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dc.contributor.authorAkrivis, Georgiosen
dc.contributor.authorPapageorgiou, Demetrios T.en
dc.contributor.authorSmyrlis, Yiorgos-Sokratisen
dc.creatorAkrivis, Georgiosen
dc.creatorPapageorgiou, Demetrios T.en
dc.creatorSmyrlis, Yiorgos-Sokratisen
dc.description.abstractWe analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified Kuramoto-Sivashinsky equations. Time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. The general case analyzed covers several physical applications arising in multiphase hydrodynamics and the emerging dynamics arise from a competition of long-wave instability (negative diffusion), short-wave damping (fourth order stabilization), nonlinear saturation (Burgers nonlinearity), and dispersive effects. The solutions of such systems typically converge to compact absorbing sets of finite dimension (i.e., global attractors) and are characterized by chaotic behavior. Our objective is to employ schemes which capture faithfully these chaotic dynamics. In the general case the dispersive term is taken to be a pseudodifferential operator which is allowed to have higher order than the familiar fourth order stabilizing term in the Kuramoto-Sivashinsky equation. In such instances we show that first and second order time-stepping schemes are appropriate and provide convergence proofs for the schemes. In physical situations when the dispersion is of lower order than the fourth order stabilization term (for example, a hybrid Kuramoto-Sivashinsky-Korteweg-deVries equation also known as the Kawahara equation in hydrodynamics), higher order time-stepping schemes can be used and we analyze and implement schemes of order six or less. We derive optimal order error estimates throughout and utilize the schemes to compute the long time dynamics and to characterize the attractors. Various numerical diagnostic tools are implemented, such as the projection of the infinite-dimensional dynamics to one-dimensional return maps that enable us to probe the geometry of the attractors quantitatively. Such results are possible only if computations are carried out for very long times (we provide examples where integrations are carried out for 10 8 time units), and it is shown that the schemes used here are very well suited for such tasks. For illustration, computations are carried out for third order dispersion (the Kawahara equation) as well as fifth order dispersion (the Benney-Lin equation) but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics. © 2012 Society for Industrial and Applied Mathematics.en
dc.sourceSIAM Journal on Scientific Computingen
dc.subjectPartial differential equationsen
dc.subjectControl nonlinearitiesen
dc.subjectDynamical systemsen
dc.subjectMathematical operatorsen
dc.subjectSecond ordersen
dc.subjectSpatial discretizationsen
dc.subjectDispersion (waves)en
dc.subjectComputational studiesen
dc.subjectError estimatesen
dc.subjectLinearly implicit schemesen
dc.subjectSpectral methodsen
dc.subjectAbsorbing seten
dc.subjectChaotic attractorsen
dc.subjectChaotic behaviorsen
dc.subjectChaotic dynamicsen
dc.subjectDiagnostic toolsen
dc.subjectDispersive effectsen
dc.subjectDispersive-dissipative systemsen
dc.subjectFinite dimensionsen
dc.subjectFourth orderen
dc.subjectFully discrete schemeen
dc.subjectGeneral classen
dc.subjectGlobal attractoren
dc.subjectHigher orderen
dc.subjectInitial value problemsen
dc.subjectKuramoto-Sivashinsky equationen
dc.subjectKuramoto-Sivashinsky equationsen
dc.subjectLong-time dynamicsen
dc.subjectLongwave instabilityen
dc.subjectNegative diffusionen
dc.subjectNonlinear saturationen
dc.subjectOrder dispersionsen
dc.subjectPeriodic initial value problemsen
dc.subjectPhysical applicationen
dc.subjectPseudo-differential operatoren
dc.subjectPseudospectral methodsen
dc.subjectReturn mapen
dc.subjectShort wavesen
dc.subjectThird order dispersionen
dc.subjectTime unitsen
dc.subjectTime-stepping schemesen
dc.titleComputational study of the dispersively modified Kuramoto-Sivashinsky equationen
dc.description.endingpageA813Σχολή Θετικών και Εφαρμοσμένων Επιστημών / Faculty of Pure and Applied SciencesΤμήμα Μαθηματικών και Στατιστικής / Department of Mathematics and Statistics
dc.description.notes<p>Cited By :12</p>en
dc.source.abbreviationSiam J.Sci.Comput.en
dc.contributor.orcidSmyrlis, Yiorgos-Sokratis [0000-0001-9126-2441]

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