We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate
finite difference approximation to its dynamics. The analysis is based upon
centre manifold theory so we are assured that the finite difference model
accurately models the dynamics and may be constructed systematically. The
theory is applied after dividing the physical domain into small elements by
introducing insulating internal boundaries which are later removed. The
Kuramoto-Sivashinsky equation is used as an example to show how holistic finite
differences may be applied to fourth order, nonlinear, spatio-temporal
dynamical systems. This novel centre manifold approach is holistic in the sense
that it treats the dynamical equations as a whole, not just as the sum of
separate terms

MacKenzie, T.

Roberts, A. J.

ANZIAM Journal

English

arXiv

T. MacKenzie

A. J. Roberts

Crossref

Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation

We analyse the nonlinear Kuramoto-Sivashinsky equation to develop  an accurate finite difference approximation to its dynamics. The  analysis is based upon centre manifold theory so we are assured  that the finite difference model accurately models the dynamics  and may be constructed systematically. The theory is applied after  dividing the physical domain into small elements by introducing  insulating internal boundaries which are later removed. The  Kuramoto-Sivashinsky equation is used as an example to show how  holistic finite differences may be applied to fourth order,  nonlinear, spatio-temporal dynamical systems. This novel centre  manifold approach is holistic in the sense that it treats the  dynamical equations as a whole, not just as the sum of separate  terms

