Linearly implicit methods for a semilinear parabolic system arising in two-phase flows

Abstract

We study the discretization of a nonlinear parabolic system arising in two-phase flows, which in a special case reduces to the Kuramoto–Sivashinsky equation, by linearly implicit methods and, in particular, by implicit–explicit multistep methods. We carry out extensive numerical experiments to investigate the accuracy and efficiency of these algorithms with extremely satisfactory results. These numerical experiments establish the analyticity of the solution and the existence of global attractors (rigorous proofs of such results for this system are not available). Our numerical experiments yield a sharp estimate for the band of analyticity of the solutions as the parameters vary. The accuracy of the schemes enables, in general, the exhaustive numerical study of such systems and the full classification of the inertial manifold. We provide numerical examples of travelling time-periodic attractors as well as quasi-periodic and chaotic attractors.