Numerical Simulation of Asymptotic States of the Damped Kuramoto

Numerical Simulation of Asymptotic States of theDamped Kuramoto-Sivashinsky Equation

Hector Gomez∗ and Jose ParısUniversity of A CorunaCampus de Elvina s/n

15071, A Coruna, Spain(Dated: February 25, 2011)

The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for the under-standing of the onset and evolution of secondary instabilities in a wide range of physical phenomena.Most existing studies about this equation deal with its asymptotic states on one-dimensional set-tings or on periodic square domains. We utilize large-scale numerical simulation to investigate theasymptotic states of the damped Kuramoto-Sivashinsky equation on annular two-dimensional ge-ometries and three-dimensional domains. To this end, we propose an accurate, efficient and robustalgorithm based on a recently introduced numerical methodology, namely, Isogeometric Analysis.Wecompared our two-dimensional results with several experiments of directed percolation on squareand annular geometries, and found qualitative agreement.

PACS numbers: 02.60.Cb,02.70.-c,47.54.-r,05.45.-a,47.11.-j

I. INTRODUCTION

A thermodynamical system far from equilibrium mayexhibit primary instabilities which drive it into a inho-mogeneous asymptotic state. In many relevant systems,these asymptotic states consist of spatially and tempo-rally ordered cellular structures. In the last decades,there has been increasing interest in the so-called sec-ondary instabilities [1], which may destroy the orderedcellular state giving rise to disordered states both in spaceand time. Prime examples of phenomena where sec-ondary instabilities exist are, for instance, the Rayleigh-Benard convection [2, 3], directional solidification [4, 5],Faraday waves [6, 7] or directed percolation [8–13].The transition from a homogeneous stationary state toasymptotic cellular states through primary instabilitiesmay be successfully analyzed by using a linear stabil-ity analysis, but more sophisticated methods are neces-sary to understand secondary instabilities [14]. Remark-ably, one of the most successful tools for the understand-ing of secondary instabilities has turned out to be thestudy of the asymptotic states of the damped Kuramoto-Sivashinsky equation [15], which has emerged as a fun-damental universal model describing the onset and evo-lution of secondary instabilities [14]. As a consequence,there is significant interest in the study of the asymp-totic states of the damped Kuramoto-Sivashinsky equa-tion. The main difficulty to achieve this goal is that thedamped Kuramoto-Sivashinsky equation is not a gradi-ent system [16]. Thus, there is no known Lyapunov func-tional for the equation. This fact significantly limits ourcapacity to study its asymptotic states using analyticaltechniques and numerical simulation appears as a very at-tractive alternative. At this point, the one-dimensional

∗ [email protected]

equation is fairly well understood [15, 17]. In the lastyears, significant progress has been made in the under-standing of the two-dimensional equation [18], but theresults are limited to periodic square domains. Giventhe strong dependence of the asymptotic states on thegeometry and the dimensionality of the domain [18, 19],the understanding of the late-time states on non-squaretwo-dimensional and three-dimensional domains is con-sidered a very relevant research topic. This is preciselyone of the objectives of this work. To investigate theasymptotic states we use numerical simulation. Thus,we propose an effective, accurate and robust numericalscheme for the damped Kuramoto-Sivashinsky equation,which is another contribution of this work.

The numerical simulation of the damped Kuramoto-Sivashinsky equation presents several challenges. This isthe reason why most calculations available in the liter-ature are restricted to one-dimensional settings [8, 20–24] and only very recently were two-dimensional sim-ulations on square domains available [18, 25–27]. Wedo not know of any three-dimensional simulation nor weare aware of two-dimensional calculations on non-squaredomains (although we know of numerical solutions to amodified Kuramoto-Sivashinsky equation on a disk [28–30]). We feel that one of the main reasons for this is thatthe damped Kuramoto-Sivashinsky equation includes afourth-order partial-differential operator. The numericalresolution of higher-order partial differential equationsis significantly less developed than that of second-orderproblems. For example, in the context of finite elementmethods, the use of conforming discretizations for fourth-order partial-differential spatial operators requires utiliz-ing globally C1-continuous basis functions. There existsome three-dimensional finite elements possessing globalC1 continuity, but they introduce a number of additionaldegrees of freedom and severely restrict the geometri-cal complexity of the domain. Thus, in the finite ele-ment context, the standard approach is to use a mixed

2

method, which for a fourth-order problem doubles thenumber of global degrees of freedom compared to theprimal variational formulation. As a consequence, themost widely used numerical methodologies for fourth-order partial-differential equations are either finite differ-ences or pseudo-spectral collocation methods, whose ap-plicability to complicated three-dimensional geometriesis limited. Thus, we feel that there is no totally satisfac-tory solution to the higher-order operator problem, yetfourth-order equations are becoming ubiquitous, primar-ily due to the fast development of phase-field modeling[31, 32].

