Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg‐Landau equations

Pan, Kejia; Jin, Xianlin; He, Dongdong

doi:10.1002/mma.5897

Cited by 11 publications

(2 citation statements)

References 72 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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“…The scheme is uniquely solvable, and the numerical solutions are bounded and unconditionally convergent. For the strongly coupled fractional Ginzburg-Landau system, a linearized three time level semi-implicit finite difference scheme in [34] was proposed to solve it. The difference scheme is unconditionally stable, fourth-order accurate in space, and second-order accurate in time.…”

Section: Introductionmentioning

confidence: 99%

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

2021

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A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

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A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

Pan

Yue

et al. 2020

Comp. Appl. Math.

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In this paper, an efficient and high-order accuracy finite difference method is proposed for solving multidimensional nonlinear Burgers' equation. The third-order three stage Runge-Kutta total variation diminishing (TVD) scheme is employed for the time discretization, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. Our method is third-order accurate in time and sixth-order accurate in space. The CCD-TVD method treats the nonlinear term explicitly thus it is very efficient and easy to implement. In addition, we prove the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments including both two-dimensional and three-dimensional problems have been conducted to demonstrate the high efficiency and accuracy of the proposed method.

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Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg‐Landau equations

Cited by 11 publications

References 72 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

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

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