A Conservative Linearly-Implicit Compact Difference Scheme for the Quantum Zakharov System

Zhang, Gengen; Su, Chunmei

doi:10.1007/s10915-021-01482-3

Cited by 6 publications

(2 citation statements)

References 41 publications

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“…It has been shown that in the numerical simulation of the collision of solitons, the solution of mass-and energy-conserving schemes cannot produce nonlinear blow-up [44,55]. 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.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

Zhang¹,

Jiang²

2022

Preprint

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“…[47] proposed a conservative linearly-implicit difference scheme for the fractional modified Zakharov system with quantum correction in one-dimension, and show some dynamical phenomena. Very recently, Zhang and Su [50] developed a highly accurate conservative method for solving QZS (1). However, tracing the literature regarding the studies of QZS (1), the numerical methods proposed in the literature are limited.…”

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confidence: 99%

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

Zhang¹

2022

DCDS-B

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In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.

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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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A Conservative Linearly-Implicit Compact Difference Scheme for the Quantum Zakharov System

Cited by 6 publications

References 41 publications

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

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

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

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

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