Fourier spectral approximation to long-time behaviour of the derivative three-dimensional Ginzburg–Landau equation

Abstract

In this paper, we consider a derivative Ginzburg–Landau equation with periodic initial-value condition in three-dimensional space. A fully discrete Galerkin–Fourier spectral approximation scheme is constructed, and then the dynamical behaviour of the discrete system is analysed. Firstly, the existence of global attractors A N τ of the discrete system are proved by a priori estimate of the discrete solution. Next, the convergence of approximate attractors is proved by error estimates of the discrete solution. Furthermore, the long-time convergence as N → ∞ and τ → 0 simultaneously as well as the numerical long-time stability of the discrete scheme are obtained.