 
Summary: COMPUTATIONAL STUDY OF THE DISPERSIVELY
MODIFIED KURAMOTOSIVASHINSKY EQUATION
G. AKRIVIS, D. T. PAPAGEORGIOU, AND Y.S. SMYRLIS§
Abstract. 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 dis
cretization. The general case analyzed covers several physical applications arising in multiphase hy
drodynamics and the emerging dynamics arise from a competition of longwave instability (negative
diffusion), shortwave damping (fourth order stabilization), nonlinear saturation (Burgers nonlinear
ity) 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 dis
persive term is taken to be a pseudodifferential operator which is allowed to have higher order than
the familiar fourth order stabilizing term in KuramotoSivashinsky equation. In such instances we
show that first and secondorder 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
