High-Order Linearly Implicit Structure-Preserving Exponential Integrators for the Nonlinear Schrödinger Equation

Jiang, Chenglong; Cui, Jin; Xu, Qian; Song, Songhe

doi:10.1007/s10915-021-01739-x

Cited by 28 publications

(8 citation statements)

References 51 publications

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“…Remark 3.2. It is well-known that the scalar auxiliary variable (SAV) approach [46,47] is also an efficient method for developing high-order accurate structure-preserving methods for the conservative systems [11,29]. However, we should note that it is challenging for introducing a special scalar auxiliary variable to construct high-order accurate methods in time which can preserve the original energy of the system.…”

Section: Time Semi-discretizationmentioning

confidence: 99%

“…Zhang and Su [57] proposed a linearly-implicit conservative compact finite difference method for the QZS (1.1), however it is shown rigorously in mathematics that the scheme is only second-order accurate in time. In [11,23,29], it is clear to see that high-order accurate structure-preserving scheme will provide much smaller numerical error and more robust than the second-order accurate one as the large time step is chose. Thus, it is desirable to propose high-order accurate mass-and energy-preserving methods for solving the QZS (1.1).…”

Section: Introductionmentioning

confidence: 99%

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

Zhang¹,

Jiang²

2022

Preprint

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In this paper, we present a new methodology to develop arbitrary high-order structure-preserving methods for solving the quantum Zakharov system. The key ingredients of our method are: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) the original system is then rewritten into a new equivalent system which inherits the quadratic energy based on the energy variational principle; (iii) the resulting system is discretized by symplectic Runge-Kutta method in time combining with the Fourier pseudo-spectral method in space. The proposed method achieves arbitrary high-order accurate in time and can preserve the discrete mass and Hamiltonian energy exactly. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, extensive numerical examples are provided to demonstrate the theoretical claims and illustrate the efficiency of our method.

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Section: Time Semi-discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhang¹,

Jiang²

2022

Preprint

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“…However, to the best of our knowledge, all of existing momentum-preserving schemes are only second-order accurate in time at most. It has been shown in [10,16] that, compared with the second-order accurate schemes, the high-order accurate ones not only provide smaller numerical errors as a large time step chosen, but also will be more advantages in the robustness. Consequently, the first aim of this paper is to present a novel paradigm for developing arbitrary high-order momentum-preserving schemes for the equation (1.1).…”

Section: Introductionmentioning

confidence: 99%

Arbitrary high-order structure-preserving schemes for the generalized Rosenau-type equation

Jiang¹,

Xu²,

Song³

et al. 2022

Preprint

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In this paper, we are concerned with arbitrarily high-order momentum-preserving and energy-preserving schemes for solving the generalized Rosenau-type equation, respectively. The derivation of the momentum-preserving schemes is made within the symplectic Runge-Kutta method, coupled with the standard Fourier pseudospectral method in space. Unlike the momentum-preserving scheme, the energypreserving one relies on the use of the quadratic auxiliary variable approach and the symplectic Runge-Kutta method, as well as the standard Fourier pseudo-spectral method. Extensive numerical tests and comparisons are also addressed to illustrate the performance of the proposed schemes.

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“…Nevertheless, all of existing energy-preserving schemes are second-order accurate in time. In [9,14], numerical experiments show that the high-order energy-preserving schemes not only provide much smaller numerical error but also more robust than the second-order accurate schemes. Therefore, it is interesting to design high-order energy-preserving schemes for solving the ZR equation (1.1), which however has not been considered in the literature.…”

Section: Introductionmentioning

confidence: 99%

Arbitrarily high-order energy-preserving schemes for the Zakharov-Rubenchik equation

Zhang¹,

Jiang²,

Hao³

2022

Preprint

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In this paper, we present a high-order energy-preserving scheme for solving Zakharov-Rubenchik equation. The main idea of the method is first to reformulate the original system into an equivalent one by introducing an quadratic auxiliary variable, and the symplectic Runge-Kutta method, together with the Fourier pseudospectral method is then employed to compute the solution of the reformulated system. The main benefit of the proposed method is that it can conserve the three invariants of the original system and achieves arbitrarily high-order accurate in time. In addition, an efficient fixed-point iteration is proposed to solve the resulting nonlinear equations of the proposed schemes. Several experiments are provided to validate the theoretical results.

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High-Order Linearly Implicit Structure-Preserving Exponential Integrators for the Nonlinear Schrödinger Equation

Cited by 28 publications

References 51 publications

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

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

Arbitrary high-order structure-preserving schemes for the generalized Rosenau-type equation

Arbitrarily high-order energy-preserving schemes for the Zakharov-Rubenchik equation

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