A structure preserving difference scheme with fast algorithms for high dimensional nonlinear space-fractional Schrödinger equations

Yin, Baoli; Wang, Jinfeng; Liu, Yang; Li, Hong

doi:10.1016/j.jcp.2020.109869

Cited by 20 publications

(9 citation statements)

References 40 publications

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“…In the future, we will apply the TT-M FE method to two-or three-dimensional coupled Schrödinger-Boussinesq models, nonlinear fractional Schrödinger equations [34][35][36][37][38][39][40][41] and the coupled nonlinear Schrödinger equations [42][43][44], and also investigate the method's conservation properties.…”

Section: Discussionmentioning

confidence: 99%

TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

Tian

Sun

Liu

et al. 2022

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In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank–Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor’s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method.

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Section: Discussionmentioning

confidence: 99%

TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

Tian

Sun

Liu

et al. 2022

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“…Noticing that the solution of (1a)-(1c) satisfies the associated preservation properties as below (cf. earlier studies 2,18 ) M(𝜓(., t))…”

Section: Introductionmentioning

confidence: 93%

“…i,𝑗 be the approximation of the splitting conservative FDM ( 12)- (18). Then the truncation errors are valid: 12)-( 16) from ( 28)- (32) gives the error equations as follows:…”

Section: Lemma 3 Let 𝜓(X 𝑦 T) Be the Analytical Solution Of (1a)-(1c)...mentioning

confidence: 99%

“…Devising accurate and efficient algorithms capable of preserving the invariant properties of the system is essential because of the merits in stability and good resolution. So far many excellent conservative algorithms have been created to mimic the nonlinear space fractional Schrödinger equation 12–23 …”

Section: Introductionmentioning

confidence: 99%

“…So far many excellent conservative algorithms have been created to mimic the nonlinear space fractional Schrödinger equation. [12][13][14][15][16][17][18][19][20][21][22][23] As we know, in terms of the advantage of saving the memory and CPU time, the splitting algorithms have been extensionally applied to numerically simulate PDEs and nonlinear problems (cf. previous work [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Time second‐order splitting conservative difference scheme for nonlinear fractional Schrödinger equation

Xie

Ali

Zhang

2022

Math Methods in App Sciences

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This paper concentrates on the error estimation of novel time second-order splitting conservative finite difference method (FDM) for high-dimensional nonlinear fractional Schrödinger equation rigorously. The discrete preservation property of our scheme is exhibited. By virtue of the cut-off technique and discrete energy method, it is shown that our scheme possesses the accuracy of (Δt 2 + h 2x + h 2 𝑦 ) in sense of L 2 -norm. Numerical experiments are exhibited to validate the accuracy and conservation property of our scheme.

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Fast L2-1σ Galerkin FEMs for generalized nonlinear coupled Schrödinger equations with Caputo derivatives

Wei

Niu

et al. 2022

Applied Mathematics and Computation

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A structure preserving difference scheme with fast algorithms for high dimensional nonlinear space-fractional Schrödinger equations

Cited by 20 publications

References 40 publications

TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

Time second‐order splitting conservative difference scheme for nonlinear fractional Schrödinger equation

Fast L2-1σ Galerkin FEMs for generalized nonlinear coupled Schrödinger equations with Caputo derivatives

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