ON SOME DISSIPATIVE FULLY DISCRETE NONLINEAR GALERKIN SCHEMES FOR THE KURAMOTO-SIVASHINSKY EQUATION

We show that two fully discrete nonlinear Galerkin schemes based on explicit approximate inertial manifolds preserve the dissipativity of the Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing ball is shown to be uniform in both the time step and number of modes, so that the result holds in the PDE limit. While the schemes are specifically designed to deal with the difficulty of the linear instability in the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Navier-Stokes equations.