is proved that

lim supt→+∞

∥∥u(·, t)∥∥ � c · ν−13/10,

with ‖ ·‖ denoting theL2-norm of 2π -periodic functions andc a positive constant, independent ofu0 andν. Similar boundedness results can be derived for any Sobolev norm of the solution. The analytthe solution has also been proved by Collet et al. [9]. In the case of the rescaledL-periodic KS equation

Ut + UUx + Uxx + Uxxxx = 0, (1.3)

it is proved that, for sufficiently larget, the functionU(·, t) is analytic in a strip of widthβ � cL−16/25

around the real axis, which in turn implies that the high frequency part of the spectrum has the fo∣∣U (j, t)∣∣ = O

(e−cL−16/25q|j |),

whereU(j, t) is thej th Fourier coefficient ofU(·, t) andq = 2π/L. A series of numerical experimenin [9] indicate the presence of a much stronger bound, namely, that there exists aβ > 0, independenof L, such that the solutions of (1.3) satisfy

lim supt→+∞

∑j∈Z

e2βq|j |∣∣U (j, t)∣∣2

< ∞, (1.4)

and numerical computations indicate thatβ ≈ 3.5. Straightforward calculations show that ifU(x, t) is asolution of (1.3), then

u(x, t) = L

2πU

(L

2πx,

L2

4π2t

),

G. Akrivis, Y.-S. Smyrlis / Applied Numerical Mathematics 51 (2004) 151–169 153

s

ericalr of thextensivetion, as

]. Whilemethodsrs (forer 5-splitar parttremely

licatedhaotic

page of

Fig. 1. The spectrum of the solution of KS for different values of the parameterν. Thelog-plot shows that the Fourier modedecay exponentially fast; slower asν decreases.

is a 2π -periodic solution of the KS equation forν = (2πL

)2. Similarly, if u(j, t) is the j th Fourier

coefficient ofu thenu(j, t) = U(j, L2

4π2 t). Therefore (1.4) implies that

lim supt→+∞

∣∣u(j, t)∣∣ � Me−β

√ν |j |, (1.5)

for some positive constantM. This bound has been extremely useful in the present work in the numexperiments. The truncation of the high Fourier coefficients becomes plausible and the numbefrequencies kept could be determined by (1.5). Nevertheless, in our work, we have carried out enumerical experiments in order to determine how many modes contribute numerically to the solushown in Fig. 1.

The KS equation has been studied numerically by many authors, see [13,14,16,18,20,23,25,26the space discretization has been consistently carried out via spectral methods, many differenthave been used for the time discretization including Runge–Kutta methods of different ordeexample, the subroutineradau5 which is based on the Runge–Kutta–Radau IIA method of ordwith step size control [12])1 and a split scheme of variable time step, according to the Strangmethod, integrating the linear and nonlinear parts separately, exploiting the fact that the linecan be integrated exactly in the Fourier space [25,26]. The numerical studies have revealed exinteresting low-dimensional dynamic behaviour, including stationary and travelling waves, comptime-periodic attractors, a variety of quasi-periodic attractors, homoclinic bursts and various c

1 This subroutine, together with its implementations, which include the KS equation, can be found in the webE. Hairer: http://www.unige.ch/math/folks/hairer/testset/testset.html.


attractors. Several transitions to chaos have been reported, including the one via period doubling cascades[25,26].

stableons onalsable butduce totion of

t

d. Time

establish

numberrtedctors, atcertain

ently the

hoiceernce for

Explicit multistep or Runge–Kutta schemes are unstable for (1.1) and may be only conditionallyin the fully discrete case, i.e., when we discretize in space as well, under very restrictive conditithe size ofkM4, with k denoting the time step andM the highest degree of trigonometric polynomiused for the space discretization. Implicit schemes on the other hand may be unconditionally sttheir implementation requires solving nonlinear problems at every time step; these problems renonlinear systems in the fully discrete case. In this paper we discretize in time by a combinaimplicit and explicit schemes leading to unconditionally stable, linearly implicit schemes.

With Av = νvxxxx + vxx + 1νv andB(v) = −vvx + 1

νv, the KS equation may be written in the form

ut + Au = B(u). (1.6)

We discretize the linear partvt + Av = 0 of (1.6) byp-step BDF schemes,p � 6, and the nonlinear parvt = B(v) by appropriate explicit schemes; this leads to unconditionally stable schemes of orderp.

