IMA Journal of Numerical Analysis (2015) Page 1 of 20 doi:10.1093/imanum/drv011 Linearly implicit schemes for multi-dimensional Kuramoto–Sivashinsky type equations arising in falling film flows Georgios Akrivis Department of Computer Science and Engineering, University of Ioannina, 45110 Ioannina, Greece akrivis@cs.uoi.gr Anna Kalogirou Department of Mathematics, Imperial College London, London SW7 2AZ, UK anna.kalogirou09@imperial.ac.uk Demetrios T. Papageorgiou ∗ Department of Mathematics, Imperial College London, London SW7 2AZ, UK ∗ Corresponding author: d.papageorgiou@imperial.ac.uk and Yiorgos-Sokratis Smyrlis Department of Mathematics and Statistics, University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus smyrlis@ucy.ac.cy [Received on 23 April 2014; revised on 24 February 2015] This study introduces, analyses and implements space-time discretizations of two-dimensional active dissipative partial differential equations such as the Topper–Kawahara equation; this is the two- dimensional extension of the dispersively modified Kuramoto–Sivashinsky equation found in falling film hydrodynamics. The spatially periodic initial value problem is considered as the size of the peri- odic box increases. The schemes utilized are implicit–explicit multistep (BDF) in time and spectral in space. Numerical analysis of these schemes is carried out and error estimates, in both time and space, are derived. Preliminary numerical experiments provided strong evidence of analyticity, thus yielding a practical rule-of-thumb that determines the size of the truncation in Fourier space. The accuracy of the BDF schemes (of order 1–6) is confirmed through computations. Extensive computations into the strongly chaotic regime (as the domain size increases), provided an optimal estimate of the size of the absorbing ball as a function of the size of the domain; this estimate is found to be proportional to the area of the periodic box. Numerical experiments were also carried out in the presence of dispersion. It is observed that sufficient amounts of dispersion reduce the complexity of the chaotic dynamics, and can organize solution into nonlinear travelling wave pulses of permanent form. Keywords: Topper–Kawahara equation; linearly implicit schemes; implicit–explicit BDF schemes; spectral methods; error estimates; dynamical systems. 1. Introduction In this study, we develop and implement numerical schemes to solve classes of multidimensional active- dissipative partial differential equations (PDEs) in the presence of dispersion. Of particular interest are two-dimensional Kuramoto–Sivashinsky type equations arising in the hydrodynamic stability of viscous c  The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. IMA Journal of Numerical Analysis Advance Access published April 9, 2015 at University of Cyprus on April 13, 2015 http://imajna.oxfordjournals.org/ Downloaded from