A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

Hao, Zhaopeng; Sun, Zhi‐zhong

doi:10.1002/num.22076

Cited by 36 publications

(20 citation statements)

References 27 publications

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“…And the second-order scheme (3.5) is easy to extend to the case of spatial fourth-order. Let A α x be the average operator defined as [28] A α…”

Section: Existence and Uniquenessmentioning

confidence: 99%

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An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

Pan

2018

Numer Algor

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In this paper, the coupled fractional Ginzburg-Landau equations are first time investigated numerically. A linearized implicit finite difference scheme is proposed. The scheme involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. The unique solvability, the unconditional stability and optimal pointwise error estimates are obtained by using the energy method and mathematical induction. Moreover, the proposed second-order method can be easily extended into the fourth-order method by using an average finite difference operator for spatial fractional derivatives and Richardson extrapolation for time variable. Finally, numerical results are presented to confirm the theoretical results.The Ginzburg-Landau equation (GLE) has been used to model a wide variety of physical systems [1]. The existence, regularity, and the long-time behavior of the exact solution for the GLE are given in [2][3][4][5]. For the global well-posedness and the existence of the global attractor of the GLE, readers can refer to Refs. [6,7]. Sakaguchi and Malomed [8] proposed the coupled Ginzburg-Landau equations (CGLE) to describe the Bose-Einstein condensates and nonlinear optical waveguides and cavities. And the linearly coupled complex cubic-quintic Ginzburg-Landau equations [9] and the coupled Ginzburg-Landau equations with higher-order nonlinearities [10] are also discussed.The fractional generalization of the GLE was first suggested by Tarasov and Zaslavsky [11,12] for fractal media. Since then, the fractional Ginzburg-Landau equation (FGLE) has been exploited to describe various physical phenomena, such as the media with fractal mass dimension [11], a class of critical phenomena when the organization of the system near the phase transition point is influenced by a competing nonlocal ordering [13] and a network of diffusively Hindmarsh-Rose neurons with long-range synaptic coupling [14]. Recently, the coupled fractional Ginzburg-Landau equation (CFGLE) with stocastic noise was discussed by [15].The dissipative mechanism of the FGLE and CFGLE are not characterized by the classical Laplacian but by the fractional power of the Laplacian, which brings some essential difficulties in theoretical analysis. Many authors have worked on the FGLE and CFGLE from the theoretical aspects [15][16][17][18][19][20]. For example, the solution of FGLE is derived by using psi-series with fractional powers [16]. Pu and Guo [17] investigated the global well-posedness and dynamics for the nonlinear FGLE. Millot and Sire [20] considered the asymptotic analysis of the FGLE in a bounded domain. Shu et al. [15] studied the random attractors for the CFGLE with stochastic noise. From the numerical point of view, there are quite a lot of numerical studies for the classical GLE, see [21-26] and references therein. But there are not too much numerical studies for the FGLE. Mvogo et al. [14] proposed a

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“…And the second-order scheme (3.5) is easy to extend to the case of spatial fourth-order. Let A α x be the average operator defined as [28] A α…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…However, the method is a nonlinear scheme, a fixed point iteration is needed at each time step, which is generally computational expensive. Hao and Sun [28] proposed a three-level linearized method for FGLE with second-order accuracy in time and fourthorder accuracy in space. However, this method is not unconditionally stable.…”

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confidence: 99%

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

Pan

2018

Numer Algor

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“…Lemma 3.1. (See [8,9]) For 1 2 < σ ≤ 1, there exists a constant c 0 > 0 depending on the parameter σ but independent of h > 0 such that…”

Section: Analysis Of the Finite Difference Schemesmentioning

confidence: 99%

An Improved Algorithm Based on Finite Difference Schemes for Fractional Boundary Value Problems with Nonsmooth Solution

Hao

Cao

2017

J Sci Comput

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In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with non-smooth solution. Two popular finite difference schemes, the weighted shifted Grünwald difference (WSGD) scheme and the fractional centered difference (FCD) scheme, are revisited and the error estimate of the schemes is provided in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, it is demonstrated that, with the use of the proposed algorithm, the improved WSGD and FCD schemes can recover the second-order accuracy for non-smooth solution. Several numerical examples are given to validate our theoretical prediction. It is shown that both accuracy and convergence rate of numerical solutions can be significantly improved by using the proposed algorithm.

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“…There are quite a lot of numerical studies for the classical GLE and FGLE, see [7][8][9][10][11][12][13][14][15] and the references therein. To the authors' best knowledge, there are not too many numerical studies for the CFGLE.…”

Section: Introductionmentioning

confidence: 99%

A fourth-order linearized difference scheme for the coupled space fractional Ginzburg–Landau equation

Yuan

Zeng

2019

Adv Differ Equ

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In this paper, the coupled space fractional Ginzburg-Landau equations are investigated numerically. A linearized semi-implicit difference scheme is proposed. The scheme is unconditionally stable, fourth-order accurate in space, and second-order accurate in time. The optimal pointwise error estimates, unique solvability, and unconditional stability are obtained. Moreover, Richardson extrapolation is exploited to improve the temporal accuracy to fourth order. Finally, numerical results are presented to confirm the theoretical results.

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A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

Cited by 36 publications

References 27 publications

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

An Improved Algorithm Based on Finite Difference Schemes for Fractional Boundary Value Problems with Nonsmooth Solution

A fourth-order linearized difference scheme for the coupled space fractional Ginzburg–Landau equation

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