This work proposes a numerical formulation based onIsogeometric Analysis [33], which is a generalization ofFinite Element Analysis with several advantages [34–41].Isogeometric Analysis is based on developments of Com-putational Geometry and consists of using Non-UniformRational B-Splines (NURBS) as basis functions in a vari-ational formulation. For an introduction to NURBS, thereader is referred to [42, 43]. Among the advantagesof Isogeometric Analysis over Finite Element Analysis[44], we mention precise geometrical modeling, simpli-fied mesh refinement, supperior approximation capabil-ities and, most importantly for the present work, C1-continuity or higher on non-trivial geometries.

In addition to space discretization, the other keydifficulty in the simulation of the damped Kuramoto-Sivashinsky equation is time integration. Since theKuramoto-Sivashinsky equation is nonlinear, noncon-tractive and essentially a non-gradient system, a funda-mental question emerges. What would be an adequatenotion of stability for the time-integrator to satisfy? Inthe absence of a clear notion of stability, we use anA-stable method that has proved very effective for theKuramoto-Sivashinsky equation in our numerical simu-lations, namely, the generalized-α method [45, 46]. Wealso make use of an adaptive time-stepping algorithm toimpose control over local errors [47].

Our space and time discretization schemes render aneffective, accurate and robust methodology. We presenttwo-dimensional numerical examples on non-square ge-ometries and three-dimensional simulations. To the bestof our knowledge these are the first simulations of theirkind.

II. THE DAMPED KURAMOTO-SIVASHINSKYEQUATION

Here we state an initial and boundary-value problemfor the damped Kuramoto-Sivashinsky equation over thetime interval [0, T ]. Let Ω ⊂ R3 be an open set. Wedenote Γ the boundary of Ω, which is assumed to have acontinuous unit outward normal vector n. The problemin stated as: given u0 : Ω 7→ R, find u : Ω × [0, T ] 7→ R

such that

∂u

∂t= −∆u−∆2u− αu + |∇u|2 in Ω× (0, T ) (1)

u(x, 0) = u0(x) in Ω (2)

with adequate boundary conditions. Periodic boundaryconditions are the standard choice in square domains. Onmore complex geometries the boundary conditions

∇(u + ∆u) · n = 0 (3)∇u · n = 0 (4)

may be utilized. In a variational formulation, equations(3)-(4) may be thought of as natural boundary conditionsfor the damped Kuramoto-Sivashinsky equation.

Although the dynamics of the Kuramoto-Sivashinskyequation [48, 49] is very complex, the four terms on theright hand side of equation (1) have a clear meaning intheir own right. The first term is destabilizing in thesense that it increases the L2 energy in the system. Usingthe same terminology, we would qualify the second andthird terms as stabilizing. Finally, the last term is anenergy-transfer operator. It transfers energy from lowerto higher frequencies [50].

Additional insight about primary instabilities of theequation may be obtained by using a linear stability anal-ysis [14]. It may be shown that the homogeneous con-stant state u = 0 is linearly stable for α > 0.25. Addition-ally, it is known that for α = 0 the Kuramoto-Sivashinskyequation leads to spatiotemporal chaotic states [51, 52].For intermediate values of α, the asymptotic states maybe different. One of the most significant developmentsin the last years is due to Paniconi and Elder [18]who identified three asymptotic states of the dampedKuramoto-Sivashinsky equation on a square domain fordifferent values of α, namely, the hexagonally ordered(0.2176 < α < 0.2500), the so-called breathing hexago-nal state (0.2070 < α < 0.2176) and the spatiotempo-rally chaotic (weakly turbulent) state (0 ≤ α < 0.2070).Given the strong dependence of late-time states on thedimensionality and topology of the domain, we aim atgeneralizing those results by performing numerical sim-ulations on two-dimensional non-square geometries andthree-dimensional calculations. The next section showsour proposed numerical formulation to achieve this goal.

