Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

Cai, Wenjun; Jiang, Chaolong; Wang, Yushun; Song, Yongzhong

doi:10.1016/j.jcp.2019.05.048

Cited by 58 publications

(24 citation statements)

References 37 publications

(62 reference statements)

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“…Various numerical schemes such as finite difference schemes [9], two-level dissipationless Maxwell-Bloch systems [10], generalized leapfrog method [6], finite element methods [7], a split cosine scheme [11], a three-time level fourth-order explicit finite-difference scheme [12], method of lines [13], a modified predictor-corrector scheme [14], dual reciprocity boundary element method [15], a numerical method based on radial basis functions (RBFs) [16], boundary element method [4], meshless local Petrov-Galerkin method [17], a local weak meshless technique based on the radial point interpolation method [18], a method based on collocation and RBFs [19], meshless local boundary integral equation method [20], interpolated coefficient finite element method [21], differential quadrature methods [22][23][24][25], and space-time spectral collocation method [26] have been developed for solving the SG equation. Recently, localized methods of approximate particular solutions [27] and structure-preserving algorithms [28] have been sublimed for simulation of the SG equation.…”

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confidence: 99%

“…The precision and capacity of the proposed algorithms are demonstrated by several 1D and 2D test problems. The main contribution of this article lies in the facts: (i) to the best of author's knowledge, barycentric rational interpolation is used first time for SG, (ii) semi-discrete problem is analyzed for truncation error and convergence analysis in L 2 and H 1 spaces, (iii) the two novel algorithms are compared to each other and already available techniques and found that the algorithms have accuracy and efficiency, and (iv) comparison point of view barycentric rational interpolation based algorithm (BRIA) and local radial basis functions based differential quadrature (LRBF-DQ) algorithms gives better accuracy than [15,16,19] and similar result to [4,15,16,18,20,22,27,28].…”

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Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine‐Gordon equation

Jiwari

2020

Numer Methods Partial Differential Eq.

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In this article, barycentric rational interpolation and local radial basis functions (RBFs) based numerical algorithms are developed for solving multidimensional sine-Gordon (SG) equation. In the development of these algorithms, the first step is to drive a semi-discretization in time with a finite difference, and then the semi-discrete problem is analyzed for truncation errors and convergence in L 2 and H 1 spaces. After that, the semi-discrete system is fully discretized by two different functions, such as barycentric rational and local RBFs. Finally, we obtain a linear system in both the algorithms and the system is solved by a MATLAB routine. In numerical experiments, 1D and 2D SG are considered with various examples of line and ring solitons. Moreover, a comparative study of present results with available numerical ones and exact solutions is also discussed. KEYWORDS barycentric rational interpolation, differential quadrature method, local radial basis functions, sine-Gordon equation, truncation error and convergence analysis 1 INTRODUCTION Nonlinear phenomena play key prefaces in problems arising in physics, applied mathematics, and engineering. These phenomena are modeled by nonlinear evolution equations [1]. Solitonary solutions of evolution equations confer better information to understand the physical mechanism of the phenomena.

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confidence: 99%

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confidence: 99%

Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine‐Gordon equation

Jiwari

2020

Numer Methods Partial Differential Eq.

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“…More recently, inspired by the scalar auxiliary variable (SAV) approach [18,19], Cai et al developed a linearly implicit energy-conserving scheme for the sine-Gordon equation [1]. The resulting scheme leads to a linear system with constant coefficients that is easy to implement.…”

Section: Introductionmentioning

confidence: 99%

A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach

et al. 2020

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In this paper, we present a linearly implicit energy-preserving scheme for the Camassa-Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes for gradient systems. The Camassa-Holm equation is first reformulated into an equivalent system by utilizing the multiple scalar auxiliary variables approach, which inherits a modified energy. Then, the system is discretized in space aided by the standard Fourier pseudo-spectral method and a semi-discrete system is obtained, which is proven to preserve a semi-discrete modified energy. Subsequently, the linearized Crank-Nicolson method is applied for the resulting semi-discrete system to arrive at a fully discrete scheme. The main feature of the new scheme is to form a linear system with a constant coefficient matrix at each time step and produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution. Several numerical results are addressed to confirm accuracy and efficiency of the proposed scheme.

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“…The IEQ method is an efficient way to construct linearly implicit energy-preserving schemes for the Hamiltonian partial differential equations (PDEs) [2,10,21]. About the idea of IEQ schemes, we refer to the review paper [23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Linearly Implicit Energy-Preserving Fourier Pseudospectral Schemes for the Complex Modified Korteweg–de Vries Equation

Yan¹,

Zheng²,

Zhu³

et al. 2020

ANZIAM J.

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We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

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Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

Cited by 58 publications

References 37 publications

Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine‐Gordon equation

Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine‐Gordon equation

A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach

Linearly Implicit Energy-Preserving Fourier Pseudospectral Schemes for the Complex Modified Korteweg–de Vries Equation

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