Long Time Behavior of Difference Approximations for the Two-Dimensional Complex Ginzburg–Landau Equation

Abstract

The finite difference approximations of the dynamical systems governed by two-dimensional complex Ginzburg–Landau equation was considered. For a fully discrete scheme, the solvability, stability and convergence are proved in discrete Sobolev spaces. Furthermore, the existence of global attractors 𝒜 _{h, τ} of the discrete system and the upper semicontinuity dist(𝒜 _{h, τ}, 𝒜) → 0 are obtained.

Keywords:

• Ginzburg–Landau equation
• Global attractors
• Semi-explicit difference approximation
• Stability and convergence

AMS Subject Classification:

• 65M06
• 37L99
• 35B40
• 35Q55
• 65P99

ACKNOWLEDGMENTS

Gratitude is extended to Prof. FaYong Zhang for his guidance. Thanks is also offered to Prof. Endre Suli and Prof. Andrew Stuart for many valuable suggestions.