Certain Nonlinear Equations, Numerical Analysis,

A generalized Kuramoto-Sivashinsky equation possessing a global attractor is considered. The existence of an attractor is one of the most important characteristics for a dissipative system.The long-time dynamics is completely determined by the attractor of the system. The dynamical properties of a class of finite diference scheme are analysed.The exsi-tence of global attractor is proved for the discrete system in Hh2 space .In order to prove the existence of an attractor , we first set up the interpolation unequality in discrete periodic functions space, from which, the simu-taneous relation of || · ||L2h between |·|Wk,p h and |·|Hnh (p ∈ Z+, 0 ≤ k < n) is derived.Then, it constructs the finite difference scheme of the equation.The existence of an absorbing set in Hh2 space , the stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomouse system case.So the existence of an attractor of the equation in H2h space is obtained by using the known conclusion. Finally, long-time stability of the finite difference scheme and convergence of the difference solution also are analysed in the nonau-tonomous system case.