The paper is organized as follows. In Section 2 we describe the numerical schemes usediscretization is obtained via stablep-step implicit–explicit BDF methods,p = 1, . . . ,6, whereas inspace we use pseudo-spectral methods. In Section 3 we present numerical experiments whichthe pth order of accuracy of thep-step BDF method via a suitably modifiedinhomogeneousproblemwhere the solution is known. We have carried out specific experiments in order to assess theof modes to be used, for different values ofν. We have also reproduced results of previously reponumerical experiments, through extensive computations. We do obtain the same universal attraa significantly smaller cost. Finally in Section 4, we give some concluding remarks and suggestother possibilities.

2. The numerical schemes

In this section we present the numerical schemes, first the time stepping schemes and subsequfully discrete schemes, and derive optimal order error estimates.

2.1. Discretization in time

Forp ∈ {1,2,3,4,5,6}, the polynomialsα, β andγ are given by

α(ζ ) :=p∑

j=1

1

jζ p−j (ζ − 1)j , β(ζ ) := ζ p and γ (ζ ) := ζ p − (ζ − 1)p.

Let αj andγj denote the coefficients ofζ j of the polynomialsα andγ, respectively. The(α,β)-schemedescribed by the polynomialsα and β is the p-step BDF scheme; these schemes are stronglyA(0)-stable and will be used for the discretization of the linear part of (1.6). The explicit scheme(α, γ )

will be used for the discretization of the nonlinear part of (1.6). Let us note that this particular cof the polynomialγ is motivated by the fact that, for the given(α,β)-scheme, it is the only choicleading to ap-step implicit–explicit(α,β, γ ) scheme of orderp, see [3]. We refer to [3] and [4] fomore information concerning the implicit–explicit BDF schemes; also see [5]. The standard referenumerical methods for parabolic equations is Thomée’s monograph [27].


Let T > 0, k denote the time step,N ∈ N be such thatNk = T , andtn := nk, n = 0, . . . ,N. We usethe(α,β, γ ) scheme to define approximationsUn to u(tn) by

a-ent

rthe

p∑i=0

αiUn+i + kAUn+p = k

p−1∑i=0

γiB(Un+i

), n = 0, . . . ,N − p, (2.1)

for given starting approximationsU0, . . . ,Up−1. Sinceαp > 0 and the operatorA is positive definite, theapproximationsUp, . . . ,UN are well defined by (2.1).

Forp = 1, . . . ,6, the scheme (2.1) takes the following form, respectively,

Un+1 + kAUn+1 = Un + kB(Un

), (2.2)

3

2Un+2 + kAUn+2 = 2Un+1 − 1

2Un + 2kB

(Un+1

) − kB(Un

), (2.3)

11

6Un+3 + kAUn+3 = 3Un+2 − 3

2Un+1 + 1

3Un + 3kB

(Un+2

) − 3kB(Un+1

) + kB(Un

), (2.4)

25

12Un+4 + kAUn+4 = 4Un+3 − 3Un+2 + 4

3Un+1 − 1

4Un + 4kB

(Un+3) − 6kB

(Un+2)

+ 4kB(Un+1) − kB

(Un

), (2.5)

137

60Un+5 + kAUn+5 = 5Un+4 − 5Un+3 + 10

3Un+2 − 5

4Un+1 + 1

5Un + 5kB

(Un+4

)− 10kB

(Un+3

) + 10kB(Un+2

) − 5kB(Un+1

) + kB(Un

), (2.6)

147

60Un+6 + kAUn+6 = 6Un+5 − 15

2Un+4 + 20

3Un+3 − 15

4Un+2 + 6

5Un+1 − 1

6Un + 6kB

(Un+5)

− 15kB(Un+4) + 20kB

(Un+3) − 15kB

(Un+2)

+ 6kB(Un+1) − kB

(Un

). (2.7)

Scheme (2.2) is obviously a combination of the implicit and the forward Euler methods.

Remark 1. In the experiments we have also discretized the equation

ut + uux + uxx + νuxxxx = f (x, t) in R × R+0 .

In this case the schemes (2.2)–(2.7) are modified by adding the termkf (tn+p) to their right sides,p beingthe order of the scheme. Alternatively,f could have been incorporated intoB.

Starting approximations. To maintain the order of accuracy of thep-step scheme, starting approximtions U1, . . . ,Up−1 to u(t1), . . . , u(tp−1), for p = 2, . . . ,6, of the same order are required. We preshere some choices leading to such approximations.

The first choice is based on a Taylor expansion and requires, forp > 2, the calculation of higher-ordederivatives of the initial valueu0. Assume thatu0 is sufficiently smooth such that one can calculate


time derivativesu(j)(0), j = 1, . . . , p − 1, of the exact solution att = 0. Then, it is easily seen thatU0 = u0 andUj = T

pu(0), with

ce it

nge–

daryo note

g withnished.

for the

s: we

j

Tp

j u(0) := u0 + jku(1)(0) + · · · + (jk)p−1

(p − 1)!u(p−1)(0), j = 1, . . . , p − 1,

approximateu(·, tj ) to the desired order. This choice might be inconvenient in applications sinrequires high order time derivatives of the solutionu.

