A time-splitting Galerkin finite element method for the Davey–Stewartson equations

Gao, Yali; Mei, Liquan

doi:10.1016/j.cpc.2015.08.004

Cited by 6 publications

(2 citation statements)

References 28 publications

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“…Some works tackle the elliptic-hyperbolic form of the DSS, and present finite-difference schemes in which the energy is conserved throughout time [16]. A numerical solution for the DSS based on Galerkin's method, finite time steps, and an extrapolated Crank-Nicolson scheme is provided in [17,18]. In that work, a decoupled semi-implicit multistep scheme is implemented to improve accuracy.…”

Section: Introductionmentioning

confidence: 99%

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

Molina-Holguín,

Urenda-Cázares,

Macías-Díaz

et al. 2024

Fractal Fract

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In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications.

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Section: Introductionmentioning

confidence: 99%

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

Molina-Holguín,

Urenda-Cázares,

Macías-Díaz

et al. 2024

Fractal Fract

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“…The split-step approach combined with various numerical methods are efficient for parabolic and Schrödinger-type equations, especially for multi-dimensional ones. The approach could be combined with the FD method, the finite element method, the spectral/pseudospectral method, the orthogonal spline collocation (OSC) method and so on (see [6][7][8][9][10][11][12][13] and related references). One obvious advantage of the split-step technique is that it can efficiently reduce computational costs by separating a multidimensional equation into several ones in one dimension.…”

Section: Introductionmentioning

confidence: 99%

Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions

Wang

Zhang

2019

Numer Algor

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In this paper, split-step cubic B-spline collocation (SS3BC) schemes are constructed by combining the split-step approach with the cubic B-spline collocation (3BC) method for the nonlinear Schrödinger (NLS) equation in one, two, and three dimensions with Neumann boundary conditions. Unfortunately, neither of the advantages of the two methods can be maintained for the multi-dimensional problems, if one combines them in the usual manner. For overcoming the difficulty, new medium quantities are introduced in this paper to successfully reduce the multi-dimensional problems into one-dimensional ones, which are essential for the SS3BC methods. Numerical tests are carried out, and the schemes are verified to be convergent with second-order both in time and space. The proposed method is also compared with the split-step finite difference (SSFD) scheme. Finally, the present method is applied to two problems of the Bose-Einstein condensate. The proposed SS3BC method is numerically verified to be effective and feasible.

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Galerkin methods for the Davey–Stewartson equations

Gao

Mei

2018

Applied Mathematics and Computation

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A time-splitting Galerkin finite element method for the Davey–Stewartson equations

Cited by 6 publications

References 28 publications

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions

Galerkin methods for the Davey–Stewartson equations

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