Time discretization and stability regions for dissipative-dispersive Kuramoto-Sivashinsky equation arising in turbulent gas flow over laminar liquid

In this study, spatio-temporal discretization for semilinear dissipative partial differential equations type is introduced, analyzed and implemented. The model studied here is the dispersively Kuramoto-Sivashinsky equation with an additional term representing the dispersive term, arising in turbulent gas flow over laminar liquid (Tseluiko and Kalliadasis, 2011). This additional term is multiplied by a parameter that represents the influence of the turbulent gas flow. Our objective is to examine the effect of this additional term on the dynamics of the Kuramoto-Sivashinsky equation characterized by its chaotic behavior. This is achieved by combining the Exponential Time Differencing Crank-Nicolson (ETD-CN) scheme derived by Kleefed et al. (2012), and Fourier pseudospectral schemes for temporal and spatial stepping, respectively. The method is known to be stable and second order convergent. In addition, a theoretical study and a plot of stability regions of ETD-CN were performed showing its effectiveness for the stiff problem studied here.