Appropriate starting approximations can be alternatively obtained by using linearly implicit RuKutta schemes of order at leastp − 1 for the computation ofU1, . . . ,Up−2, cf. [21,17]; Up−1 can becomputed by the(p − 1)-step implicit–explicit scheme. Let us note that, due to the periodic bounconditions, the Runge–Kutta schemes do not suffer from order reduction in this case, cf. [1]. Alsthat schemes of orderp − 1, when applied a fixed number of steps, yield approximations of orderp.

Let us also mention that a further, suitable and popular approach is to “bootstrap” by startinlow-order schemes but with sufficiently small time steps such that the overall accuracy is not dimi

As already mentioned, for the second order scheme (2.3), we may computeU1 by performing one stepby the first order scheme (2.2), i.e., we letU1 be given by

U1 + kAU1 = u0 + kB(u0

). (2.3(1))

Let us also briefly discuss two convenient ways leading to appropriate starting approximationsthird order scheme. In this case, we need third order starting approximationsU1 andU2. OnceU1 hasbeen calculated, we may perform one step with the second-order scheme (2.3) to getU2. For U1, westart with the second order approximationsu0 + kut(0) to u(k) andu0u0

x + k(uut)x(0) to u(k)ux(k), andperform one step with a linearly implicit modified second order Crank–Nicolson scheme to obtainU1 by

U1 + 1

2kAU1 = u0 − 1

2kAu0 + kB

(u0

) + 1

2νk2ut(0) − 1

2k2(uut)x(0). (2.4(1))

The next starting approximationU2 may then be calculated by

3

2U2 + kAU2 = 2U1 − 1

2u0 + 2kB

(U1

) − kB(u0

). (2.4(2))

In (2.4(1)) the time derivative ofu enters. An alternative way of defining a third order approximationU1

to u(k), avoiding the use of time derivatives, is by means of implicit–explicit Runge–Kutta methodbegin with a second order approximationU1, computed by the implicit–explicit Euler scheme (2.2),

U1 + kAU1 = u0 + kB(u0

), (2.4(1′))

and correct it to a third order approximation by the linearly implicit Crank–Nicolson scheme,

1

k

(U1 − u0) + 1

2A

(U1 + u0) = B

(1

2

(U1 + u0)),

i.e., by

U1 + 1

2kAU1 = u0 − 1

2kAu0 + kB

(1

2

(U1 + u0)). (2.4(1′′))


2.2. Discretization in space

esent

enever

ential

luesunded

ns ofd

Fourier

The spatially 2π -periodic initial data assumption of the initial value problem enables us to reprthe solutionu of the KS equation in the form

u(x, t) = α0(t) +∞∑

j=1

(αj(t)cosjx + βj (t)sinjx

).

The termα0(t) remains constant due to the conservative nature of KS. On the other hand, whu(x, t) is a solution then so isu(x − ct, t) + c, which allows us to assumeα0(t) = 0, for simplicity.Replacingu(·, t) by its Fourier series in the PDE we obtain

ut + uux + uxx + νuxxxx

=∞∑

j=1

[(α′

j − (j2 − νj4

)αj − Aj

)cosjx + (

β ′j − (

j2 − νj4)βj − Bj

)sinjx

],

with

Aj = −j

2

∑m+n=j

αmβn + j

2

∑m−n=j

(αmβn − αnβm),

Bj = j

4

∑m+n=j

(αmαn − βmβn) + j

2

∑m−n=j

(αmαn + βmβn),

for j ∈ N. Thus KS equation is transformed into an infinite dimensional system of ordinary differequations (ODEs):{

α′j = −λjαj + Aj,

β ′j = −λjβj + Bj,

for j ∈ N,

and

Aj = Aj(α1, α2, . . . , β1, β2, . . .) and Bj = Bj(α1, α2, . . . , β1, β2, . . .),

andλj = νj4 − j2 are theeigenvaluesof the linear operator. The algebraic growth of the eigenvamakes the system stiff. All the spatial Sobolev norms of the solution of the KS equation remain bofor all times, see [8,11,15]. Thus

ωj = lim supt→+∞

∣∣α2j + β2

j

∣∣1/2, j ∈ N,

remain bounded for all times. In particular, the Collet et al. [9] proof of the analyticity of the solutiothe KS equation, implies that theωj ’s decay exponentially with respect toj. They have also presentenumerical evidence in [9] that the number of determining Fourier coefficients is proportional toν−1/2.This work enables one to approximate the solution of the KS equation by truncation of the higherfrequencies.