III. NUMERICAL FORMULATION

In this section we present our numerical formulationfor the damped Kuramoto-Sivashinsky equation. Wefirst derive a semidiscrete formulation and then use anadaptive time-stepping method to advance the solutionin time.

3

A. Semidiscrete formulation

Our starting point is the weak formulation of thecontinuous problem. At this point we assume periodicboundary conditions in all directions. Let us call V thespace of trial and weighting functions which are assumedto be the same. We suppose V ⊂ H2, where H2 is theSobolev space of square integrable functions with squareintegrable first and second derivatives. The problem maybe stated as follows: find u ∈ V such that for all w ∈ V

B(w, u) = 0 (5)

where

B(w, u) =(w, ∂u

∂t

)− (∇w, ∇u)+

(∆w, ∆u) +(w, αu− |∇u|2)

(6)

and (·, ·) is the L2-inner product with respect to the do-main Ω.

To perform the space discretization of (5) we make useof the Galerkin method. We approximate (5) by the fol-lowing finite-dimensional problem over the finite elementspace Vh ⊂ V : find uh ∈ Vh such that for all wh ∈ Vh

B(wh, uh) = 0 (7)

In equation (7) uh takes on the form

uh(x, t) =nb∑

A=1

uA(t)NA(x) (8)

where nb is the dimension of the discrete space, the NA’sare the basis functions of the discrete space, and the uA’sare the coordinates of uh on Vh.

Note that the condition Vh ⊂ V mandates our discretespace to be H2-conforming. This condition is satisfied byC1-continuous NURBS basis functions.

B. Time integration

The time integration of the Kuramoto-Sivashinskyequation constitutes a significant challenge. Explicitmethods have to face severe limitations on the timestep due to the fourth-order spatial derivatives. Semi-implicit methods may be attractive because the fourth-order term is linear. Thus, treating implicitly this termand explicitly the rest may permit taking somewhatlarger time-steps, while avoiding the use of a nonlinearsolver. However, we favor the use of a fully-implicit al-gorithm, namely, the generalized-α method. The reasonfor this is that using the fully implicit algorithm we wereable to take significantly larger time steps compared tothe semi-implicit scheme. This is due to the fact thatthe most straightforward semi-implicit methods are onlyfirst-order accurate and, as a consequence, may be in-accurate for large time steps. Another reason to use

the fully-implicit method is that our numerical examplesshowed a very fast convergence of the nonlinear solver.

To define our time integration scheme, we introducethe following residual vector,

R = RA (9)RA = B(NA, uh) (10)

Let us call U and U the vector of global degrees of free-dom of the scalar field u and its time derivative, respec-tively. The time-stepping algorithm may be described asfollows: given Un, Un, find Un+1, Un+1 such that

R(Un+αm , Un+αf) = 0 (11)

where

Un+αm = Un + αm(Un+1 − Un) (12)Un+αf

= Un + αf (Un+1 −Un) (13)

Un+1 = Un + ∆tUn + γ∆t(Un+1 − Un) (14)

Note that, although U and U are independently treatedin the algorithm, Un+1 and Un+1 are related throughequation (14) and, thus, they are not independent un-knowns.

To complete the description of the method, it remainsto define αm, αf and γ. These are real-valued parametersthat define the accuracy and the stability properties ofthe algorithm. Jansen et al. [46] proved that, for a linearmodel problem, second-order accuracy is attained if

γ =12

+ αm − αf (15)

while unconditional A-stability requires

αm ≥ αf ≥ 1/2 (16)

We are interested in second-order accurate uncondition-ally A-stable methods, so we will take values of αm, αf

and γ that satisfy equations (15) and (16) simultaneously.One of the key features of the generalized-α method isthat αm and αf can be parametrized in terms of ρ∞, thespectral radius of the amplification matrix that controlshigh frequency dissipation [46]. Thus,

αm =12

(3− ρ∞1 + ρ∞

); αf =

11 + ρ∞

(17)

As a consequence, if we set ρ∞, and then select αm

and αf using (17) and calculate γ utilizing (15), wehave a family of second-order accurate unconditionally A-stable methods with optimal control over high frequencydissipation. The details of the implementation of thegeneralized-α method for a nonlinear problem may befound in [47].