Australian Mathematical Society (AustMS): E-Journals

ANZIAM J. 42 (E) ppC918{C935, 2000 C918Holistic nite dierences accurately model thedynamics of the Kuramoto-SivashinskyequationT. MacKenzie A. J. Roberts(Received 7 August 2000)AbstractWe analyse the nonlinear Kuramoto-Sivashinsky equation to de-velop an accurate nite dierence approximation to its dynamics. Theanalysis is based upon centre manifold theory so we are assured that∗Dept. Maths. & Comp., University of Southern Queensland, Toowoomba, QLD 4350,Australia; mailto:mackenzi@usq.edu.au and mailto:aroberts@usq.edu.aurespectively.0See http://anziamj.austms.org.au/V42/CTAC99/Mack for this article and ancillaryservices, c© Austral. Mathematical Soc. 2000. Published 27 Nov 2000.Contents C919the nite dierence model accurately models the dynamics and maybe constructed systematically. The theory is applied after dividing thephysical domain into small elements by introducing insulating inter-nal boundaries which are later removed. The Kuramoto-Sivashinskyequation is used as an example to show how holistic nite dierencesmay be applied to fourth order, nonlinear, spatio-temporal dynamicalsystems. This novel centre manifold approach is holistic in the sensethat it treats the dynamical equations as a whole, not just as the sumof separate terms.Contents1 Introduction C9202 Centre manifold theory underpins the delity C9233 Numerical comparisons show the eectiveness C9274 Conclusion C934References C9341 Introduction C9201 IntroductionWe continue exploring the new approach to nite dierence approximationintroduced by Roberts [10] by approximating the dynamics of solutions to theKuramoto-Sivashinsky equation [4, 3, 5]. In some non-dimensional form wetake the following partial dierential equation (pde) to govern the evolutionof u(x; t):@u@t+ u@u@x+ R@2u@x2+@4u@x4= 0 : (1)This model equation includes the mechanisms of linear growth uxx controlledby the parameter R, high-order dissipation, uxxxx, and nonlinear advec-tion/steepening, uux. Consider implementing the method of lines by dis-cretising in x and integrating in time as a set of ordinary dierential equa-tions. A nite dierence approximation to (1) on a regular grid in x isstraightforward, say xj = jh for some grid spacing h. For example, thelinear term@2u@x2=uj+1 − 2uj + uj−1h2+Oh2:However, there are diering valid alternatives for the nonlinear term uux:two possibilities areu@u@x=uj(uj+1 − uj−1)2h+Oh2(2)or u@u@x=u2j+1 − u2j−14h+Oh2: (3)1 Introduction C921Which is better? The answer depends upon how the discretisation of thenonlinearity interacts with the dynamics of other terms. The conventionalapproach of considering the discretisation of each term separately does nottell us. Instead, in order to nd the best discretisation we consider theinfluence of all terms in the equation in a holistic approach.As introduced in [10] and discussed in x2, centre manifold theory [1, 8,e.g.] has appropriate characteristics to do this. It addresses the evolution ofa dynamical system in a neighbourhood of a marginally stable xed point;based upon the linear dynamics the theory guarantees that an accurate low-dimensional description of the nonlinear dynamics may be deduced. Forexample the analysis herein supports the approximation (2) but with higher-order and nonlinear corrections. The analysis of the Kuramoto-Sivashinskyequation (1) in x 3 favours the discretisationdujdt+"uj(−uj+2 + 9uj+1 − 9uj−1 + uj−2)16h+ u2j+1 − u2j−116h!−uj+2uj+1 − uj−2uj−148h#+ R−uj+2 + 16uj+1 − 30uj + 16uj−1 − uj−212h2+uj+2 − 4uj+1 + 6uj − 4uj−1 + uj−2h4= 0 ; (4)as a low-order approximation. Provided the initial conditions are not tooextreme, centre manifold theory assures us that such a discretisation models1 Introduction C922the dynamics of (1) to errors O (kuk3; h2). Such accuracy on a relativelycoarse grid is extremely useful for such sti pdes. Further, because thecentre manifold is composed of actual solutions to the dynamical system, weare assured that equation (4) models the whole of the Kuramoto-Sivashinskyequation, independent of its algebraic form.The discretisation (4) is a low-order approximation, centre manifold the-ory also provides systematic corrections. Analysis to higher orders in thenonlinearity, discussed in x 3, shows higher order corrections to the discreti-sation of the nonlinear terms. The specic nite dierence models presentedhere were derived by a computer algebra program. Computer algebra is aneective tool because of the systematic nature of centre manifold theory [9].In this preliminary exploration of the approximation of the Kuramoto-Sivashinsky equation (1) we only consider an innite domain or strictly pe-riodic solutions in nite domains. Then all elements of the discretisationare identical by symmetry and the analysis of all elements is simultaneous.However, if physical