By applying the Fourier transform tou(x, t) we have

d

dtu(j) = λj u(j) − (uux )(j).


Thus forj > ν−1/2 we easily obtain

.

ncated

calllow to

lim supt→+∞

∣∣u(j)∣∣ � 1

λj

lim supt→+∞

∣∣(uux )(j)∣∣ = j

2λj

lim supt→+∞

∣∣(u2)(j)

∣∣.The above inequality provides a first (crude) estimate of the number ofsignificantFourier coefficientsNevertheless, according to our numerical experiments with various schemes, a small multiple of[ν−1/2]suffices for calculations in double precision. In our numerical experiments, we use at least 4ν−1/2 Fouriermodes. In fact the number of modes in our experiments is an integer of the form 2m or 3× 2m, so that theFFT is efficiently implemented for the nonlinear term.

The computed value of the nonlinear termsAj andBj via the pseudo-spectral method is

ANj = −j

2

j−1∑m=1

αmβj−m + j

2

N−j∑m=1

(αm+j βm − αmβm+j ),

BNj = j

4

j−1∑m=1

(αmαj−m − βmβj−m) + j

2

N−j∑m=1

(αmαm+j + βmβm+j ).

Note that in the nonlinear part of the system of ODEs corresponding to high frequencies in the trusystem, i.e.,AN

j , BNj , most of the contribution comes from the low frequencies, even forj large. This

implies that higher frequencies areslavedby the low frequencies, which is one of the most typicharacteristics of dissipative infinite dimensional dynamical systems. Nevertheless, we wish to athe high frequencies as much freedom as possible in order to develop individual behaviour.

2.3. Fully discrete schemes

Let µj := νj4 − j2 + 1ν. We now present the fully discrete schemes. Forp � 3, we also give the

starting approximations, except for the first one which is the initial value.The first order scheme is

(1+ kµj )un+1j =

(1+ 1

νk

)un

j − k(unun

x

)j. (2.8)

The second order scheme takes the form

(1+ kµj )u1j =

(1+ 1

νk

)u0

j − k(u0u0

x

)j, (2.9(1))

cf. (2.3(1)), and(3

2+ kµj

)un+2

j = 2

(1+ 1

νk

)un+1

j −(

1

2+ 1

νk

)un

j − 2k(un+1un+1

x

)j+ k

(unun

x

)j. (2.9)

The approximationu1 for the third order scheme can be calculated from(1+ 1

2kµj

)u1

j =(

1+(

1

ν− 1

2µj

)k

)u0

j − k(u0u0

x

)j

+ 1

2νk2(ut (0)

)j+ i

1

2k2(u0ut (0)

)j, (2.10(1))


cf. (2.4(1)), or, alternatively, from( )

ctively.

(1+ kµj )(u1)

j= 1+ 1

νk u0

j − k(u0u0

x

)j, (2.10(1′))

and (1+ 1

2kµj

)u1

j =(

1+ 1

2

(1

ν− µj

)k

)u0

j + 1

2ν

(u1)

j− 1

4k((

u1 + u0)(u1x + u0

x

))j, (2.10(1′′))

cf. (2.4(1′)) and (2.4(1′′)), the approximationu2 by(3

2+ kµj

)u2

j = 2

(1+ 1

νk

)u1

j −(

1

2+ 1

νk

)u0

j − 2k(u1u1

x

)j+ k

(u0u0

x

)j, (2.10(2))

cf. (2.4(2)), and the other approximations by(11

6+ kµj

)un+3

j = 3

(1+ 1

νk

)un+2

j − 3

(1

2+ 1

νk

)un+1

j +(

1

3+ 1

νk

)un

j

− 3k(un+2un+2

x

)j+ 3k

(un+1un+1

x

)j− k

(unun

x

)j. (2.10)

The fourth, fifth and sixth order schemes can be readily obtained from (2.5), (2.6) and (2.7), respe

2.4. Error estimates

For s ∈ N0, let Hsper denote the periodic Sobolev space of orders, consisting of the 2π -periodic

elements ofHsloc(R), and let‖ · ‖Hs

perbe the norm over a period inHs

per,

‖v‖Hsper

:=(

s∑=0

2π∫0

∣∣v()(x)∣∣2

dx

)1/2

.

