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Time discretization and stability regions for dissipative-dispersive Kuramoto-Sivashinsky equation arising in turbulent gas flow over laminar liquid. (English) Zbl 1432.76047

Summary: 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 [D. Tseluiko and S. Kalliadasis, J. Fluid Mech. 673, 19–59 (2011; Zbl 1225.76044)]. 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 B. Kleefed et al. [Numer. Methods Partial Differ. Equations 28, No. 4, 1309–1335 (2012; Zbl 1253.65128)], 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.

MSC:

76A20 Thin fluid films
76N15 Gas dynamics (general theory)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Software:

SERK2v3; SERK2
PDFBibTeX XMLCite
Full Text: DOI

References:

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