boundaries to the domain of the pde are present, thenthose elements near a physical boundary will need special treatment. Furtherresearch is needed on this and other issues.2 Centre manifold theory underpins the delity C9232 Centre manifold theory underpins the -delityHere we describe in detail one way to place the discretisation of the Kura-moto-Sivashinsky equation (1) within the purview of centre manifold theory.The discretisation is established via an equi-spaced grid of collocationpoints, xj = jh say, for some small spacing h. Here we scale the Kuramoto-Sivashinsky to the scale of the grid spacing h. Thus our view of the dynamicsshrinks with h. This is dierent to the analysis in [10] and allows the lineardynamics to be dominated by simply the highest order spatial derivativeterm. We work on the scale of the grid by transforming (1) to the followingspace and time scales:  = x=h (so that for example j = j) and  = t=h4,giving@u@+ h3u@u@+ h2R@2u@2+@4u@4= 0 : (5)Then the crucial step: at midpoints j+1=2 = (j+j+1)=2 articial boundariesare introduced: 24 @u+@− @u−@@3u+@3− @3u−@335 ="00#; (6)1− γ224 @u+@+ @u−@@3u+@3+ @3u−@335 = γA24 u+ − u−@2u+@2− @2u−@235 ; (7)2 Centre manifold theory underpins the delity C924where u+ is just to the right of a midpoint and u− to the left. These bound-aries divide the domain into a set of elements, the jth element centred uponj and of width  = 1. When γ = 0 the right hand side of (7) disappears,thus the elements are eectively insulated from each other and so the solu-tion is particularly simple, namely u is constant in each element. We usethis simple class of solutions as a basis for analysing the γ 6= 0 case whenthe elements are coupled together. We are particularly interested in the ap-proximation at γ = 1 when the grid scaled Kuramoto-Sivashinsky pde (5)is eectively restored over the whole domain because (6{7) then ensure suf-cient continuity between adjacent elements. The introduction of the nearidentity operatorA = 1 + @212− @4720+@630240+    = @2coth @2!; (8)ensures that high-order approximations to linear terms are obtained exactlyas discussed in [10, x4]: it is remarkable that the exactly equivalent operatorworks for both Burgers’ equation and the Kuramoto-Sivashinsky equation.The following application of centre manifold theory to rigorously developthe above ideas is based upon a linear picture of the dynamics. Adjoin thedynamically trivial equations@γ@=@h@= 0 ; (9)and consider the dynamics in the extended state space (u(); γ; h). Thisis a standard trick used to unfold bifurcations [1, x1.5] or to justify long-wave approximations [6]. Within each element u = γ = 0 is a xed point.2 Centre manifold theory underpins the delity C925Linearized about each xed point, that is to an error O (kuk2 + γ2 + h2), thepde is@u@=@4u@4; s.t.@u@=1=2=@3u@3=1=2= 0 ;namely the hyperdiusion equation with essentially insulating boundary con-ditions. There are thus linear eigenmodes associated with each element:γ = 0 ; u /(en cos[n( − j−1=2)] ; j−1=2 <  < j+1=2 ;0 ; otherwise ;(10)for n = 0; 1; : : :, where the decay rate of each mode is n = −n44 ; togetherwith the trivial modes γ = const, h = const and u = 0. In a domain withm elements, evidentally all eigenvalues are negative, −4 or less, except form+2 zero eigenvalues: 1 associated with each of the m elements and 2 fromthe trivial (9). Thus, provided the nonlinear terms in (5) are suciently wellbehaved, the existence theorem ([2, p281] or [11, p96]) guarantees that am + 2 dimensional centre manifold M exists for (5{9). The centre manifoldM is parameterized by γ, h and a measure of u in each element, say uj:using u to denote the collection of such parameters, M is written asu(; ) = v(;u; γ; h) : (11)In this the analysis has a very similar appearance to that of nite elements.The theorem also asserts that on the centre manifold the parameters uj evolvedeterministically_uj = gj(u; γ; h) ; (12)2 Centre manifold theory underpins the delity C926where _uj denotes duj=d , and gj is the restriction of (5{9) to M. In thisapproach the parameters of the description of the centre manifold may beanything that sensibly measures the size of u in each element|we simplychoose the value of u at the grid points, uj() = u(j; ). This providesthe necessary amplitude conditions, namely that uj = v(j;u; γ; h). Theabove application of the theorem establishes that in principle we may ndthe dynamics (12) of the interacting elements of the discretisation. A loworder approximation written in unscaled variables is given in (4).The next outstanding question to answer is: how can we be sure thatsuch a description of the interacting elements does actually model the dy-namics of the original system (5{9)? Here, the relevance theorem of centremanifolds, [2, p282] or [11, p128], guarantees that all solutions of (5{9) whichremain in the neighbourhood of the origin in (u(); γ; h) space are exponen-tially quickly attracted to a solution of the m nite dierence equations (12).For practical purposes the rate of attraction is estimated by the leading neg-ative eigenvalue, here −4. Centre manifold theory also guarantees that thestability near the origin is the same in both the model and the original.Thus the nite dierence model will be stable if the original dynamics arestable. After exponentially quick transients have died out, the nite dier-ence equation (12) on the centre manifold accurately models the completesystem (5{9).The last piece of theoretical support tells us how to approximate the shapeof the centre manifold and the evolution thereon. Approximation theoremssuch as that by Carr & Muncaster [2, p283] assure us that upon substituting3 Numerical comparisons show the eectiveness C927the ansatz (11{12) into the original (5{9) and solving to some order of errorin kuk, γ and h, then M and the evolution thereon will be approximated tothe same order. The catch with this application is that we need to evaluatethe approximations at γ = 1 because it is only then that the articial internalboundaries are removed. In some applications of such an articial homotopygood convergence in the parameter γ [7] has been found. Thus although theorder of error estimates do provide assurance, the actual error due to theevaluation at γ = 1 should be also assessed otherwise. Here we have craftedthe interaction (7) between elements so that low order terms in γ recover theexact nite dierence formula for linear terms. Note that although centremanifold theory \guarantees" useful properties near the origin in (u(); γ; h)space, because of the need to evaluate asymptotic expressions at γ = 1, wehave used a weaker term elsewhere, namely \assures".3 Numerical comparisons show the eective-nessWe now turn to a detailed description of the centre manifold model for theKuramoto-Sivashinsky equation (1).The algebraic details of the derivation of the centre manifold model (11{12) are handled by computer algebra.1 In an algorithm introduced in [9], the1The reduce computer algebra source code is available from the authors upon request.3 Numerical comparisons show the eectiveness C928program iterates to drive to zero the residuals of the governing dierentialequation (5) and its boundary conditions (6{8). Hence by the Approximationtheorems we assuredly construct appropriate approximations to the centremanifold model.The nite dierence model is given by the evolution on the centre mani-fold. In order to represent the spatial fourth derivative in the Kuramoto-Siva-shinsky equation we need to determine the interactions between next-nearestneighbouring elements. Thus the rst approximation we can consider involvesquadratic terms in γ. After returning to x and t variables it isdujdt= −γRh2(uj+1 − 2uj + uj−1)− γ2huj(uj+1 − uj−1)− γ2h4(uj+2 − 4uj+1 + 6uj − 4uj−1 + uj−2)+γ2R12h2(uj+2 − 4uj+1 + 6uj − 4uj−1 + uj−2)+γ248h(uj+2uj+1 + 3uj+2uj − 3uj+1 − 3uj+1uj+3uj−1uj + 3uj−1 − 3uj−2uj − uj−2uj−1)+γh2u2j120(uj+1 − 2uj + uj−1)+γ2h260480hu2j (−30uj+2 − 170uj+1 + 256uj − 170uj−1 − 30uj−2)+ uj (−126uj+2uj+1 − 54uj+1uj−1 − 126uj−2uj−1)+ u2j+1 (10uj+2 − 20uj+1 + 235uj)3 Numerical comparisons show the eectiveness C929+u2j−1 (10uj−1 − 20uj−1 + 235uj)i+Okuk4; γ3; h4: (13)Recall that γ = 1 is the case of interest because the internal boundarycondition (6{7) evaluated at γ = 1 ensures sucient continuity to recoverthe original problem. The rst two lines recorded here, when evaluated forγ = 1, form the conventional second-order nite dierence equation for theKuramoto-Sivashinsky equation (1). The third line when evaluated for γ = 1gives the fourth order accurate corrections to the Ruxx term. The fourth andfurther lines above start accounting systematically for the variations in theeld u within each element and how they aect the evolution through thenonlinear term.The application of centre manifold theory ensures that the approxima-tion (13) models well the nonlinear dynamics. The nite dierence approx-imation is also independent of any valid rewriting of the pde because theanalysis exploits solutions of the equation not the algebraic form of the equa-tions. Finite dierence equations derived via this approach holistically modelall the interacting dynamics of the entire pde.To show the eectiveness of the approach we compare the nite dierencemodel obtained from various truncations of (13) to accurate numerical solu-tions obtained on a much ner grid. Choosing m intervals on [0; 2) givesan element length h = 2=m and grid points xj = jh for j = 0; : : : ; m − 1.Because of the antisymmetry in u(x; t) about x = k, when starting from theinitial condition u(x; 0) = 10 sinx, we only display the interval [0; ]. There3 Numerical comparisons