The inner product inH := L2per = H 0

per is denoted by(·, ·), and the induced norm by‖ · ‖. As alreadymentioned, we letA :H 4

per→ H be defined byAv := νvxxxx + vxx + 1νv. ThenA is a selfadjoint, positive

definite operator,V := D(A1/2) = H 2per, and the norm||| · |||, |||v||| := ‖A1/2v‖,

|||v||| =(

ν‖vxx‖2 − ‖vx‖2 + 1

ν‖v‖2

)1/2

,

is equivalent to the norm ofV. Indeed, it is readily seen that

‖vx‖2 � ν

2‖vxx‖2 + 1

2ν‖v‖2 for all v ∈ V,

therefore

(Av, v) � 1

2

(ν‖vxx‖2 + 1

ν‖v‖2

)for all v ∈ V, (2.11)

and thus,A is positive definite.


Furthermore, letB :V → H be given byB(v) := −vvx + 1νv. Then,

B(v) − B(w) = −1

2

(v2 − w2

)x+ 1

ν(v − w),

and thus by periodicity, forω ∈ V, we have(B(v) − B(w),ω

) = 1

2

(v2 − w2,ωx

) + 1

ν(v − w,ω),

and therefore,(B(v) − B(w),ω

)� 1

2‖v + w‖‖v − w‖‖ωx‖L∞ + 1

ν‖v − w‖‖ω‖. (2.12)

Now, sinceωx vanishes at some point in the interval(0,2π), it is easily seen that

‖ωx‖L∞ �√

2π ‖ωxx‖,and (2.12) yields

(B(v) − B(w),ω

)�

√2π

2‖v + w‖‖v − w‖‖ωxx‖ + 1

ν‖v − w‖‖ω‖. (2.13)

We identifyH with its dual, and denote byV ′ the dual ofV with respect to the inner product ofH, againby (·, ·) the duality pairing betweenV ′ andV, and by||| · ||| the dual norm onV ′. Since, obviously,√

ν

2‖ωxx‖ � |||ω|||, 1√

2ν‖ω‖ � |||ω|||, for all ω ∈ V,

we conclude from (2.13) that∣∣∣∣∣∣B(v) − B(w)∣∣∣∣∣∣

� 1√

ν

[√π ‖v + w‖ + √

2]‖v − w‖. (2.14)

Let Tu be a tube around the solutionu defined in terms of the norm ofH , i.e.,

Tu :={v ∈ V : min

t

∥∥u(t) − v∥∥ � 1

}.

Obviously, forv,w ∈ Tu, we have

‖v + w‖ � 2+ 2 max0�t�T

∥∥u(t)∥∥,

and (2.14) yields∣∣∣∣∣∣B(v) − B(w)∣∣∣∣∣∣

� µ‖v − w‖ (2.15)

with

µ := 1√ν

[2√

π(1+ max

0�t�T

∥∥u(t)∥∥)

+ √2]. (2.16)

Let M ∈ N andSM := {ϕ: = −M + 1, . . . ,M} with ϕ(x) := eix.


Let PM :V ′ → SM denote the orthogonalL2-projection operator ontoSM, i.e., (v − PMv,χ) = 0,χ ∈ SM. If we expandv ∈ L2 in a Fourier series,

tor

per

v =∞∑

=−∞vϕ,

thenPMv corresponds to the partial sum

PMv =M∑

=−M+1

vϕ.

This projection has the following approximation property:There exists a constantc, independent ofvandM, such that, forv ∈ Hm

per and = 0, . . . ,m,

‖v − PMv‖Hper

� cM−m∥∥v(m)

∥∥L2

per. (2.17)

Since differentiation commutes withPM, we havePMA = APM. Further, we define the discrete operaBM :H 2

per→ SM, BM := PMB.

In the semidiscreteproblem corresponding to (1.1), (1.2), we seek a functionuM , uM(·, t) ∈ SM,

satisfying{∂tuM(·, t) + AuM(·, t) = BM

(uM(·, t)), 0 < t < T,

uM(·,0) = u0M,

(2.18)

with u0M ∈ SM a given approximation tou0.

Let us now rewrite (2.18) in a more convenient for computational purposes way. LetuM,j (t) denotethe Fourier coefficients ofuM(·, t),

uM(·, t) =M∑

=−M+1

uM,ϕ.

It is easily seen that

PM(uMuMx)(·, t) = iM∑

j=−M+1

(M∑

=−M+1

(j − )uM,uM,j−

)ϕj ,

and thus, lettingχ = ϕj in (2.18), we may rewrite it in the form{∂t uM,j (t) + µj uM,j (t) = 1

νuM,j (t) + i

∑M=−M+1(j − )uM,(t)uM,j−(t),

uM,j (0) = u0M,j ,

(2.18′)

j = −M + 1, . . . ,M, with µj = νj4 − j2 + 1ν.

