Finite Difference Format Of Two-dimensional Ginzburg-landau Equation For A Long Time Behavior

Ginzburg-Landau equation arises in physical problems. It is a very important equation in many fields. A lot of works studied the eristence, the uniqueness of solution of Ginzburg-Landau equation ect. And there are many works studied the numerical methods for it. With the development of infinite dimensional dynamic system, the dynamical behavior of Ginzburg-Landau equation has been investigated. The existence of absorbing sets and global attractors for the dynamic system are obtained. At the same time,there are some results about the long time computational methods for one dimensional Ginzburg-Landau equation.In this paper, a fully discrete finite difference approximation to two dimensional Ginzburg-Landau equation is considered. At first, A fully discrete finite difference scheme was constructed. The existence and uniqueness of the difference solution are proved. The dynamical properties of the discrete dynamical system which is generated by the finite difference scheme are analyzed. By a priori estimate of the discrete solution, the existence of absorbing sets in discrete L~2-spaces,discrete H~1-spaces and discrete H~2-spaces are obtained; existence of global attractors of the discrete system follows. Furthermore, the stability and convergence of the difference scheme are proved.