4

C. Time-step adaptivity

The undamped Kuramoto-Sivashinsky equation isknown to amplify exponentially small perturbations infinite time intervals. For small values of the linear sta-bilizing term, the damped equation retains this feature.Thus, accurate time-integration is key to perform reliablelong-time computations. We feel that these argumentsrecommend the use of adaptive time-step control. Weemploy a recently proposed adaptive algorithm that canbe used in conjunction with the generalized-α method.The details of this algorithm may be found in [47].

IV. NUMERICAL SIMULATIONS

In this section we present some two- and three-dimensional numerical examples. The purpose of theseexamples is threefold: first, we aim at illustrating theeffectiveness and robustness of our numerical formula-tion; second, we investigate the asymptotic states ofthe damped Kuramoto-Sivashinsky equation on non-square domains in two-dimensions and on cubic three-dimensional domains; third, we compare our simulationswith directed percolation experiments.

Throughout this paper, for the computation of theasymptotic states, we take as initial condition a randomperturbation of the homogeneous state u = 0. The per-turbations are directly applied to control variables andare uniformly distributed on [−0.05, 0.05]. For the com-parison with experiments we may take different initialconditions that will be specified in each case.

For the space discretization we employ C1 quadraticNURBS for all the numerical examples.

A. Numerical simulations on a periodic square

1. Asymptotic states

Here we present the numerical solution to the dampedKuramoto-Sivashinsky equation on the domain Ω =[0, 100]2. We use periodic boundary conditions and acomputational mesh composed of 1282 C1-quadratic el-ements. Following [18, 25], we present simulations forα = 0.225, α = 0.210 and α = 0.195. Our resultsshow the hexagonal (α = 0.225), breathing hexagonal(α = 0.210) and disordered (α = 0.195) states foundby Paniconi and Elder [18]. The hexagonal state isa spatially ordered stationary solution of the dampedKuramoto-Sivashinsky equation found for relatively largevalues of α. The breathing hexagonal state is an unsteadysolution characterized by a quasi-periodic oscillation ofthe hexagonal pattern in which each cell oscillates out ofphase with its closest neighbor. In Figure 1 we plot thenumerical solutions at time t = 15000 for α = 0.225 (a),α = 0.210 (b) and α = 0.195 (c). To further illustrate

the difference between these three types of asymptoticstates we make use of the L2 energy, defined as,

E = ||u||2 = (u, u)1/2 (18)

The L2 energy has been identified as a fundamentalquantity to understand the dynamics of the Kuramoto-Sivashinsky equation [52, 53, 55]. In Figure 2 we plot thetime evolution of the L2 energy for α = 0.225, α = 0.210,and α = 0.195. In the same figure, we also depict threesubplots corresponding to the evolution of the L2 energyduring the time interval t ∈ [12000, 13000]. These sub-plots show the complexity of the evolution of the L2 en-ergy and illustrate the difference between the steady state(α = 0.225), the breathing hexagonal state (α = 0.210)and the chaotic state (α = 0.195). Thus, for α = 0.225the L2 energy is constant in time, which is consistentwith a stationary solution. For α = 0.210, the L2 energyexhibits a two-scale behavior. The small scale compo-nent is quasi-periodic and its frequency approximatelycoincides with the temporal frequency of the oscillatingpattern. Finally, for α = 0.195 we observe a complexevolution consistent with a chaotic state.

Additional insight may be obtained by plotting the L2

energy phase plane, which is the set of points (E , E) fora given time interval (here E denotes the time deriva-tive of E). The L2 energy phase plane is regarded as afundamental tool for the understanding of the dampedKuramoto-Sivashinsky equation [52, 53]. In Figure 3we plot the L2 energy phase plane for α = 0.225 (a),α = 0.210 (b) and α = 0.195 (c). Note that the verticalscales of the three subfigures are different. For α = 0.225we observe a typical phase plane of a stationary solu-tion. For α = 0.210 and α = 0.195 we plot the phaseplanes during the time interval t ∈ [14900, 15000], whichcorresponds to the last 100 units of time of the simu-lation. We observe that for α = 0.195, E takes valuesone order of magnitude larger than those achieved forα = 0.210. This is a consequence of wilder and roughervariations of the L2 energy, which are consistent with amore chaotic behavior. We also observe that the phaseplane for α = 0.210 reveals a quasi-periodic structurethat manifests itself through ordered loops. This behav-ior is consistent with the breathing hexagonal state, and,thus, it is not present in the phase plane that correspondsto α = 0.195.