We recursively define a sequence of approximationsU ∈ SM to u(·, t) by

p∑i=0

αiUn+i + kAUn+p = k

p−1∑i=0

γiBM

(Un+i

). (2.19)

Let W(·, t) ∈ SM denote theL2-projection ofu(·, t) in SM, W(·, t) = PMu(·, t), t ∈ [0, T ].


Let EM(t) ∈ SM denote theconsistency errorof the semidiscrete equation (2.18) forW ,( )

lowing

e

EM(t) := Wt(·, t) + AW(·, t) − BM W(·, t) , t ∈ [0, T ]. (2.20)

Obviously

EM(t) = Wt(·, t) + PMAu(·, t) − PMB(W(·, t)),

and thus, in view of (1.6),

EM(t) = PM

[B

(u(·, t)) − B

(W(·, t))],

and in view of (2.15) and (2.17), under obvious regularity assumptions, we easily obtain the foloptimal order estimate for the consistency errorEM,

max0�t�T

∣∣∣∣∣∣EM(t)∣∣∣∣∣∣

� C(u)M−m. (2.21)

We can now derive an optimal order error estimate:

Theorem 2.1. Assume that we are given starting approximationsU0, U1, . . . , Up−1 ∈ SM to u(·, t0), . . . ,

u(·, tp−1) such that

max0�j�p−1

∥∥u(·, tj ) − Uj

∥∥ � c(kp + M−m

). (2.22)

Let Un ∈ SM, n = p, . . . ,N, be recursively defined by(2.19). Then, if the solutionu of (1.1)–(1.2)issufficiently smooth, there exists a constantC, independent ofk andM, such that, fork sufficiently smallandM sufficiently large,

max0�n�N

∥∥u(·, tn) − Un

∥∥ � C(kp + M−m

). (2.23)

Proof. Let W j := W(·, tj ), j = 0, . . . , p − 1, and defineWn ∈ SM , n = p, . . . ,N, by applying the timestepping scheme to Eq. (2.20), i.e., by

p∑i=0

αiWn+i + kAWn+p = k

p−1∑i=0

γi

[BM

(Wn+i

) + EM

(tn+i

)]. (2.24)

Then, according to [2, Theorem 4.2], under the assumption

max0�t�T

∣∣∣∣∣∣∂jt u(·, t)∣∣∣∣∣∣ � C, j = 1, . . . , p + 1,

for a constantC, we have

max0�n�N

∥∥W(·, tn) − Wn

∥∥ � Ckp. (2.25)

In view of (2.17) and (2.25), it remains to estimateϑn := Wn − Un. Subtracting (2.19) from (2.24), wobtain

p∑i=0

αiϑn+i + kAϑn+p = k

p−1∑i=0

γi

[BM

(Wn+i

) − BM

(Un+i

)] + k

q−1∑i=0

γiEM

(tn+i

).


Since on the right side of (2.15) only the norm‖ · ‖ appears, i.e., in the notation of [2] we haveλ = 0,

and alsoβ(ζ ) = ζ p, i.e., all coefficients ofβ but the one ofζ p vanish, see [2, Remark 7.2], we may use

lue

mplicit

he KS

e stepation of

besifferent

uble

the stability estimate of [2, (5.16′)], to obtain

∥∥ϑn∥∥2 + k

n∑=0

∣∣∣∣∣∣ϑ∣∣∣∣∣∣2 � Cecµ2tn

{p−1∑j=0

∥∥ϑj∥∥2 + k

n−p∑=0

∣∣∣∣∣∣EM

(t

)∣∣∣∣∣∣2

}. (2.26)

From this estimate, (2.21) and (2.22), we easily conclude, fork sufficiently small andM sufficientlylarge,

max0�n�N

∥∥Wn − Un∥∥ � C

(kp + M−m

). (2.27)

From (2.17), (2.25) and (2.27) the desired estimate (2.23) follows and the proof is complete.�Remark 3. It is well known thatu(·, t) is analytic for positivet, cf. [9]. Therefore, the exponentm forthe spacial accuracy in the error estimate (2.23) is determined by the regularity of the initial vau0,u0 ∈ Hm

per. In the case of the equationut + uux + uxx + νuxxxx = f (x, t) the smoothness of bothu0 andf determinem.

Remark 4. The results of this subsection can be easily extended to the general class of linearly ischemes considered in [2].

We refer to [6] and [7] for spectral methods.

3. Numerical experiments

In this section we present some numerical results of the implicit–explicit BDF schemes for tequation. We have performed the following experiments.

