Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto–Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(Ah,τ,A)→0d(Ah,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.