(untitled) (Advances in Applied Mathematics and Mechanics, 2024)
In this paper, we consider an implicit method for solving the nonlinear time-space fractional Ginzburg-Landau equation. The scheme is based on the L2-1_σ formula to approximate the Caputo fractional derivative and the weighted and shifted Grünwald difference method to approximate the Riesz space fractional derivative. In order to overcome the non-local property of Riesz space fractional derivatives and the historical dependence brought by Caputo time fractional derivatives, this paper introduces the fractional Sobolev norm and the fractional Sobolev inequality. It is proved in detail that the difference scheme is stable and uniquely solvable by the discrete energy method. In particular, the difference scheme is unconditionally stable when γ≤0, where γ is a coefficient of the equation. Moreover, the scheme is shown to be convergent in l^2_h norm at the optimal order of O(τ^2+h^2) with time step τ and mesh size h. Finally, we provide a linearized iterative algorithm, and the numerical results are presented to verify the accuracy and efficiency of the proposed scheme.