3.1. Accuracy tests in a given time interval[0, T ]

We have carried out tests to establish thepth order accuracy, forp ∈ {1,2,3,4,5,6}, of thecorrespondingp-step scheme for various time steps. We have thus obtained bounds for the timwhich is required in order to achieve satisfactory accuracy. In these tests the numerical integrthe scheme was performed in the interval[0, T ], for a relatively small timeT . All these experimentsapproximate the solution of aninhomogeneousproblem of the form{

ut + uxx + uux + νuxxxx = f (x, t),

u(x,0) = g(x),(3.1)

with a known solution. Bothf andg are 2π -periodic in space. The exact solution of (3.1) is taken tou(x, t) = sin(x + t). In all experimentsT was chosen to be 1 whereasν = 1/2 and the number of mode16. This number was determined from preliminary experiments, see Section 3.3. We used six dtime steps, namelyk = 1

10,120,

140,

180,

1160,

1320, for all six schemes. We run the experiments using a do

precision FORTRAN code (without external subroutine calls) on an IBM-6000 workstation.


Table 1Implementation of all six schemes for a known solution: The maximum error in theL2-norm (i.e., max0�nk�T ‖un − Un‖) of

ottepaccuracy..

rf the

the approximate solution for various time steps is presented

Order k = 110 k = 1

20 k = 140 k = 1

80 k = 1160 k = 1

320

1 0.3639 0.1979 0.1034 0.5291e−1 0.2676e−1 0.1346e−12 0.3903e−1 0.1017e−1 0.2584e−2 0.6505e−3 0.1631e−3 0.4085e−43 0.4013e−2 0.5102e−3 0.6430e−4 0.8070e−5 0.1011e−5 0.1265e−64 0.4567e−3 0.2862e−4 0.1790e−5 0.1119e−6 0.6997e−8 0.4374e−95 0.6826e−4 0.2123e−5 0.6610e−7 0.2061e−8 0.6437e−10 0.2010e−116 0.1245e−4 0.1996e−6 0.3100e−8 0.4838e−10 0.7578e−12 0.5415e−12

Fig. 2.loglog plot of the maximum errors in theL2-norm over the interval[0, T ] for various time steps.

Table 1 contains all the maximum errors in theL2-norm, over the interval[0, T ], for all six schemesand all six time steps. All these data are presented in theloglog Fig. 2. In both the table and the plone can observe thepth order of accuracy of thep-step method. The error in the case of the time s

1320 for the 6-step method has not decreased as expected, since that would be beyond machineIt is noteworthy that the computation cost of BDF methods is independent of the number of steps

3.2. Accuracy in the approximation of periodic attractors

We have also carried out tests to establish thepth order accuracy, of thep-step method, fop ∈ {1,2,3,4,5,6}, of the periodic universal attractors of the KS equation for various values oparameterν. Indeed, certaincomputablecharacteristics of universal attractors, such as theL2-norm in


Table 2The order of accuracy in the approximation of the attractors: calculated period of the periodic attractor whenν = 0.13

663536635366353663536635366353

066353066353066353066353

ximatedith time

y

f the

.s of thed

dting ofientsides the

omena

Time step Number of steps of the BDF-scheme

p = 1 p = 2 p = 3 p = 4 p = 5 p = 6

4/1000 1.057880776 0.997326720 0.996054681 0.996066198 0.996066355 0.99602/1000 1.026733486 0.996380184 0.996064879 0.996066343 0.996066353 0.99601/1000 1.011334827 0.996144639 0.996066168 0.996066353 0.996066353 0.99601/2000 1.003683672 0.996085902 0.996066330 0.996066353 0.996066353 0.99601/4000 0.999870702 0.996071238 0.996066350 0.996066353 0.996066353 0.99601/8000 0.997967440 0.996067578 0.996066353 0.996066353 0.996066353 0.99601/16000 0.997016623 0.996066647 0.996066353 0.996066353 0.996066353 0.9961/32000 0.996541420 0.996066419 0.996066353 0.996066353 0.996066353 0.9961/64000 0.996303873 0.996066372 0.996066353 0.996066353 0.996066353 0.9961/128000 0.996185096 0.996066358 0.996066353 0.996066353 0.996066353 0.996

the case of travelling waves and the period, in the case of periodic attractors, appear to be approwith the expected order of accuracy. Their qualitative characteristics are apparently obtainable wsteps larger than time steps in any other scheme used so far.

