Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

Zhang, Zuo‐Feng; Lin, Xiaoman; Pan, Kejia; Ren, Yunzhu

doi:10.1016/j.camwa.2020.05.027

Cited by 37 publications

(10 citation statements)

References 49 publications

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“…In the end, we implemented the difference scheme through two numerical tests, which showed a perfect consistency with our theoretical findings. In addition, our numerical method and the analysis technique of the optimal pointwise error estimates can be easily extended to the cases with spatial fourth-order accuracy [54], the 2D coupled SFNSE [40], 2D space fractional Ginzburg-Landau equation [37,53] and some other space fractional diffusion equation in 2D and 3D [15,51,52,56], which will be our future work. In addition, for the resulting systems of algebraic equations, the coefficient matrices have Toeplitz structure, which can be solved by adopting a super-fast solver with preconditioner [23,[27][28][29] or multigrid methods [36,38] to reduce the CPU time and storage requirement in future.…”

Section: Discussionmentioning

confidence: 99%

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

Jin

et al. 2021

Numerical Methods Partial

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In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at the discrete level, with an unconditional stability and a second-order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.

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Section: Discussionmentioning

confidence: 99%

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

Jin

et al. 2021

Numerical Methods Partial

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“…Then, we denote rate τ = log τ1/τ2 relerr(τ 1 , h) relerr(τ 2 , h) and rate h = log h1/h2 relerr(τ, h 1 ) relerr(τ, h 2 ) . The numerical solution ((N, M ) = (512, 10000)) computed by the linearized second-order backward differential (LBDF2) scheme [24] is used as the reference solution. Moreover, we denote the numerical solution obtained by the LBDF2 scheme [24] as the LBDF2 solution.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In [19], the authors used an exponential Runge-Kutta method to solve the two-dimensional (2D) FGLE. Other related work can be found in [20][21][22][23][24] and references therein.…”

Section: Introductionmentioning

confidence: 99%

A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

Zhao

Ostermann

2020

Preprint

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Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is used. The convergence of our method is proved rigorously.Numerical examples are reported which show that the proposed method is robust and accurate.

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“…Zhang et al [11] established the numerical asymptotic stability result of the compact θ -method for the generalized delay diffusion equation. More researches on delay fractional problems can be referred to [12,13] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

Qin

Yang

Ren

et al. 2021

Journal of Function Spaces

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In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.

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Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

Cited by 37 publications

References 49 publications

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

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