Computational study of the dispersively modified KuramotoSivashinsky equation
Date
2012ISSN
10648275Source
SIAM Journal on Scientific ComputingVolume
34Issue
2Pages
A792A813Google Scholar check
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We analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified KuramotoSivashinsky 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 longwave instability (negative diffusion), shortwave 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 KuramotoSivashinsky equation. In such instances we show that first and second order timestepping 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 KuramotoSivashinskyKortewegdeVries equation also known as the Kawahara equation in hydrodynamics), higher order timestepping 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 infinitedimensional dynamics to onedimensional 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 BenneyLin equation) but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics. © 2012 Society for Industrial and Applied Mathematics.
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