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

We show that two fully discrete nonlinear Galerkin schemes based on ex plicit approximate inertial manifolds preserve the dissipativity of th e Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing bal l 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 specifi cally designed to deal with the difficulty of the linear instability i n the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Nav ier-Stokes equations.