Arbitrarily high-order structure-preserving schemes for the Gross–Pitaevskii equation with angular momentum rotation

Cui, Jin; Wang, Yushun; Jiang, Chenglong

doi:10.1016/j.cpc.2020.107767

Cited by 20 publications

(18 citation statements)

References 35 publications

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“…We conclude this paper with two remarks. First, the proposed energy-preserving scheme involves solving linear equations with complicated variable coefficients at every time step and this limitation can be removed by using the idea of the scalar auxiliary variable (SAV) approach [10,43,44]. For more details, please refer to [25].…”

Section: Discussionmentioning

confidence: 99%

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

Jiang¹,

Xu²,

Song³

et al. 2021

Preprint

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In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, our key ideas mainly follow the extrapolation/prediction-correction technique and symplectic Runge-Kutta (RK) methods in time combined with the standard Fourier pseudo-spectral method in space. We show that it is uniquely solvable, unconditionally stable and can exactly preserve the momentum of the system. Subsequently, based on the energy quadratization approach and the analogous linearized idea used in the construction of the linear momentum-preserving scheme, the energy-preserving scheme is presented and it is proven to preserve both the discrete mass and quadratic energy. Numerical results are addressed to demonstrate the accuracy and efficiency of the schemes.

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Section: Discussionmentioning

confidence: 99%

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

Jiang¹,

Xu²,

Song³

et al. 2021

Preprint

Self Cite

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“…When considering the mKdV (p = 3) case, u p+1 dx + C 0 > 0 will automatically hold for any positive constant C 0 , since p + 1 = 4 is even. This is also true for considering the nonlinear Schrödinger (NLS) equations or the Gross-Pitaevskii (GP) equations in [21] and [46], as the authors there only consider the cubic nonlinearity |u| 2 u, and thus, |u| 4 dx + C 0 will automatically hold for any constant C 0 > 0. However, when considering the KdV (p = 2) case, a large enough constant C 0 has to be set in the beginning of the simulation to guarantee v 2 := u p+1 dx + C 0 > 0 for all t ∈ [0, T ].…”

Section: 2mentioning

confidence: 99%

“…Before illustrating the examples, we describe a fast solver for solving the resulting nonlinear system from the IRK methods. This fast solver is similar to the one in [21], which is based on the fixed point iteration. It is on the same order of the computational cost in solving the original equation, despite of the new auxiliary variable being introduced.…”

Section: 1mentioning

confidence: 99%

“…The SAV approach was proposed for minimizing the free energy by gradient flows, aiming to keep the energy stable in the numerical time flow, see [63], [62] and convergence analysis in [64]. It is then applied to the Hamiltonian PDEs in [16] and [21]. Inspired by the same idea, we reformulate the equation (1.1) into an equivalent system by introducing an auxiliary variable.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5, we first describe the numerical algorithm for solving the resulting nonlinear system from the symplectic Runge-Kutta method. This is based on the fixed point iteration similar to [21]. Then, we list our numerical examples.…”

Section: Introductionmentioning

confidence: 99%

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Arbitrarily high-order conservative schemes for the generalized Korteweg-de Vries equation

Yang¹

2021

Preprint

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This paper proposes a new class of arbitarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge-Kutta method will preserve both the mass and the reformulated energy in the discrete time flow. With the Fourier pseudo-spectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the fully discrete sense. The discrete mass possesses the precision of the spectral accuracy.

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Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Zhang,

Jiang

2023

Adv Comput Math

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Arbitrarily high-order structure-preserving schemes for the Gross–Pitaevskii equation with angular momentum rotation

Cited by 20 publications

References 35 publications

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

Arbitrarily high-order conservative schemes for the generalized Korteweg-de Vries equation

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

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