Compact Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Wave Operator

Lang-yang, Huang; Tian, Zhenfu; Cai, Yaoxiong

doi:10.1155/2020/4345278

Cited by 4 publications

(4 citation statements)

References 28 publications

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“…where D i is the partial derivative with respect to the i-th argument. The corresponding formula of the formulation ( 17) is the formula (6). By eliminating p n , the evolution formulation of x n can be obtained, which is ψ b , as follows…”

Section: The Symplectic Propertymentioning

confidence: 99%

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Highly Efficient Numerical Integrator for the Circular Restricted Three-Body Problem

Wang

Tang

2022

Symmetry

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The dynamic equation of a mass point in the circular restricted three-body problem is governed by Coriolis and centrifugal force, in addition to a co-rotating potential relative to the frame. In this paper, we provide an explicit, symmetric integrator for this problem. Such an integrator is more efficient than the symplectic Euler method and the Gauss Runge–Kutta method as regards this problem. In addition, we proved the integrator is symplectic by the discrete Hamilton’s principle. Several groups of numerical experiments demonstrated the precision and high efficiency of the integrator in the examples of the quadratic potential and the bounded orbits in the circular restricted three-body problem.

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Section: The Symplectic Propertymentioning

confidence: 99%

“…Therefore, the symmetric, symplectic algorithms [1][2][3][4][5] are the standard methods to such problems. In addition, efficient structure-preserving methods [6,7] are also a research focus. The well-known Boris algorithm [8][9][10][11][12][13] in the plasma dynamics has some good geometric properties.…”

Section: Introductionmentioning

confidence: 99%

Highly Efficient Numerical Integrator for the Circular Restricted Three-Body Problem

Wang

Tang

2022

Symmetry

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“…Brugnano et al applied the Hamiltonian boundary value methods to the NLSW in [7]. A local energy-preserving method was introduced in [16] by Huang et al However, the aforementioned numerical schemes are fully implicit, and a nonlinear system must be solved by some iterative methods which makes them time-consuming. To improve the efficiency, linear energy-preserving schemes based on the leapfrog method were devised for the NLSW in [22,23,26,31,44,47].…”

Section: Introductionmentioning

confidence: 99%

Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator and convergence analysis

Gu¹,

Cai²,

Jiang³

et al. 2022

Preprint

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In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the H 1 norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori L ∞ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

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“…There are many other related works in this active research area which we cannot enumerate here, including the high-order energy-preserving and momentum-preserving algorithms etc. [25,5,14,22,4,35,23,6].…”

mentioning

confidence: 99%

Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection

Lü¹,

Wang²,

Song³

et al. 2021

DCDS-B

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Based on the local energy dissipation property of the molecular beam epitaxy (MBE) model with slope selection, we develop three, second order fully discrete, local energy dissipation rate preserving (LEDP) algorithms for the model using finite difference methods. For periodic boundary conditions, we show that these algorithms are global energy dissipation rate preserving (GEDP). For adiabatic, physical boundary conditions, we construct two GEDP algorithms from the three LEDP ones with consistently discretized physical boundary conditions. In addition, we show that all the algorithms preserve the total mass at the discrete level as well. Mesh refinement tests are conducted to confirm the convergence rates of the algorithms and two benchmark examples are presented to show the accuracy and performance of the methods.

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Compact Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Wave Operator

Cited by 4 publications

References 28 publications

Highly Efficient Numerical Integrator for the Circular Restricted Three-Body Problem

Highly Efficient Numerical Integrator for the Circular Restricted Three-Body Problem

Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator and convergence analysis

Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection

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