Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC

Kong, Linghua; Hong, Jialin; Fu, F. C. L.; Chen, Jing

doi:10.1016/j.cam.2011.04.019

Cited by 12 publications

(3 citation statements)

References 22 publications

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“…However, the system is required to solve nonlinear algebraic equations at each time step. To reduce the computational cost, we introduce the splitting idea [48] for the symplectic integrators. The splitting method was originally designed to improve the speed, accuracy, and stability of numerical simulation.…”

Section: Splitting Symplectic Integratorsmentioning

confidence: 99%

Symplectic‐preserving Fourier spectral scheme for space fractionalKlein–Gordon–Schrödingerequations

Wang

2020

Numerical Methods Partial

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In the paper, the symplectic‐preserving Fourier spectral scheme is presented for space fractional Klein–Gordon–Schrödinger equations involving fractional Laplacian. First, we validate space fractional Klein–Gordon–Schrödinger equations that can be expressed as an infinite dimension Hamiltonian system. We apply the Fourier spectral method in space, and the semi‐discrete system preserves the mass and energy conservation laws. Second, by introducing some variables, the semi‐discrete system can be expressed as a large Hamiltonian ordinary differential system. We use the midpoint rule in time to semi‐discrete system, and obtain a symplectic approximation scheme of these equations. Moreover, we can prove that the scheme is convergent. To reduce the computational cost, we introduce the splitting idea for the symplectic integrators. Finally, we give numerical experiments to show the efficiency of the scheme.

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Section: Splitting Symplectic Integratorsmentioning

confidence: 99%

Symplectic‐preserving Fourier spectral scheme for space fractionalKlein–Gordon–Schrödingerequations

Wang

2020

Numerical Methods Partial

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“…we now consider the repulsive GP equation in space [−8, 8] × [−8, 8] (see [22]). Let N = M = 36, ∆t = 0.1, we is an external field, and A is a constant.…”

Section: Numerical Experiments For 2d Nonlinear Schrödinger Equationsmentioning

confidence: 99%

General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs

2015

Journal of Computational Physics

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In this paper we propose and investigate a general approach to constructing local energy-preserving algorithms which can be of arbitrarily high order in time for solving Hamiltonian PDEs. This approach is based on the temporal discretization using continuous Runge-Kutta-type methods, and the spatial discretization using pseudospectral methods or Gauss-Legendre collocation methods. The local energy conservation law of our new schemes is analyzed in detail. The effectiveness of the novel local energy-preserving integrators is demonstrated by coupled nonlinear Schrödinger equations and 2D nonlinear Schrödinger equations with external fields. Our new schemes are compared with some classical multi-symplectic and symplectic schemes in numerical experiments. The numerical results show the remarkable long-term behaviour of our new schemes.

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“…Very interesting problems in quantum physics consist of multi-particle systems that can be modeled by the multi-particle Shrödinger equations. These equations are important in many different fields of science such as wave propagation, relativistic quantum mechanics, quantum field theory and mathematical physics [2,8,10,12,16,18,19].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Shrödinger Equations Based on the Meshless Methods

RAHIMI¹,

KARBASSI²,

HOOSHMANDASL³

2022

KgJMat

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In this work, two-dimensional time-dependent quantum equation problems are studied. We introduce a numerical algorithm for solving the two-dimensional nonlinear complex quantum system with MLS and FDM methods. An eﬃcient and accurate computational algorithm based on both, the moving least squares (MLS) and the ﬁnite diﬀerence (FDM) methods is proposed for solving it. The results demonstrate that the proposed algorithm is a robust algorithm with good accuracy. This is developed on MLS and FDM methods using numerical simulation for solving these kind of problems.

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Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC

Cited by 12 publications

References 22 publications

Symplectic‐preserving Fourier spectral scheme for space fractionalKlein–Gordon–Schrödingerequations

Symplectic‐preserving Fourier spectral scheme for space fractionalKlein–Gordon–Schrödingerequations

General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs

Numerical Solution of Shrödinger Equations Based on the Meshless Methods

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