2. Comparison with experiments

The breathing hexagonal state has been experimen-tally observed in a pattern of two-dimensional jets [9],obtained by way of a directed percolation experiment. Inthe same work, the authors observe a stationary hexago-nal state with several topological defects. They identifythe so-called penta-hepta defect which consists of a spotsurrounded by seven (rather than six) dots and a neigh-bor of it enclosed by five. Here we aim to show that our

5

(a)α = 0.225 (b)α = 0.210 (c)α = 0.195

FIG. 1. (Color online) Numerical solution to the damped Kuramoto-Sivashinsky equation on the domain [0, 100]2 at timet = 15000 for α = 0.225 (a), α = 0.210 (b) and α = 0.195 (c). Boundary conditions are periodic. The computational mesh iscomposed of 1282 C1-quadratic elements. For α = 0.225 we observe a hexagonal pattern. The solution for α = 0.210 correspondsto the so-called breathing hexagonal state. For α = 0.195 we observe a disordered state consistent with a chaotic behavior.

simulations of the damped Kuramoto-Sivashinsky equa-tion reproduce this topological defect. For this purpose,we performed a simulation for α = 0.225 on a signifi-cantly larger domain, namely, Ω = [0, 512]2. Figure 4shows a snapshot of the solution with several penta-heptadefects marked (a) and a detailed view of the area wherethe penta-hepta defects are located (b).

As an additional comparison, we perform a statisti-cal study of the average time before the system reachesa stable state of constant energy from a chaotic initialcondition. The initial conditions correspond to differentrealizations of a chaotic asymptotic state calculated us-ing α = 0.195. Then, we suddenly increase the value ofα and measure the time before the system reaches a sta-ble state of constant energy. We sampled seven valuesof α, which lead to stationary solutions from α = 0.225to α = 0.240. For each value of α, we performed elevencalculations corresponding to different chaotic initial con-ditions. Figure 5(a) shows the evolution of the L2 energyfrom a chaotic state to a stable state for several values ofα and different initial conditions. From these curves wecalculated the average time before stabilization <τ >. InFigure 5(b) we plot α versus < τ >−1. The data fits astraight line with a coefficient of determination 0.9982.The behavior is the qualitatively the same as that foundin a experiment of directed percolation [13], where thesame scaling has been measured. We remark that thisscaling has also been observed in statistical studies ofrelaminarization in pipe or shear flows [54].

B. Numerical simulations on an annular surface

1. Asymptotic states

In this example we calculate the numerical solution tothe damped Kuramoto-Sivashinsky equation on an an-nular surface. This geometrical setting has been very

recently analyzed under the assumption of radial sym-metry [19]. Here we remove this hypothesis. This ex-ample also shows that our numerical formulation can beapplied to non-square geometries, while maintaining itsaccuracy, stability and robustness. Our study suggeststhat the hexagonal, breathing hexagonal and disorderedstates found in the periodic square also exist in this geo-metrical setting.

The exterior radius of the annular surface is re = 50.0,while the interior is ri = 12.5. On the boundary, weimpose the conditions (3)-(4). We construct the com-putational mesh joining four NURBS patches. Each ofthese patches corresponds to a quarter of the annular sur-face and is composed of C1 quadratic NURBS elements.The patches are joint as to maintain C1 continuity ofthe solution over the whole domain. This can be accom-plished applying linear restriction operators to the solu-tion and weighting functions spaces. The resulting mesh(comprised of four patches) is composed of a total of 256elements in the circumferential direction and 64 in theradial direction. We note that our formulation achievesexact geometrical modeling of this problem. The reasonfor this is that NURBS can represent all conic sectionsexactly [33].