Table 2 provides the period of the periodic attractor (see Fig. 4) atν = 0.13 for different time stepsand for all thep-step methods. One can easily observe, at least forp = 1,2,3, that the order of accuracin the calculation of the period is the same as the order of accuracy of the scheme. (Forp = 4,5,6 thecalculation of the period isunfortunatelytoo accurate, and thus we cannot observe any decay oerror.) It is noteworthy that the period of the attractor is measured from the period of theL2-norm ofthe solution. The periodicity of theEnergyor theL2-norm, i.e.,E(t) = ‖u(·, t)‖, can be seen in Fig. 3However, Fig. 5 provides a much better evidence of periodicity since it contains six phase planeEnergy, i.e., points(E(t),E′(t)), wheret ∈ [T1, T2]. The derivative ofE(t) can be efficiently calculatefrom

d

dt

∥∥u(·, t)∥∥2 = 2(ut , u) = 2(uux + uxx + νuxxxx, u) = −2(ux, ux) + 2ν(uxx, uxx)

and thus

E′(t) = ν‖uxx‖2 − ‖ux‖2

‖u‖ ,

which is computed from the Fourier modes.The phase planes correspond to six different values of the parameterν. Closed curves correspon

to periodic orbits. Finally, the period is measured by an extremely accurate method, consisapproximatingE(t), near at where its derivative vanishes, by a suitable polynomial, the coefficof which are chosen via a weighted least square method. Newton’s method subsequently provroot, where the derivative of the polynomial vanishes.

Finally, we should report that, as with previous numerical experiments, certain chaotic phenhave been identified. Specifically, period doubling cascade leading to chaos atν ≈ 0.121228053894. . . ,following at least 14 period-doublings. In Fig. 5, five period-doublings can be observed.


Fig. 3. Three period doubling of theL2-norm.

Fig. 4. The evolution of the spatial shape of the approximate solution in a periodic attractor.


ble due5). Westimateses, seenumberouble

es usede casesntil

he KSpreviouslyhaviour.sionalof high

Fig. 5. Phase plane of theL2-norm: Five period doublings.

3.3. Determining the number of Fourier modes

The space discretization is achieved by truncating the high frequencies. This is quite plausito the analyticity of the solution and the exponential decay of the Fourier coefficients, see (1.already have a rough idea of the rate of decay of these Fourier modes from the numerical ein [9]. Nevertheless, we have performed our own experiments in order to obtain similar estimatFig. 1. These specific preliminary experiments were performed in order to get a better idea of theof modes with a significant numerical contribution to the solution. (Given that our codes are in dprecision.) Clearly, this number is much larger than the number of the determining modes.

In order to achieve this, we used 64 modes and time steps significantly smaller than the onlater and thus enabling even the high frequencies to develop individual behaviour. We tested thν = 0.9,0.5,0.3,0.2,0.13,0.1,0.08,0.05,0.03, and in each case we kept subdividing the time step uthelog-plot of the spectrum stabilized. In the case ofν = 0.03, we kept subdividing until the timestep becamek = 10−8 . Fig. 1 provides theselog-plots for selected cases.

4. Final remarks and possible extensions

The implementation of the BDF schemes for the numerical approximation of the solution of tequation has been very successful. These schemes allow us to use time steps larger than thoseused in order to achieve the desired accuracy and reproduce the already known asymptotic beHowever, when we use a larger time step for the approximation of the solution of infinite dimendynamical systems, which are characterized by stiffness, we lose the accurate approximationfrequencies. Though stability in the integration of


d

dtuj = λj uj + (uux )j ,

uracy

tributely lessall time

dividualand veryindeed

that the.reviousr many

finiteations

Anal.

635.

r.

SIAM

Part 2),

ies in

Comm.

(1993)

Partial

cond

for j large, is immediate due to the implicit integration of the linear part, in order to obtain good accwe need that

k|λj | = �t∣∣j2 − νj4

∣∣ 1, (4.1)

which is extremely restrictive. However, higher order terms, due to their exponential decay, congradually (asj grows) less to the solution and thus their relative accuracy becomes gradualimportant. This fact compensates to the inherent stiffness of the system. Runs with extremely smsteps have been carried out with our schemes, in order to allow high frequencies to develop inbehaviour. However, no change in the qualitative characteristics of the attractors was observed,little change in the quantitative ones. (See Table 2.) It turns out that the high frequencies areslaved.

One could assert that, ideally, the largest allowed time step is the one having the propertyrelative error in any mode is inversely proportional to the contribution of this mode to the solution

In our experiments, the time steps, though significantly larger than the ones used in pexperiments, are still small enough so that (4.1) holds for the determining modes and in fact fomore modes than the determining ones.

These methods could be efficiently applied for the numerical approximation of most indimensional dynamical systems. In future work the method will be used in nonlinear evolution equof parabolic type where the linear part can be integrated exactly.

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Implicit–explicit BDF methods for the Kuramoto–Sivashinsky equation

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