5. Conclusions

In this chapter, we developed and analyzed a local discontinuous Galerkin method for solving the nonlinear Ginzburg-Landau equation and have proven the stability of this method. Numerical experiments confirm that the optimal order of convergence is recovered. As a last example, the Ginzburg-Landau equation with initial condition is solved for different values of γ and results show that the parameter γ dramatically affects the wave shape. In addition, the solution decays rapidly with time evolution especially for γ<0.
