Numerical analysis of a fourth-order linearized difference method for nonlinear time-space fractional Ginzburg-Landau equation

An efficient difference method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the $L2$- $1_{\sigma }$ scheme to handle Caputo fractional derivative, while a fourth-order difference method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the difference scheme is unconditionally convergent in pointwise sense with the rate of $O({\tau }^2+h^4)$, where $\tau$ and $h$ are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis.