Conservative linearly-implicit difference scheme for a class of modified Zakharov systems with high-order space fractional quantum correction

Xiao, Aiguo; Wang, Chenxi; Wang, Junjie

doi:10.1016/j.apnum.2019.07.019

Cited by 13 publications

(5 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Following recent reports available in the literature [24], the determination of conserved positive quantities for the finite-difference method (17) would be helpful in bounding the numerical solutions [7]. The authors have attempted to employ the discrete energy method to derive such quantities [33,34]. Unfortunately, their efforts have not yielded conserved positive quantities to this day.…”

Section: Discussionmentioning

confidence: 99%

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

2021

View full text Add to dashboard Cite

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

show abstract

Section: Discussionmentioning

confidence: 99%

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

2021

View full text Add to dashboard Cite

show abstract

“…When H 0, α = β = 2, the system (3)-( 4) reduces to the quantum Zakharov system (1)- (2). In fact, it is easy to show that the system (3)-( 6) satisfies the mass and energy conserved laws [38,39] d dt…”

Section: Introductionmentioning

confidence: 99%

“…To the best of the author's knowledge, there exists few reports on numerical methods for fractional quantum Zakharov system (3)- (6). Only in [39], we presented an efficient conservative difference scheme to solve the system (3)- (6). As a class of high accuracy methods, Fourier spectral methods are often chosen to solve differential equations with periodic boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Conservative Fourier spectral method for a class of modified Zakharov systems with high-order space fractional quantum correction $\dagger$

Guo,

Xiao,

wang

et al. 2024

Preprint

View full text Add to dashboard Cite

In this paper, we consider the Fourier spectral method and numerical investigation for a class of modified Zakharov systems with high-order space fractional quantum correction. First, the Fourier spectral scheme of the system is developed with periodic boundary condition based on the Crank-Nicolson/leap-frog methods in time and the Fourier spectral method in space. Moreover, it is shown that the scheme preserves simultaneously mass and energy conservation laws. Second, we analyze stability and convergence of the numerical scheme. Last, the numerical experiments are given, and the results show the correctness of theoretical results and the efficiency of the conservative scheme.

show abstract

“…The blow-up in finite time of the solution for high-dimensional classical ZS was investigated in [38]. Along the numerical front, the numerical studies of the classical or generalized ZS are very rich, such as time splitting method [3,2,31,32], scalar auxiliary variable approach [48], finite difference method [1,8,20,52], discontinuous Galerkin method [51], etc. Recently, there has been growing interest in developing accurate and efficient numerical methods for the QZS (1.1).…”

Section: Introductionmentioning

confidence: 99%

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

Zhang¹,

Jiang²

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Conservative linearly-implicit difference scheme for a class of modified Zakharov systems with high-order space fractional quantum correction

Cited by 13 publications

References 37 publications

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

Conservative Fourier spectral method for a class of modified Zakharov systems with high-order space fractional quantum correction $\dagger$

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

Contact Info

Product

Resources

About