TY - JOUR
T1 - A Second-Order Alikhanov Type Implicit Scheme for the Time-Space Fractional Ginzburg-Landau Equation
AU - Gao , Yufang
AU - Chang , Shengxiang
AU - Lu , Changna
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1451
EP - 1473
PY - 2024
DA - 2024/10
SN - 16
DO - http://doi.org/10.4208/aamm.OA-2022-0097
UR - https://global-sci.org/intro/article_detail/aamm/23474.html
KW - Time-space fractional Ginzburg-Landau equation, Caputo fractional derivative, Riesz fractional derivative, $L2−1_σ$ formula, convergence.
AB - <p style="text-align: justify;">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 $\gamma≤0,$ where $\gamma$ is a coefficient of the equation. Moreover, the scheme is shown to be convergent in $l^2_h$ norm at the optimal order of $\mathcal{O}(\tau^2+h^2)$ with time step $\tau$ 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.</p>