Computational Study of the Dispersively Modified Kuramoto-Sivashinsky Equation

We 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 multi-phase 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 pseudo-differential operator which is allowed to have higher order than the familiar fourth order stabilizing term in 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 only possible if computations are carried out for very long times (we provide examples where integrations are carried out for 10 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.