In Figure 6 we plot the solution for α = 0.225, α =0.210 and α = 0.195 at time t = 15000. To better un-derstand the asymptotic states we make use again of theL2 energy time evolution (see Figure 7). The blue, redand black lines correspond to α = 0.225, α = 0.210 andα = 0.195, respectively (see also the labels appended tothe lines). The dynamics of the L2 energy is qualitativelysimilar to the behavior exhibited in the last example. Inthe subplots displayed in Figure 7 we observe that forα = 0.225 the L2 energy is fairly constant, which is con-sistent with a stationary solution. For α = 0.210 weobserve a quasi-periodic behavior, which is the manifes-

6

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

t

E

α = 0.225

α = 0.210

α = 0.195

12000 12200 12400 12600 12800 130001800

1850

1900

1950

2000

α = 0.195

12000 12200 12400 12600 12800 130001550

1600

1650

1700

1750

α = 0.210

12000 12200 12400 12600 12800 130001150

1200

1250

1300

1350

α = 0.225

FIG. 2. (Color online) Time evolution of the L2 energy for the damped Kuramoto-Sivashinsky equation on the square [0, 100]2.Boundary conditions are periodic. The black, red and blue lines correspond to α = 0.195, α = 0.210 and α = 0.225, respectively(the lines are also labeled with the value of α to which they correspond). We represent three subplots that show to the evolutionof the L2 energy in the time period t ∈ [12000, 13000].

1217 1217.5 1218 1218.5 1219

−2

0

2

4

6

8

10x 10

−3

E

E

(a)α = 0.225

1591 1592 1593 1594 1595 1596 1597−1

−0.5

0

0.5

1

E

E

E

E

E

E

(b)α = 0.210

1880 1890 1900 1910 1920 1930−10

−5

0

5

10

E

E

E

E

E

E

E

E

E

E

(c)α = 0.195

FIG. 3. (Color online) L2 energy phase planes for the damped Kuramoto-Sivashinsky equation on the square [0, 100]2 forα = 0.225 (a), α = 0.210 (b) and α = 0.195 (c). Note that the vertical scales of the three subplots are different. The solidsquares in the plots indicate where the phase planes start.

tation of the breathing structure in the L2 energy. Fi-nally, for α = 0.195, the plot shows a complex behaviorconsistent with a chaotic state. This is again confirmedby the L2 energy phase planes, which are shown in Figure8. The phase plane for α = 0.225 clearly correponds to astationary solution. For α = 0.210 and α = 0.195 we plot

the phase planes in the time interval t ∈ [14900, 15000],which corresponds to the last 100 units of time of thesimulation. Observe that the vertical scales are differentfor each subfigure. We note that for α = 0.195, E takesvalues an order of magnitude larger than those taken forα = 0.210. Also, for α = 0.210 there is a quasi-periodic

7

(a)General view (b)Detail

FIG. 4. (Color online) Numerical solution to the damped Kuramoto-Sivashinsky equation on the square [0, 512]2 at timet=15000 for α = 0.225 (a) and detail of the solution (b). The plot on the right-hand side corresponds to the area marked witha box on the left-hand side. The plots highlight the appearance of the penta-hepta defects experimentally observed in a patternof two-dimensional jets [9].

102

104

106

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

α = 0.2400

α = 0.2375

α = 0.2350

α = 0.2325

α = 0.2300

α = 0.2275

α = 0.2250

τ

E

(a)Energy decay

0.225 0.23 0.235 0.24

6

7

8

9

10

11

12

13x 10

−5

α

<τ>−1

(b)Stabilization time

FIG. 5. (Color online) On the left-hand side we plot the evolution of the L2 energy from a chaotic initial state to a stable stateof constant energy for seven values of α and different chaotic initial conditions. On the right-hand side we plot α versus thereciprocal of the average time before the system achieves a stable state of constant energy. The data fits a straight line with acoefficient of determination 0.9982.

structure in the phase plane, which is not present forα = 0.195. We conclude that the snapshots of the solu-tion (Figure 6) and the phase planes (Figure 8) suggestthat the asymptotic states in the annular surface are thesame as those found on the periodic square, although this

result may not hold if we change the ratio of the exteriorto the interior radii.

