A linear discrete scheme for the Ginzburg–Landau equation

Abstract

This paper considers a 2D Ginzburg–Landau equation with a periodic initial-value condition. A fully discrete Galerkin–Fourier spectral approximation scheme, which is a linear scheme, is constructed and the dynamical behaviour of the discrete system is then analysed. First, the existence and convergence of global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution without any restriction on the time step, and the convergence of the discrete scheme is then obtained. The numerical stability of the discrete scheme is proved.

Keywords:

• Ginzburg–Landau equation
• Global attractor
• Spectral method
• Convergence
• Numerical stability

AMS Subject Classification :

• 65M60
• 65N35
• 65N30

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 10432010 and 10571010).