An energy conservative difference scheme for the nonlinear fractional Schrödinger equations

Wang, Pengde; Huang, Chengming

doi:10.1016/j.jcp.2014.03.037

Cited by 195 publications

(77 citation statements)

References 34 publications

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“…Along the numerical front, different reliable and efficient numerical methods have been developed, such as finite difference methods [8][9][10][11][12][13][14][15][16][17], spectral or collocation methods [18][19][20][21][22][23][24][25][26] and finite element methods [27][28][29]. It is well known that for differential equations with geometric structures, the structure-preserving schemes always perform better than the general-purpose ones [30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that for differential equations with geometric structures, the structure-preserving schemes always perform better than the general-purpose ones [30][31][32][33]. For the FSE, extensive invariant-preserving numerical methods have been constructed, for example, the schemes in [8,12,14,22,28] preserve the mass, and the schemes in [9,13,15,21,24,25,27,29] preserve both the mass and energy.…”

Section: Introductionmentioning

confidence: 99%

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Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

Fei

Huang

Wang

2019

Numerical Methods Partial

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This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L 2 norm with the second-order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results. KEYWORDSconservation law, error estimate, Fourier pseudospectral method, fractional Schrödinger equation, structure-preserving method 1 Numer Methods Partial Differential Eq. 2020;36:369-393. wileyonlinelibrary.com/journal/num

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

Fei

Huang

Wang

2019

Numerical Methods Partial

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“…For the time-fractional Schrödinger equations, Mohebbi et al [23] employed a meshless technique based on collocation methods and radial basis functions, Khan et al [16] derived approximating solutions by homotopy analysis methods, and Wei et al [35] gave discrete solution via a rigorous analysis of implicit fully discrete local discontinuous Galerkin method. For the space-fractional Schrödinger equations, some fully or linearly implicit difference methods were introduced and discrete conservation properties were analyzed in [30,31,33]. Two-dimensional problems were considered and a fourth-order ADI scheme was presented in [40].…”

Section: Introductionmentioning

confidence: 99%

Galerkin finite element method for nonlinear fractional Schrödinger equations

2016

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In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

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“…The convergence order is O(τ 2 + h 2 ). Recently, Wang and Huang [26] studied the nonlinear Riesz space fractional Schrödinger equations and analyzed the mass conservation and energy conservation in the discrete sense. The convergence order of the proposed scheme was also proved to be O(τ 2 + h 2 ).…”

Section: Introductionmentioning

confidence: 99%

A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation

Zhao¹,

Sun²,

Hao³

2014

SIAM J. Sci. Comput.

246

105

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In this paper, a novel compact operator is derived for the approximation of the Riesz derivative with order α ∈ (1, 2]. The compact operator is proved with fourth-order accuracy. Combining the compact operator in space discretization, a linearized difference scheme is proposed for a two-dimensional nonlinear space fractional Schrödinger equation. It is proved that the difference scheme is uniquely solvable, stable, and convergent with order O(τ 2 + h 4 ), where τ is the time step size, h = max{h 1 , h 2 }, and h 1 , h 2 are space grid sizes in the x direction and the y direction, respectively. Based on the linearized difference scheme, a compact alternating direction implicit scheme is presented and analyzed. Numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly. Introduction.Fractional quantum mechanics is a theory used to discuss quantum phenomena in fractal environments. In quantum physics the first successful attempt to apply the fractality concept was the Feynman path integral approach to quantum mechanics. Feynman and Hibbs reformulated the nonrelativistic quantum mechanics as a path integral over Brownian paths [1,2]. Thus the Feynman-Hibbs fractional background leads to standard (nonfractional) quantum mechanics. Laskin [3] extended the fractality concept in quantum physics to construct a fractional path integral and formulate the fractional quantum mechanics as a path integral over the paths of the Lévy flights. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Lévy paths leads to fractional quantum mechanics and fractional statistical mechanics, namely, if the path integral over Brownian trajectories leads to the well-known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.The space fractional Schrödinger equation includes a space fractional derivative of order α (1 < α < 2) instead of the Laplacian in the standard Schrödinger equation [4]. Assumption that an anomalous relation between energy and the angular frequency is of fractional Planck quantum energy relation leads to a time fractional Schrödinger equation [5], which replaces iu t in the classical time dependent Schrödinger equation by i α ∂ α u ∂t α , where i 2 = −1 and α is the order of fractional derivative.

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An energy conservative difference scheme for the nonlinear fractional Schrödinger equations

Cited by 195 publications

References 34 publications

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

Galerkin finite element method for nonlinear fractional Schrödinger equations

A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation

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