8

(a)α = 0.225 (b)α = 0.210 (c)α = 0.195

FIG. 6. (Color online) Numerical solution to the damped Kuramoto-Sivashinsky equation on an annular surface at t = 15000for α = 0.225 (a), α = 0.210 (b) and α = 0.195 (c). Boundary conditions are defined in equations (3)-(4). The computationalmesh is composed of 256 elements in the circumferential direction and 64 in the radial direction. For α = 0.225 we observea hexagonal pattern. The solution for α = 0.210 corresponds to the so-called breathing hexagonal state. For α = 0.195 weobserve a disordered state consistent with a chaotic behavior.

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

100

200

300

400

500

600

700

800

900

1000

t

E α = 0.225

α = 0.210

α = 0.195

12000 12200 12400 12600 12800 13000760

780

800

820

840

α = 0.195

12000 12200 12400 12600 12800 13000650

670

690

710

730

α = 0.210

12000 12200 12400 12600 12800 13000440

460

480

500

520

α = 0.225

FIG. 7. (Color online) Time evolution of the L2 energy for the damped Kuramoto-Sivashinsky equation on an annular surface.The black, red and blue lines correspond to α = 0.195, α = 0.210 and α = 0.225, respectively (the lines are also labeled withthe value of α to which they correspond). We represent three subplots corresponding to the evolution of the L2 energy in thetime interval t ∈ [12000, 13000].

9

482 483 484 485−5

0

5

10

15x 10

−3

E

E

(a)α = 0.225

686 686.5 687 687.5−0.2

−0.1

0

0.1

0.2

0.3

E

E

(b)α = 0.210

735 740 745 750 755 760 765−3

−2

−1

0

1

2

3

E

E

(c)α = 0.195

FIG. 8. (Color online) L2 energy phase planes for the damped Kuramoto-Sivashinsky equation on an annular surface forα = 0.225 (a), α = 0.210 (b) and α = 0.195 (c). Note that the vertical scales of the three subplots are different. The solidsquares in the plots indicate where the phase planes start.

2. Comparison with experiments

The annular geometry has recently received the at-tention from experimentalists both in the context ofRayleigh-Benard convection [56] and directed percolation[11]. Here we compare our numerical simulations with theexperiements presented in [11]. In particular, we showthat the damped Kuramoto-Sivashinsky equation repro-duces the transition from a stationary liquid curtain toa pattern of columns, as predicted by the experiments.The annular geometry is defined by an exterior radiusre = 61 and a interior radius ri = 51. This geome-try corresponds to one of the experiments presented in[11]. The computational mesh is composed of 256 C1-quadratic elements in the circumferential direction and16 elements in the radial direction. Boundary conditionsare defined by equations (3)–(4). We simulate the sta-tionary liquid curtain by taking a constant initial condi-tion u0(x) = 10. Then, we let the solution evolve un-til a pattern of columns develops. Figure 9 shows theequilibrium arrangement, which is in agreement with thecellular pattern found in [11].

C. Asymptotic states on three-dimensionaldomains

In this section we present the three-dimensional coun-terpart of the simulation presented in section IV A1. Thecomputational domain is Ω = [0, 100]3 and we employ auniform mesh composed of 1283 C1-quadratic elements.In this example we will not be able to run the calculationsuntil such long times as in the two-dimensional simula-tions due to excessive computational cost. We ran theexamples up to t = 3500, which required about 20000time steps. We assume that the asymptotic states arereached before this time. In Figures 10, 11 and 12 we plotthe numerical solutions at time t = 3500 for α = 0.225,

FIG. 9. (Color online) Numerical solution to the dampedKuramoto-Sivashinsky equation on an annular domain.Boundary conditions are defined in equations (3)-(4). Thisexample shows qualitative agreement with an experiment ofdirected percolation presented in [11].

