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 multi-phase hy-
drodynamics and the emerging dynamics arise from a competition of long-wave 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 pseudo-differential 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

Source: Akrivis, Georgios - Department of Computer Science, University of Ioannina

Collections: Computer Technologies and Information Sciences