Finite-dimensional approximations of random attractor for stochastic discrete complex Ginzburg-Landau equations

TL;DR

This paper develops finite-dimensional approximations for random attractors in stochastic discrete complex Ginzburg-Landau equations, demonstrating convergence and stability of these approximations.

Contribution

It introduces a numerical scheme for discretizing the stochastic complex Ginzburg-Landau equation and proves the existence and convergence of various attractors, including random attractors.

Findings

Existence of a numerical attractor for the discretized system

Upper semicontinuity of the numerical attractor with respect to the global attractor

Finite-dimensional approximations of global, numerical, and random attractors

Abstract

In this paper, we apply an implicit Euler scheme to discretize the complex Ginzburg-Landau equation and prove the existence of a numerical attractor for the discrete Ginzburg-Landau system. We establish the upper semicontinuity of the numerical attractor with respect to the global attractor as the time step tends to zero. Furthermore, we provide finite-dimensional approximations for three types of attractors (global, numerical, and random), and demonstrate the existence of truncated attractors along with their convergence as the dimension of the state space tends to infinity. Finally, we prove the existence of a random attractor and establish the upper semi-continuity both of the global random attractor and the truncated random attractor.