α = 0.210 and α = 0.195, respectively. On the left-handside of each figure we plot isosurfaces of the solution,while on the right-hand side we present slices along sev-eral planes, which clearly show disordered states. Wealso note that there are no qualitative differences betweenthe solutions for different values of α. Figure 13 showsthe evolution of the L2 energy for α = 0.225, α = 0.210and α = 0.195. In all cases we observe a complex evo-lution without a clear structure. Additionally, Figure 13shows that the trend in the evolution of the L2 energy isfairly constant from t = 300 till the end of the computa-tion. This supports our hypothesis about the asymptoticstates being reached before t = 3500. To further ana-lyze the evolution of the L2 energy we make use again

10

FIG. 10. (Color online) Numerical solution to the Kuramoto-Sivashinsky equation at time t = 3500 for α = 0.225. Thecomputational domain is [0, 128]3 and boundary conditions are periodic in all directions. The computational mesh is composedof 1283 C1 quadratic elements. The left-hand side shows isosurfaces of the solution, while the right-hand side presents slicesalong several planes.


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of the phase planes, which are shown in Figure 14. Thephase planes correspond to the last 100 units of time ofthe simulation. Unlike in the two-dimensional examples,the vertical scale is the same for all subplots. We do notobserve qualitative differences between the phase planesfor the three values of α. We conclude that, althoughfurther study is warranted, the snapshots of the solution(Figures 10–12) and the L2 energy phase planes (Figure14) suggest that the hexagonal and the breathing hexag-onal states may not exist on three-dimensional domains.

V. CONCLUSION

We presented a computational approach to investi-gate the asymptotic states of the damped Kuramoto-Sivashinsky equation. We applied our numerical tech-nique to problems on non-square two-dimensional geome-tries and three-dimensional domains. Thus, our workextends previous studies on the topic which were al-most invariably restricted to one-dimensional settingsor square domains. Our study suggests that for thetwo-dimensional domain that we analyze the asymptoticstates are the same as those found on a periodic square.However, in three-dimensional domains we consistentlyfound chaotic asymptotic states. Since the dampedKuramoto-Sivashinsky equation is a fundamental model

that describes the onset and evolution of secondary insta-bilities, our study may contribute to a better understand-ing of physical phenomena exhibiting this behavior. Wepresented several comparisons of our numerical simula-tions with experiments of directed percolation. We con-clude that the damped Kuramoto-Sivashinsky equationreproduces the hexagonal and breathing hexagonal statesfound on directed percolation experiments on squares,and it also exhibits the penta-hepta defects found in ex-periments. We have also computed numerically the scal-ing of the stabilization time of chaotic solutions withrespect to the control parameter α. Our study agreesqualitatively with an experiment of directed percolation.We have also presented a qualitative comparison of oursimulations with a directed percolation experiment on anannular geometry.

VI. ACKNOWLEDGEMENTS

The authors were partially supported by Xuntade Galicia (grants No. 09REM005118PR and No.09MDS00718PR), Ministerio de Ciencia e Innovacion(grants No. DPI2009-14546-C02-01 and No. DPI2010-16496) cofinanced with FEDER funds, and Universidadde A Coruna.

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

t

Eα = 0.225

α = 0.210

α = 0.195

2500 2550 2600 2650 2700 2750 28004700

4800

4900

5000

5100

α = 0.195

2500 2550 2600 2650 2700 2750 28003600

3700

3800

3900

4000

α = 0.210

2500 2550 2600 2650 2700 2750 28002750

2850

2950

3050

3150

α = 0.225

FIG. 13. (Color online) Time evolution of the L2 energy for the damped Kuramoto-Sivashinsky equation on the cube [0, 100]3.Boundary conditions are periodic in all directions. The black, red and blue lines correspond to α = 0.195, α = 0.210 andα = 0.225, respectively (the lines are also labeled with the value of α to which they correspond). We represent three subplotscorresponding to the evolution of the L2 energy in the time period t ∈ [2500, 2800].

2850 2900 2950−15

−10

−5

0

5

10

15

20

E

E

(a)α = 0.225

3750 3800 3850 3900−15

−10

−5

0

5

10

15

20

E

E

(b)α = 0.210

4800 4850 4900 4950−15

−10

−5

0

5

10

15

20

E

E

(c)α = 0.195

FIG. 14. (Color online) L2 energy phase planes for the Kuramoto-Sivashinsky equation on cube at the time interval t ∈[3400, 3500] for α = 0.225 (a), α = 0.210 (b), and α = 0.195 (c). Note that the vertical scale is the same for all subplots. Thesolid squares in the plots indicate where the phase planes start.

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Numerical Simulation of Asymptotic States of the Damped Kuramoto