Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

Guo, Shenyang; Yan, Wenjing; Li, Can; Mei, Liquan

doi:10.1007/s10915-022-02008-1

Cited by 7 publications

(4 citation statements)

References 37 publications

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“…Thus, the efficient conservative time-stepping schemes are desired, such that they not only preserve the inherent structures in long-time computations, but also admit the high computation efficiency for the systems in multiple dimensions. As far as we know, although there exist some numerical schemes on the nonlocal coupled Gordontype equations [12][13][14], the convergence analyses of the associated schemes are not well covered. This gap in the current research is the motivation for the present work.…”

Section: Introductionmentioning

confidence: 99%

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Fully decoupled, linear, and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

Hu,

Kong,

Cai

et al. 2023

Numer Algor

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In this paper, we intend to utilize the generalized scalar auxiliary variable (GSAV) approach proposed in recent paper [SIAM J. Numer. Anal., 60 (2022), 1905-1931 for the nonlocal coupled sine-Gordon equation to construct a class of fully decoupled, linear and second-order accurate energy-preserving scheme. The unconditional unique solvability and discrete energy conservation law of the proposed scheme are rigorously discussed, and the unconditional convergence is then proved by the mathematical induction. Particularly, the convergence analysis covers the proposed scheme in multiple dimensions due to the corresponding nonlinear terms satisfy the global Lipschitz continuity straightforwardly. Finally, time evolution of dynamical behaviour of the governing equation with different nonlocal parameters are observed, and ample numerical comparisons demonstrate that the proposed scheme enjoys the high efficiency in long-time computations.

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Section: Introductionmentioning

confidence: 99%

“…al. [13,14] studied the energy-preserving schemes by the ESAV method and the Crank-Nicolson method for the nonlocal coupled Gordon-type wave equation and the nonlocal Klein-Gordon-Schrödinger equation, respectively. The proposed schemes are fully decoupled and linearly implicit, but the convergence analyses have not been developed.…”

Section: Introductionmentioning

confidence: 99%

Fully decoupled, linear, and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

Hu,

Kong,

Cai

et al. 2023

Numer Algor

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“…Cai and Chen [13] combined the Runge-Kutta method (RKM), the integral factor method, and the supplementary variable method to propose a prediction-correction scheme to study the dynamic behavior of the NLSE with fractional-in-space Laplacian operators [14]. Cai et al [15] adopted the dissipation-preserving Galerkin approach to numerically solve the space fractional order NLSE with a strongly damping term. The differential quadrature method is applied by Ali et al [16] to numerically solve the NLSE.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

Tian,

Wang,

2024

Fractal Fract

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The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given.

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“…Hu et al [11] employed a dissipation-preserving Crank-Nicolson pseudo-spectral method to solve the fractional Laplacian sine-Gordon equation with damping. Guo et al [8] combined the Crank-Nicolson scheme with the exponential scalar auxiliary variable technique in the time direction and employed spectral-Galerkin method in the space direction to solve the coupled fractional Laplacian Klein-Gordon equation. However, the error estimations in those works are incomplete.…”

Section: Introductionmentioning

confidence: 99%

Splitting ADI Scheme for Fractional Laplacian Wave Equations

Tao Sun,

Hai-Wei Sun

2024

NMTMA

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In this paper, we investigate the numerical solution of the two-dimensional fractional Laplacian wave equations. After splitting out the Riesz fractional derivatives from the fractional Laplacian, we treat the Riesz fractional derivatives with an implicit scheme while solving the rest part explicitly. Thanks to the tensor structure of the Riesz fractional derivatives, a splitting alternative direction implicit (S-ADI) scheme is proposed by incorporating an ADI remainder. Then the Gohberg-Semencul formula, combined with fast Fourier transform, is proposed to solve the derived Toeplitz linear systems at each time integration. Theoretically, we demonstrate that the S-ADI scheme is unconditionally stable and possesses second-order accuracy. Finally, numerical experiments are performed to demonstrate the accuracy and efficiency of the S-ADI scheme.

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Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

Cited by 7 publications

References 37 publications

Fully decoupled, linear, and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

Fully decoupled, linear, and energy-preserving GSAV difference schemes for the nonlocal coupled sine-Gordon equations in multiple dimensions

Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

Splitting ADI Scheme for Fractional Laplacian Wave Equations

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