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

Mingfa Fei; Wenhao Li; Yulian Yi; Mingfa Fei; Wenhao Li; Yulian Yi

doi:10.3934/era.2022186

Electronic Research Archive

2022, Volume 30, Issue 10: 3635-3659. doi: 10.3934/era.2022186

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

1.
School of Mathematics, Changsha University, Changsha 410022, China
2.
Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received: 08 June 2022 Revised: 21 July 2022 Accepted: 25 July 2022 Published: 01 August 2022

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.
- Ginzburg-Landau equation,
- fractional derivative,
- $ L2 $-$ 1_{{\sigma}} $ scheme,
- difference method,
- convergence
Citation: Mingfa Fei, Wenhao Li, Yulian Yi. Numerical analysis of a fourth-order linearized difference method for nonlinear time-space fractional Ginzburg-Landau equation[J]. Electronic Research Archive, 2022, 30(10): 3635-3659. doi: 10.3934/era.2022186

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Abstract

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.

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