A Second-Order Alikhanov Type Implicit Scheme for the Time-Space Fractional Ginzburg-Landau Equation
Year: 2024
Author: Yufang Gao, Shengxiang Chang, Changna Lu
Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 6 : pp. 1451–1473
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2022-0097
Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 6 : pp. 1451–1473
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Time-space fractional Ginzburg-Landau equation Caputo fractional derivative Riesz fractional derivative $L2−1_σ$ formula convergence.