show the eectiveness C9300 0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.80.91xtFigure 1: Contours of an accurate solution u(x; t), |{, to compare withnumerical approximations (13):   , the conventional approximation, errorsO (h2);− − −, the rst correction, errors O (kuk3);−  −  − , the secondcorrection,errors O (kuk4). Kuramoto-Sivashinsky equation (1) with param-eter R = 2 is discretised on just m = 8 elements in [0; 2) and drawn withcontour interval u = 3.3 Numerical comparisons show the eectiveness C931are three dierent approximations from (13), with γ = 1, depending uponwhere the expansion is truncated. The rst two lines form a model withO (h2) errors (a conventional nite dierence approximation), the rst velines provide the rst correction with O (kuk3) errors, and all shown termsform the model with O (kuk4) errors.The solutions of these models over 0 < t < 1 with m = 8 and R = 2are shown in Figure 1. Observe that the conventional approximation (dot-ted) is signicantly in error whereas the next two renements (dot-dashedand dashed) are overall more accurate, especially near the peak. Such ac-curacy is remarkable considering the nonlinearity, and the few points in thediscretisation, m = 8.Figure 2 is a plot of the solution of the models at t = 1 with m = 8 andR = 2. Observe that the conventional approximation undershoots the accu-rate solution and even has the incorrect sign at x = =4. The conventionalapproximation also overshoots the accurate solution and is particularly poorat x = 3=4. Overall the holistic approximations are more accurate than theconventional nite dierence approximation.The solution of the conventional approximation (dotted) and the rstholistic correction (dashed) over 0 < t < 1 with m = 16 and R = 2 areshown in Figure 3. The second holistic correction was not plotted because itis not discernable from the rst correction with m = 16. Observe the holisticapproximation is again more accurate near the peak.3 Numerical comparisons show the eectiveness C9320 0.5 1 1.5 2 2.5 3 3.5−505101520uxFigure 2: Plot of an accurate solution u(x; t), |{, at t = 1 to comparewith numerical approximations (13):  , the conventional approximation, er-rors O (h2);− − −, the rst correction, errors O (kuk3);−  −  − , thesecond correction,errors O (kuk4). Kuramoto-Sivashinsky equation (1) withparameter R = 2 is discretised on just m = 8 elements in [0; 2).3 Numerical comparisons show the eectiveness C9330 0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.80.91xtFigure 3: Contours of an accurate solution u(x; t), |{, to compare withnumerical approximations (13):  , the conventional approximation, errorsO (h2);− − −, the rst correction, errors O (kuk3). Kuramoto-Sivashinskyequation (1) with parameter R = 2 is discretised on m = 16 elements in[0; 2) and drawn with contour interval u = 3.4 Conclusion C9344 ConclusionCentre manifold theory is a powerful new approach to deriving nite dier-ence models of dynamical systems. Many details need to researched for ageneral application of the theory. However, there are many promising fea-tures of this application to the Kuramoto-Sivashinsky equation (1) and theearlier example of Burgers’ equation [10].References[1] J. Carr. Applications of centre manifold theory, volume 35 of AppliedMath Sci. Springer-Verlag, 1981. C921, C924[2] J. Carr and R.G. Muncaster. The application of centre manifoldtheory to amplitude expansions. II. innite dimensional problems. J.Di. Eqns, 50:280{288, 1983. C925, C926, C926[3] J.M. Hyman and B. Nicolaenko. The kuramoto-sivashinsky equation:A bridge between pdes and dynamical systems. Physica D, 18:113{126,1986. C920[4] Y. Kuramoto. Diusion induced chaos in reactions systems. Progr.Theoret. Phys. Suppl., 64:346{367, 1978. C920References C935[5] Y. Pomeau and S. Zaleski. The kuramoto-sivashinsky equation: Acaricature of hydrodynamic turbulence? Lecture Notes In Physics, 230,1984. C920[6] A.J. Roberts. The application of centre manifold theory to theevolution of systems which vary slowly in space. J. Austral. Math. Soc.B, 29:480{500, 1988. C924[7] A.J. Roberts. Low-dimensional models of thin lm fluid dynamics.Phys. Letts. A, 212:63{72, 1996. C927[8] A.J. Roberts. Low-dimensional modelling of dynamical systems.Technical report, [http://xxx.lanl.gov/abs/chao-dyn/9705010],1997. C921[9] A.J. Roberts. Low-dimensional modelling of dynamics via computeralgebra. Comput. Phys. Comm., 100:215{230, 1997. C922, C927[10] A.J. Roberts. Holistic nite dierences ensure delity to burger’sequation. Technical report,[http://xxx.lanl.gov/abs/chao-dyn/9901011], 1998. C920, C921,C923, C924, C934[11] A. Vanderbauwhede. Centre manifolds. Dynamics Reported, pages89{169, 1989. C925, C926

Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation

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