A Second-Order Alikhanov Type Implicit Scheme for the Time-Space Fractional Ginzburg-Landau Equation (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 shiftedGrünwald difference method to approximate the Riesz space fractional derivative. In order to overcome the non-local property of Riesz space fractional derivatives and thehistorical dependence brought by Caputo time fractional derivatives, this paper introduces the fractional Sobolev norm and the fractional Sobolev inequality. It is provedin 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 $\mathcal{O}(τ^2+h^2)$ with time step $τ$ and mesh size $h.$ Finally, we provide a linearized iterative algorithm, and the numerical results arepresented to verify the accuracy and efficiency of the proposed scheme.